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:M f0_12 sf (1)S gR gS 0 0 552 730 rC 125 89 :M f2_14 sf (Automated Discovery of Linear Feedback Models)S 422 86 :M f2_8 sf (1)S 270 121 :M f0_12 sf (by)S 185 151 :M (Thomas Richardson and Peter Spirtes)S 60 184 :M f2_14 sf (1)S 67 184 :M (.)S 70 184 :M ( )S 96 184 :M (Introduction)S 60 211 :M f0_12 sf .244 .024(The introduction of statistical models represented by directed acyclic graphs \(DAGs\) has)J 60 229 :M 1.745 .175(proved fruitful in the construction of expert systems, in allowing efficient updating)J 60 247 :M 2.155 .215(algorithms that take advantage of conditional independence relations \(Pearl, 1988,)J 60 265 :M .806 .081(Lauritzen )J 111 265 :M f4_12 sf .628 .063(et al.)J f0_12 sf .895 .089( 1993\), and in inferring causal structure from conditional independence)J 60 283 :M .927 .093(relations \(Spirtes and Glymour, 1991, Spirtes, Glymour and Scheines, 1993, Pearl and)J 60 301 :M .252 .025(Verma, 1991, Cooper, 1992\). As a framework for representing the combination of causal)J 60 319 :M -.005(and statistical hypotheses, DAG models have shed light on a number of issues in statistics)A 60 337 :M .101 .01(ranging from Simpson\325s Paradox to experimental design \(Spirtes, Glymour and Scheines,)J 60 355 :M 2.034 .203(1993\). The relations of DAGs with statistical constraints, and the equivalence and)J 60 373 :M 2.503 .25(distinguishability properties of DAG models, are now well understood, and their)J 60 391 :M .107 .011(characterization and computation involves three properties connecting graphical structure)J 60 409 :M .674 .067(and probability distributions: \(i\) a local directed Markov property, \(ii\) a global directed)J 60 427 :M .354 .035(Markov property, \(iii\) and factorizations of joint densities according to the structure of a)J 60 445 :M (graph \(Lauritizen, )S 150 445 :M f4_12 sf (et al.)S 174 445 :M f0_12 sf (, 1990\).)S 60 469 :M 2.017 .202(Recursive structural equation models are one kind of DAG model. However, non-)J 60 487 :M 1.701 .17(recursive structural equation models are not DAG models, and are instead naturally)J 60 505 :M 1.386 .139(represented by directed )J f4_12 sf .364(cyclic)A 212 505 :M f0_12 sf 1.952 .195( graphs in which a finite series of edges representing)J 60 523 :M 1.399 .14(influence leads from a vertex representing a variable back to that same vertex. Such)J 60 541 :M 1.605 .16(graphs have been used to model feedback systems in electrical engineering \(Mason,)J 60 559 :M .501 .05(1953, 1956\), and to represent economic processes \(Haavelmo, 1943, Goldberger, 1973\).)J 60 577 :M .036 .004(In contrast to the acyclic case, almost nothing general is known about how directed cyclic)J 60 595 :M -.002(graphs \(DCGs\) represent conditional independence constraints, or about their equivalence)A 60 613 :M 1.155 .116(or identifiability properties, or about characterizing classes of DCGs from conditional)J 60 642 :M ( )S 60 639.48 -.48 .48 204.48 639 .48 60 639 @a 60 651 :M f0_8 sf (1)S 64 654 :M f0_10 sf .039 .004( Research for this paper was supported by the National Science Foundation through grant 9102169 and the)J 60 665 :M .979 .098(Navy Personnel Research and Development Center and the Office of Naval Research through contract)J 60 676 :M .298 .03(number N00014-93-1-0568. We are indebted to Clark Glymour, Richard Scheines, Christopher Meek, and)J 60 687 :M (Marek Druzdel for helpful conversations.)S endp %%Page: 2 2 %%BeginPageSetup initializepage (peter; page: 2 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 540 26 :M f0_12 sf (2)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf .795 .079(independence relations or other statistical constraints. This paper addresses all of these)J 60 74 :M .145 .014(issues. The issues turn on how the relations among properties \(i\), \(ii\) and \(iii\) essential to)J 60 92 :M 2.568 .257(the acyclic case generalize)J 202 92 :M f1_12 sf 1.634A f0_12 sf 2.549 .255(or fail to generalize)J 323 92 :M f1_12 sf 1.535A f0_12 sf 2.594 .259(to directed cyclic graphs and)J 60 110 :M .516 .052(associated families of distributions. It will be shown that when DCGs are interpreted by)J 60 128 :M 2.959 .296(analogy with DAGs as representing functional dependencies with independently)J 60 146 :M 1.211 .121(distributed noises or "error variables," the equivalence of the fundamental global and)J 60 164 :M .513 .051(local Markov conditions characteristc of DAGs no longer holds, even in linear systems.)J 60 182 :M 3.236 .324(For linear systems associated with DCGs with independent errors or noises, a)J 60 200 :M .327 .033(characterisation of conditional independence constraints is obtained, and it is shown that)J 60 218 :M .735 .073(the result generalizes in a natural way to systems in which the error variables or noises)J 60 236 :M (are statistically dependent.)S 60 260 :M 1.107 .111(We also present a correct polynomial time \(on sparse graphs\) discovery algorithm for)J 60 278 :M 3.22 .322(linear cyclic models that contain no latent variables. This algorithm outputs a)J 60 296 :M 2.421 .242(representation of a class of non-recursive linear structural equation models given)J 60 314 :M 1.17 .117(observational data as input. Under the assumption that all conditional independencies)J 60 332 :M 1.432 .143(found in the observational data are true for structural reasons rather than because of)J 60 350 :M .055 .006(particular parameter values, the algorithm discovers causal features of the structure which)J 60 368 :M 1.204 .12(generated the data. A simple modification of the algorithm can be used as a decision)J 60 386 :M .219 .022(procedure \(whose runtime is polynomial in the number of vertices\) for determining when)J 60 404 :M .896 .09(two directed graphs \(cyclic or acyclic\) entail the same set of conditional independence)J 60 422 :M (relations.)S 60 446 :M 2.786 .279(The remainder of this paper is organized as follows: Section 2 defines relevant)J 60 464 :M 1.41 .141(mathematical ideas and gives some necessary technical results on DAGs and DCGs.)J 60 482 :M .666 .067(Section 3 obtains results for non-recursive linear structural equations models. Section 4)J 60 500 :M .274 .027(describes a discovery algorithm. Section )J 260 500 :M .3 .03(5 describes an algorithm for deciding when two)J 60 518 :M .614 .061(graphs \(cyclic or acyclic\) entail the same set of conditional indepenence relations \(or in)J 60 536 :M .933 .093(the linear case entail the same zero partial correlations\), and Section )J 407 536 :M .878 .088(6 describes some)J 60 554 :M (open research problems. All proofs are in Section )S 301 554 :M (7.)S 60 581 :M f2_14 sf (2)S 67 581 :M (.)S 70 581 :M ( )S 96 581 :M (Directed Graphs and Probability Distributions)S 60 608 :M f0_12 sf 1.191 .119(A )J 73 608 :M f2_12 sf .639 .064(directed acyclic graph)J 189 608 :M f0_12 sf 1.008 .101( \(DAG\) )J 232 608 :M f4_12 sf (G)S 241 608 :M f0_12 sf .947 .095( with a set of vertices )J 353 608 :M f2_12 sf (V)S 362 608 :M f0_12 sf .874 .087( can be given two distinct)J 60 626 :M .87 .087(interpretations. \(We place sets of variables and defined terms in boldface.\) On the one)J 60 644 :M .499 .05(hand, such graphs can be used to represent causal relations between variables, where an)J 60 662 :M .23 .023(edge from A to B in )J f4_12 sf (G)S 170 662 :M f0_12 sf .238 .024( means that A is a direct cause of B relative to )J 398 662 :M f2_12 sf (V)S 407 662 :M f0_12 sf .29 .029(. A )J 426 662 :M f2_12 sf .159 .016(causal graph)J 60 680 :M f0_12 sf (is a DAG given such an interpretation.)S endp %%Page: 3 3 %%BeginPageSetup initializepage (peter; page: 3 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 540 26 :M f0_12 sf (3)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf .345 .034(On the other hand, a DAG with a set of vertices )J f2_12 sf (V)S 306 56 :M f0_12 sf .33 .033( can also represent a set of probability)J 60 74 :M 2.065 .207(measures over )J 139 74 :M f2_12 sf (V)S 148 74 :M f0_12 sf 2.414 .241(. Following the terminology of Lauritzen )J f4_12 sf 1.832 .183(et al.)J f0_12 sf 1.785 .178( \(1990\) say that a)J 60 92 :M 2.131 .213(probability measure over a set of variables )J 291 92 :M f2_12 sf (V)S 300 92 :M f0_12 sf 1.357 .136( satisfies the )J f2_12 sf 2.775 .277(local directed Markov)J 60 110 :M .022(property)A f0_12 sf .06 .006( for a directed acyclic graph \(or DAG\) )J f4_12 sf (G)S 302 110 :M f0_12 sf .078 .008( with vertices )J 370 110 :M f2_12 sf (V)S 379 110 :M f0_12 sf .082 .008( if and only if for every)J 60 128 :M 2.266 .227(W in )J 93 128 :M f2_12 sf (V)S 102 128 :M f0_12 sf 1.746 .175(, W is independent of )J f2_12 sf (V)S 230 128 :M f0_12 sf 1.046 .105(\\\(Descendants\(W\) )J 323 128 :M f1_12 sf .466A f0_12 sf 1.304 .13( Parents\(W\)\) given Parents\(W\),)J 60 146 :M 1.416 .142(where Parents\(W\) is the set of parents of W in G, and Descendants\(W\) is the set of)J 60 164 :M .298 .03(descendants of W in G. \(A glossary of graph theoretic terminology is given in Section )J 483 164 :M (7.)S 60 182 :M 1.133 .113(Note that a vertex is its own ancestor and descendant, although not its own parent or)J 60 200 :M 1.658 .166(child.\) A DAG )J 142 200 :M f4_12 sf (G)S 151 200 :M f0_12 sf 2.377 .238( )J 157 200 :M f2_12 sf .363(represents)A f0_12 sf 1.225 .122( the set of probability measures which satisfy the local)J 60 218 :M (directed Markov property for )S f4_12 sf (G)S 212 218 :M f0_12 sf (.)S 60 242 :M .432 .043(The use of DAGs to simultaneously represent a set of causal hypotheses and a family of)J 60 260 :M .385 .039(probability distributions extends back to the path diagrams introduced by Sewell Wright)J 60 278 :M .205 .02(\(1934\). Variants of probabilistic DAG models were introduced in the 1980\325s in Wermuth)J 60 296 :M .855 .085(\(1980\), Wermuth and Lauritzen \(1983\), Kiiveri, Speed, and Carlin \(1984\), Kiiveri and)J 60 314 :M .053 .005(Speed \(1982\), and Pearl \(1988\). )J 216 311 :M f0_8 sf (2)S 220 314 :M f0_12 sf .055 .005( In Section 4 we will present assumptions which link the)J 60 332 :M (two interpretations of directed graphs.)S 60 356 :M 2.147 .215(Pearl\(1988\) defines a global directed Markov property that has been shown to be)J 60 374 :M .419 .042(equivalent to the local directed Markov property for DAGs, and can be used to calculate)J 60 392 :M .367 .037(the consequence of the local directed Markov property. \(See e.g. Lauritzen )J 429 392 :M f4_12 sf .341 .034(et al.)J f0_12 sf .456 .046( 1990.)J 484 389 :M f0_8 sf (3)S 488 392 :M f0_12 sf <29>S 60 410 :M .284 .028(Several preliminary notions are required. Vertex X is a )J 331 410 :M f2_12 sf (collider)S 370 410 :M f0_12 sf .285 .029( on an acyclic undirected)J 60 428 :M .252 .025(path )J f4_12 sf (U)S 93 428 :M f0_12 sf .319 .032( in directed graph )J f4_12 sf (G)S 191 428 :M f0_12 sf .362 .036( if and only if there are two adjacent edges on )J 419 428 :M f4_12 sf (U)S 428 428 :M f0_12 sf .313 .031( directed into)J 60 446 :M .049 .005(X \(e.g. A )J 108 446 :M f1_12 sf S 120 446 :M f0_12 sf .058 .006( X )J 135 446 :M f1_12 sf S 147 446 :M f0_12 sf .041 .004( B\). Every other vertex on )J f4_12 sf (U)S 284 446 :M f0_12 sf .055 .005( is a )J 307 446 :M f2_12 sf (non-collider)S 369 446 :M f0_12 sf .036 .004( on )J f4_12 sf (U)S 396 446 :M f0_12 sf .043 .004(. In a directed graph)J 60 464 :M f4_12 sf (G)S 69 464 :M f0_12 sf .252 .025(, if X and Y are not in )J f2_12 sf .194(Z)A f0_12 sf .408 .041(, then an acyclic undirected path )J f4_12 sf (U)S 359 464 :M f0_12 sf ( )S f2_12 sf .052(d-connects)A 417 464 :M f0_12 sf .37 .037( X and Y given)J 60 482 :M f2_12 sf .106(Z)A f0_12 sf .179 .018( if and only if every collider on )J 223 482 :M f4_12 sf (U)S 232 482 :M f0_12 sf .159 .016( has a descendant in )J f2_12 sf .091(Z)A f0_12 sf .179 .018(, and no non-collider on )J f4_12 sf (U)S 468 482 :M f0_12 sf .213 .021( is in)J 60 500 :M f2_12 sf (Z)S f0_12 sf .093 .009(. For three disjoint sets )J 182 500 :M f2_12 sf (X)S 191 500 :M f0_12 sf .051 .005(, )J f2_12 sf (Y)S 206 500 :M f0_12 sf .092 .009(, and )J f2_12 sf .073(Z)A f0_12 sf .046 .005(, )J f2_12 sf (X)S 255 500 :M f0_12 sf .124 .012( and )J 279 500 :M f2_12 sf (Y)S 288 500 :M f0_12 sf .124 .012( are )J 309 500 :M f2_12 sf (d-connected)S 371 500 :M f0_12 sf .115 .011( given )J 404 500 :M f2_12 sf .102(Z)A f0_12 sf .085 .009( in )J 428 500 :M f4_12 sf (G)S 437 500 :M f0_12 sf .109 .011( if and only)J 60 518 :M .614(if)A f2_12 sf .503 .05( )J 72 518 :M f0_12 sf 1.26 .126(there is a path )J 149 518 :M f4_12 sf (U)S 158 518 :M f0_12 sf 1.121 .112( that d-connects some X in )J f2_12 sf (X)S 308 518 :M f0_12 sf 1.375 .137( to some Y in )J 385 518 :M f2_12 sf (Y)S 394 518 :M f0_12 sf 1.332 .133( given )J 431 518 :M f2_12 sf .481(Z)A f0_12 sf .994 .099(. For three)J 60 536 :M .302 .03(disjoint sets )J f2_12 sf (X)S 130 536 :M f0_12 sf .2 .02(, )J f2_12 sf (Y)S 145 536 :M f0_12 sf .47 .047(, and )J 173 536 :M f2_12 sf .207(Z)A f0_12 sf .13 .013(, )J f2_12 sf (X)S 196 536 :M f0_12 sf .343 .034( and )J f2_12 sf (Y)S 229 536 :M f0_12 sf .489 .049( are )J 252 536 :M f2_12 sf (d)S 259 536 :M (-separated)S 312 536 :M f0_12 sf .452 .045( given )J 346 536 :M f2_12 sf .294(Z)A f0_12 sf .235 .023( in )J f4_12 sf (G)S 379 536 :M f0_12 sf .452 .045( if and only of )J 453 536 :M f2_12 sf (X)S 462 536 :M f0_12 sf .452 .045( is not)J 60 554 :M 1.546 .155(d-connected to )J f2_12 sf (Y)S 149 554 :M f0_12 sf 2.324 .232( given )J 188 554 :M f2_12 sf .69(Z)A f0_12 sf 1.689 .169(. A probability distribution P satisfies the global directed)J 60 572 :M (Markov property for directed graph )S 234 572 :M f4_12 sf (G)S 243 572 :M f0_12 sf -.001( if and only if for any three disjoint sets of variables)A 60 620 :M ( )S 60 617.48 -.48 .48 204.48 617 .48 60 617 @a 60 629 :M f0_8 sf (2)S 64 632 :M f0_10 sf .784 .078(It is often the case that some further restrictions are placed on the set of distributions represented by a)J 60 643 :M 1.378 .138(DAG. For example, one could also require the Minimality Condition, i.e. that for any distribution P)J 60 654 :M .802 .08(represented by )J f4_10 sf .332(G)A f0_10 sf .659 .066(, P does not satisfy the local directed Markov Condition for any proper subgraph of )J f4_10 sf .332(G)A f0_10 sf (.)S 60 665 :M (This condition, and others are discussed in Pearl\(1988\) and Spirtes, Glymour, and Scheines\(1993\).)S 60 673 :M f0_8 sf (3)S 64 676 :M f0_10 sf .016 .002( However, in Section 3 we show that the local and global directed Markov properties are not equivalent for)J 60 687 :M (cyclic directed graphs.)S endp %%Page: 4 4 %%BeginPageSetup initializepage (peter; page: 4 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 540 26 :M f0_12 sf (4)S gR gS 0 0 552 730 rC 60 56 :M f2_12 sf (X)S 69 56 :M f0_12 sf (, )S f2_12 sf (Y)S 84 56 :M f0_12 sf (, and )S f2_12 sf (Z)S f0_12 sf (, if )S f2_12 sf (X)S 143 56 :M f0_12 sf ( is d-separated from )S 241 56 :M f2_12 sf (Y)S 250 56 :M f0_12 sf ( given )S 283 56 :M f2_12 sf (Z)S f0_12 sf ( in )S f4_12 sf (G)S 315 56 :M f0_12 sf (, then )S 345 56 :M f2_12 sf (X)S 354 56 :M f0_12 sf ( is independent of )S 443 56 :M f2_12 sf (Y)S 452 56 :M f0_12 sf ( given)S 482 56 :M f2_12 sf ( Z)S 60 74 :M f0_12 sf (in P.)S 60 98 :M .377 .038(The following theorems relate the global directed Markov property to factorizations of a)J 60 116 :M .148 .015(density function. Denote a density function over )J 297 116 :M f2_12 sf (V)S 306 116 :M f0_12 sf .148 .015( by )J f4_12 sf .066(f)A f0_12 sf <28>S 331 116 :M f2_12 sf (V)S 340 116 :M f0_12 sf .171 .017(\), where for any subset )J 454 116 :M f2_12 sf (X)S 463 116 :M f0_12 sf .217 .022( of )J 480 116 :M f2_12 sf (V)S 489 116 :M f0_12 sf (,)S 60 134 :M f4_12 sf (f)S f0_12 sf <28>S 67 134 :M f2_12 sf (X)S 76 134 :M f0_12 sf .134 .013(\) denotes the marginal of )J f4_12 sf (f)S f0_12 sf <28>S 207 134 :M f2_12 sf (V)S 216 134 :M f0_12 sf .123 .012(\). If )J f4_12 sf .051(f)A f0_12 sf <28>S 244 134 :M f2_12 sf (V)S 253 134 :M f0_12 sf .133 .013(\) is the density function for a probability measure)J 60 152 :M .52 .052(over a set of variables )J f2_12 sf (V)S 181 152 :M f0_12 sf .616 .062( and An\()J 225 152 :M f2_12 sf (X)S 234 152 :M f0_12 sf .55 .055(\) is the set of ancestors of members of )J f2_12 sf (X)S 436 152 :M f0_12 sf .537 .054( in directed)J 60 170 :M (graph )S f4_12 sf (G)S 99 170 :M f0_12 sf (, say that )S 145 170 :M f4_12 sf (f)S f0_12 sf <28>S 152 170 :M f2_12 sf (V)S 161 170 :M f0_12 sf (\) )S 168 170 :M f2_12 sf (factors according to directed graph)S 349 170 :M f0_12 sf ( )S 352 170 :M f4_12 sf (G)S 361 170 :M f0_12 sf ( with vertices )S 429 170 :M f2_12 sf (V)S 438 170 :M f0_12 sf ( if and only)S 60 188 :M (if for every subset )S 150 188 :M f2_12 sf (X)S 159 188 :M f0_12 sf ( of )S 175 188 :M f2_12 sf (V)S 184 188 :M f0_12 sf (,)S 190 221 172 26 rC 362 247 :M psb currentpoint pse 190 221 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5504 div 832 3 -1 roll exch div scale currentpoint translate 64 40 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (f) 56 344 sh (g) 2710 344 sh 384 /Times-Roman f1 (\() 226 344 sh (\() 842 344 sh (\)\)) 1263 344 sh (\(V,) 3096 344 sh (\(V\)) 4759 344 sh (\)) 5290 344 sh 224 ns (V) 2896 440 sh (\() 2561 714 sh (\)) 2831 714 sh 384 /Times-Roman f1 (An) 366 344 sh (Parents) 3637 344 sh 224 ns (An) 2272 714 sh 384 /Times-Bold f1 (X) 979 344 sh 224 ns (V) 1953 714 sh (X) 2653 714 sh 384 /Symbol f1 (=) 1623 344 sh 224 ns (\316) 2130 714 sh 576 ns (\325) 2187 432 sh end MTsave restore pse gR gS 0 0 552 730 rC 60 274 :M f0_12 sf .428 .043(where )J 93 274 :M f4_12 sf .128(g)A f0_10 sf 0 2 rm .154(V)A 0 -2 rm f0_12 sf .39 .039( is a non-negative function. The following result was proved in Lauritzen )J f4_12 sf .341 .034(et al.)J 60 292 :M f0_12 sf (\(1990\).)S 60 316 :M f2_12 sf .861 .086(Theorem )J 112 316 :M (1:)S 122 316 :M f0_12 sf 1.272 .127( If )J 139 316 :M f2_12 sf (V)S 148 316 :M f0_12 sf .985 .098( is a set of random variables with a probability measure P that has a)J 60 334 :M .288 .029(density function )J 142 334 :M f4_12 sf (f)S f0_12 sf <28>S 149 334 :M f2_12 sf (V)S 158 334 :M f0_12 sf .384 .038(\), then )J 193 334 :M f4_12 sf (f)S f0_12 sf <28>S 200 334 :M f2_12 sf (V)S 209 334 :M f0_12 sf .316 .032(\) factors according to DAG )J f4_12 sf (G)S 354 334 :M f0_12 sf .363 .036( if and only if P satisfies the)J 60 352 :M (global directed Markov property for )S f4_12 sf (G)S 245 352 :M f0_12 sf (.)S 60 376 :M 1.003 .1(As in the case of acyclic graphs, the existence of a factorization according to a cyclic)J 60 394 :M .204 .02(directed graph )J 133 394 :M f4_12 sf (G)S 142 394 :M f0_12 sf .21 .021( does entail that a measure satisfies the global directed Markov property)J 60 412 :M 2.233 .223(for )J 80 412 :M f4_12 sf (G)S 89 412 :M f0_12 sf 1.935 .193(. The proof given in Lauritzen )J 255 412 :M f4_12 sf 1.935 .193(et al.)J 282 412 :M f0_12 sf 1.847 .185( \(1990\) for the acyclic case carries over)J 60 430 :M 1.071 .107(essentially unchanged to the cyclic case. \(Lauritzen )J 323 430 :M f4_12 sf 1.245 .124(et al.)J 349 430 :M f0_12 sf 1.181 .118( use a different definition of)J 60 448 :M .438 .044(d-separation that in Section )J 197 448 :M .497 .05(7 is shown to be equivalent to Pearl\325s in both the cyclic and)J 60 466 :M (the acylic case.\))S 60 490 :M f2_12 sf .861 .086(Theorem )J 112 490 :M (2:)S 122 490 :M f0_12 sf 1.272 .127( If )J 139 490 :M f2_12 sf (V)S 148 490 :M f0_12 sf .985 .098( is a set of random variables with a probability measure P that has a)J 60 508 :M .555 .056(density function )J 143 508 :M f4_12 sf (f)S f0_12 sf <28>S 150 508 :M f2_12 sf (V)S 159 508 :M f0_12 sf .606 .061(\) and )J f4_12 sf .192(f)A f0_12 sf <28>S 195 508 :M f2_12 sf (V)S 204 508 :M f0_12 sf .612 .061(\) factors according to directed \(cyclic or acyclic\) graph )J 480 508 :M f4_12 sf (G)S 489 508 :M f0_12 sf (,)S 60 526 :M (then P satisfies the global directed Markov property for )S 329 526 :M f4_12 sf (G)S 338 526 :M f0_12 sf (.)S 60 550 :M .088 .009(However, unlike the case of acyclic graphs, if a probability measure over a set of variable)J 60 568 :M f2_12 sf (V)S 69 568 :M f0_12 sf 1.24 .124( satisfies the global directed Markov property for cyclic graph )J 390 568 :M f4_12 sf (G)S 399 568 :M f0_12 sf 1.419 .142( and has a density)J 60 586 :M (function )S 103 586 :M f4_12 sf (f)S f0_12 sf <28>S 110 586 :M f2_12 sf (V)S 119 586 :M f0_12 sf (\), it does not follow that )S f4_12 sf (f)S f0_12 sf <28>S 244 586 :M f2_12 sf (V)S 253 586 :M f0_12 sf (\) factors according to )S f4_12 sf (G)S 367 586 :M f0_12 sf (.)S endp %%Page: 5 5 %%BeginPageSetup initializepage (peter; page: 5 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 540 26 :M f0_12 sf (5)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf 1.198 .12(The following weaker result relating factorization of densities and the global directed)J 60 74 :M (Markov property does hold for both cyclic and acyclic directed graphs.)S 60 98 :M f2_12 sf .861 .086(Theorem )J 112 98 :M (3:)S 122 98 :M f0_12 sf 1.272 .127( If )J 139 98 :M f2_12 sf (V)S 148 98 :M f0_12 sf .985 .098( is a set of random variables with a probability measure P that has a)J 60 116 :M .934 .093(positive density function )J 188 116 :M f4_12 sf (f)S f0_12 sf <28>S 195 116 :M f2_12 sf (V)S 204 116 :M f0_12 sf 1.048 .105(\), and P satisfies the global directed Markov property for)J 60 134 :M (directed \(cyclic or acyclic\) graph )S 222 134 :M f4_12 sf (G)S 231 134 :M f0_12 sf (, then )S 261 134 :M f4_12 sf (f)S f0_12 sf <28>S 268 134 :M f2_12 sf (V)S 277 134 :M f0_12 sf (\) factors according to )S f4_12 sf (G)S 391 134 :M f0_12 sf (.)S 60 161 :M f2_14 sf (3)S 67 161 :M (.)S 70 161 :M ( )S 96 161 :M (Non-recursive Linear Structural Equation Models)S 60 188 :M f0_12 sf .314 .031(The problem considered in this section is to investigate the generalization of the Markov)J 60 206 :M 1.091 .109(properties to linear, non-recursive structural equation models. First we must relate the)J 60 224 :M (social scientific terminology to graphical representations, and clarify the questions.)S 60 248 :M 1.45 .145(The variables in a structual equation model \(SEM\) can be divided into two sets, the)J 60 266 :M 1.608 .161(\322error\323 variables and the \322substantive\323 variables. Corresponding to each substantive)J 60 284 :M .028 .003(variable X)J f0_7 sf 0 3 rm (i)S 0 -3 rm 112 284 :M f0_12 sf .038 .004( is an equation expressing X)J 248 287 :M f0_7 sf (i)S 250 284 :M f0_12 sf .05 .005( as a )J 275 284 :M f4_12 sf (linear)S 304 284 :M f0_12 sf .039 .004( function of the direct causes of X)J f0_7 sf 0 3 rm (i)S 0 -3 rm 469 284 :M f0_12 sf .042 .004( plus)J 60 302 :M .819 .082(a unique error variable )J f1_12 sf .247(e)A f9_7 sf 0 3 rm .099(i)A 0 -3 rm f0_12 sf .871 .087(. \(We will not consider non-linear models.\) )J 404 302 :M .919 .092(Since we have no)J 60 320 :M .672 .067(interest in first moments, without loss of generality each variable can be expressed as a)J 60 338 :M (deviation from its mean.)S 60 362 :M .218 .022(Consider, for example, two SEMs S)J 235 365 :M f0_7 sf (1)S 239 362 :M f0_12 sf .246 .025( and S)J f0_7 sf 0 3 rm (2)S 0 -3 rm 273 362 :M f0_12 sf .209 .021( over )J f2_12 sf (X)S 310 362 :M f0_12 sf .298 .03( = {X)J 338 365 :M f0_7 sf (1)S 342 362 :M f0_12 sf .222 .022(, X)J f0_7 sf 0 3 rm (2)S 0 -3 rm 361 362 :M f0_12 sf .298 .03(, X)J 377 365 :M f0_7 sf (3)S 381 362 :M f0_12 sf .238 .024(}, where in both SEMs)J 60 380 :M (X)S 69 383 :M f0_7 sf (1)S 73 380 :M f0_12 sf 1.14 .114( is a direct cause of X)J f0_7 sf 0 3 rm (2)S 0 -3 rm 190 380 :M f0_12 sf 1.304 .13( and X)J 226 383 :M f0_7 sf (2)S 230 380 :M f0_12 sf 1.14 .114( is a direct cause of X)J f0_7 sf 0 3 rm (3)S 0 -3 rm 347 380 :M f0_12 sf .889 .089(. The structural equations)J 474 377 :M f0_8 sf (4)S 478 380 :M f0_12 sf 1.358 .136( in)J 60 398 :M (Figure 1 are common to both S)S 209 401 :M f0_7 sf (1)S 213 398 :M f0_12 sf ( and S)S 243 401 :M f0_7 sf (2)S 247 398 :M f0_12 sf (.)S 258 420 :M (X)S 267 423 :M f0_7 sf (1)S 271 423 :M .815 0 rm f9_7 sf ( )S f9_12 sf 0 -3 rm (= )S 0 3 rm f1_12 sf 0 -3 rm (e)S 0 3 rm f0_7 sf (1)S 235 443 :M f0_12 sf (X)S 244 446 :M .432 0 rm f9_7 sf ( )S f0_7 sf (2)S 250 443 :M 1.122 0 rm f9_12 sf ( = )S f1_12 sf (b)S 272 446 :M f0_7 sf (1)S 276 443 :M .739 0 rm f9_12 sf ( )S f0_12 sf (X)S 289 446 :M f0_7 sf (1)S 293 443 :M .122 0 rm f9_12 sf ( + )S f1_12 sf (e)S f0_7 sf 0 3 rm (2)S 0 -3 rm 235 466 :M f0_12 sf (X)S 244 469 :M .432 0 rm f9_7 sf ( )S f0_7 sf (3)S 250 466 :M 1.122 0 rm f9_12 sf ( = )S f1_12 sf (b)S 272 469 :M f0_7 sf (2)S 276 466 :M .739 0 rm f9_12 sf ( )S f0_12 sf (X)S 289 469 :M f0_7 sf (2)S 293 466 :M .122 0 rm f9_12 sf ( + )S f1_12 sf (e)S f0_7 sf 0 3 rm (3)S 0 -3 rm 146 491 :M f2_12 sf (Figure )S 183 491 :M (1: Structural Equations for SEMs S)S 366 494 :M f2_7 sf (1)S 370 491 :M f2_12 sf ( and S)S f2_7 sf 0 3 rm (2)S 0 -3 rm 60 533 :M f0_12 sf .609 .061(where )J f1_12 sf (b)S 100 536 :M f0_7 sf .401 .04(1 )J f0_12 sf 0 -3 rm 1.435 .143(and )J 0 3 rm 128 533 :M f1_12 sf (b)S 135 536 :M f0_7 sf (2)S 139 533 :M f0_12 sf .723 .072( are free parameters ranging over real values, and )J f1_12 sf .224(e)A f0_7 sf 0 3 rm (1)S 0 -3 rm 398 533 :M f0_12 sf .741(,)A f0_7 sf 0 3 rm .432 .043( )J 0 -3 rm 404 533 :M f1_12 sf .401(e)A f0_7 sf 0 3 rm .333 .033(2 )J 0 -3 rm f0_12 sf 1.19 .119(and )J 437 533 :M f1_12 sf (e)S f0_7 sf 0 3 rm (3)S 0 -3 rm 446 533 :M f0_12 sf .809 .081( are error)J 60 551 :M .543 .054(variables. In addition suppose that )J 232 551 :M f1_12 sf (e)S f0_7 sf 0 3 rm (1)S 0 -3 rm 241 551 :M f0_12 sf .583(,)A f0_7 sf 0 3 rm .34 .034( )J 0 -3 rm 247 551 :M f1_12 sf .315(e)A f0_7 sf 0 3 rm .262 .026(2 )J 0 -3 rm f0_12 sf .935 .094(and )J 278 551 :M f1_12 sf (e)S f0_7 sf 0 3 rm (3)S 0 -3 rm 287 551 :M f0_12 sf .543 .054( are distributed as multivariate normal. In)J 60 569 :M (S)S 67 572 :M f0_7 sf (1)S 71 569 :M f0_12 sf 1.167 .117( we will assume that the correlation between each pair of distinct error variables is)J 60 587 :M 1.306 .131(fixed at zero. The free parameters of S)J 259 590 :M f0_7 sf .77 .077(1 )J f0_12 sf 0 -3 rm 2.39 .239(are )J 0 3 rm 286 587 :M f3_12 sf 1.205 .121(q )J f0_12 sf 2.154 .215(= <)J 316 587 :M f3_12 sf (b)S 323 587 :M f0_12 sf .593 .059(, )J f2_12 sf .869(P)A f0_12 sf 1.843 .184(>, where )J f3_12 sf 1.033 .103(b )J 399 587 :M f0_12 sf 1.366 .137(is the set of linear)J 60 605 :M .811 .081(coefficients {)J 127 605 :M f1_12 sf (b)S 134 608 :M f0_7 sf (1)S 138 605 :M f0_12 sf 1.696 .17(, )J 146 605 :M f1_12 sf (b)S 153 608 :M f0_7 sf (2)S 157 605 :M f0_12 sf 1.493 .149(} and )J 190 605 :M f2_12 sf .586(P)A f0_12 sf 1.218 .122( is the set of variances of the error variables. We will use)J 60 623 :M f3_12 sf (S)S f2_7 sf 0 3 rm (S1)S 0 -3 rm f0_12 sf <28>S 78 623 :M f3_12 sf (q)S f3_7 sf 0 3 rm (1)S 0 -3 rm 88 623 :M f0_12 sf .398 .04(\) to denote the covariance matrix parameterized by the vector )J f3_12 sf .135(q)A f3_7 sf 0 3 rm (1)S 0 -3 rm 402 623 :M f0_12 sf .487 .049( for model S)J 464 626 :M f0_7 sf (1)S 468 623 :M f0_12 sf .476 .048(, and)J 60 651 :M ( )S 60 648.48 -.48 .48 204.48 648 .48 60 648 @a 60 660 :M f0_8 sf (4)S 64 663 :M f0_10 sf .199 .02( We realize that it is slightly unconventional to write the trivial equation for the exogenous variable X)J 477 665 :M f0_9 sf (1)S 482 663 :M f0_10 sf .267 .027( in)J 60 676 :M .272 .027(terms of its error, but this serves to give the error variables a unified and special status as providing all the)J 60 687 :M (exogenous sources of variation for the system.)S endp %%Page: 6 6 %%BeginPageSetup initializepage (peter; page: 6 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 540 26 :M f0_12 sf (6)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf .701 .07(occasionally leave out the model subscript if the context makes it clear which model is)J 60 74 :M .213 .021(being referred to. If all the pairs of error variables in a SEM S are uncorrelated, we say S)J 60 92 :M (is a SEM with )S f2_12 sf (uncorrelated errors)S 232 92 :M f0_12 sf (.)S 60 116 :M (S)S 67 119 :M f0_7 sf (2)S 71 116 :M f0_12 sf .072 .007( contains the same structural equations as S)J f0_7 sf 0 3 rm (1)S 0 -3 rm 284 116 :M f0_12 sf .083 .008(, but in S)J f0_7 sf 0 3 rm (2)S 0 -3 rm 331 116 :M f0_12 sf .078 .008( we will allow the errors between)J 60 134 :M (X)S 69 137 :M f9_7 sf (2)S 74 134 :M f0_12 sf .417 .042( and X)J f9_7 sf 0 3 rm (3)S 0 -3 rm 111 134 :M f0_12 sf .114 .011( to be correlated, i.e., we make the correlation between the errors of X)J 450 137 :M f9_7 sf (2)S 455 134 :M f0_12 sf .417 .042( and X)J f9_7 sf 0 3 rm (3)S 0 -3 rm 60 152 :M f0_12 sf (a free parameter, instead of fixing it at zero, as in S)S 306 155 :M f9_7 sf (1)S 311 152 :M f0_12 sf (. In S)S 337 155 :M f9_7 sf (2)S 342 155 :M f0_7 sf ( )S 344 152 :M f0_12 sf (the free parameters are )S 457 152 :M f3_12 sf (q)S f0_12 sf ( = <)S 483 152 :M f3_12 sf (b)S 490 152 :M f0_12 sf (,)S 60 170 :M f2_12 sf .156<50D5>A f0_12 sf .429 .043(>, where )J f3_12 sf .241 .024(b )J 128 170 :M f0_12 sf .388 .039(is the set of linear coefficients {)J 285 170 :M f1_12 sf (b)S 292 173 :M f0_7 sf (1)S 296 170 :M f0_12 sf (,)S f1_12 sf (b)S 306 173 :M f0_7 sf (2)S 310 170 :M f0_12 sf .345 .034(} and )J f2_12 sf .174<50D5>A f0_12 sf .422 .042( is the set of variances of the)J 60 188 :M .648 .065(error variables and the correlation between )J 276 188 :M f1_12 sf .296(e)A f0_7 sf 0 3 rm .246 .025(2 )J 0 -3 rm f0_12 sf .815 .082(and )J f1_12 sf .296(e)A f0_7 sf 0 3 rm (3)S 0 -3 rm 317 188 :M f0_12 sf .702 .07(. If the correlations between any of)J 60 206 :M .334 .033(the error variables in a SEM are not fixed at zero, we will call it a SEM with )J 439 206 :M f2_12 sf (correlated)S 60 224 :M (errors)S 92 224 :M f0_12 sf (.)S 95 221 :M f0_8 sf (5)S 60 248 :M f0_12 sf 1.248 .125(If the coefficients in the linear equations are such that the substantive variables are a)J 60 266 :M .593 .059(unique linear function of the error variables alone, the set of equations is said to have a)J 60 284 :M f2_12 sf 1.553 .155(reduced form)J f0_12 sf .884 .088(. A linear SEM with a reduced form also determines a joint distribution)J 60 302 :M 3.092 .309(over the substantive variables. We will consider only linear SEMs which have)J 60 320 :M .417 .042(coefficients for which there is a reduced form, all variances and partial variances among)J 60 338 :M 1.066 .107(the substantive variables are finite and positive, and all partial correlations among the)J 60 356 :M (substantive variables are well defined \(e.g. not infinite\).)S 60 380 :M .697 .07(It is possible to associate with each SEM with uncorrelated errors a directed graph that)J 60 398 :M 1.192 .119(represents the causal structure of the model and the form of the linear equations. For)J 60 416 :M .206 .021(example, the directed graph associated with the substantive variables in S)J 417 419 :M f0_7 sf (1)S 421 416 :M f0_12 sf .247 .025( is X)J f0_7 sf 0 3 rm (1)S 0 -3 rm 448 416 :M f1_12 sf S 460 416 :M f0_12 sf .325 .032( X)J 473 419 :M f0_7 sf (2)S 477 416 :M f0_12 sf .066 .007( )J f1_12 sf S 60 434 :M f0_12 sf (X)S 69 437 :M f0_7 sf (3)S 73 434 :M f0_12 sf .565 .056(, because X)J 131 437 :M f0_7 sf (1)S 135 434 :M f0_12 sf .554 .055( is the only substantive variable that occurs on the right hand side of the)J 60 452 :M (equation for X)S 130 455 :M f0_7 sf (2)S 134 452 :M f0_12 sf (, and X)S f0_7 sf 0 3 rm (2)S 0 -3 rm 173 452 :M f0_12 sf .009 .001( is the only substantive variable that appears on the right hand side)J 60 470 :M .838 .084(of the equation for X)J 166 473 :M f0_7 sf (3)S 170 470 :M f0_12 sf .794 .079(. We generally do not include error variables in the causal graph)J 60 488 :M .035 .004(associated with a SEM unless the errors are correlated. When the distinction is relevant to)J 60 506 :M .877 .088(the discussion, we enclose measured variables in boxes, latent variables in circles, and)J 60 524 :M (leave error variables unenclosed.)S 60 675 :M ( )S 60 672.48 -.48 .48 204.48 672 .48 60 672 @a 60 684 :M f0_8 sf (5)S 64 687 :M f0_10 sf (We do not consider SEMs with other sorts of constraints on the parameters, e.g., equality constraints.)S endp %%Page: 7 7 %%BeginPageSetup initializepage (peter; page: 7 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 540 26 :M f0_12 sf (7)S gR gS 181 45 19 15 rC 181 54 :M f0_12 sf (X)S gR gS 190 51 14 12 rC 190 61 :M f0_10 sf (1)S gR gS 273 45 19 15 rC 273 54 :M f0_12 sf (X)S gR gS 282 51 14 12 rC 282 61 :M f0_10 sf (2)S gR gS 359 45 19 15 rC 359 54 :M f0_12 sf (X)S gR gS 368 51 14 12 rC 368 61 :M f0_10 sf (3)S gR gS 275 88 14 15 rC 275 101 :M f1_12 sf (e)S gR gS 281 95 15 12 rC 281 105 :M f0_10 sf (2)S gR gS 359 87 14 15 rC 359 100 :M f1_12 sf (e)S gR gS 365 94 14 12 rC 365 104 :M f0_10 sf (3)S gR gS 170 41 212 82 rC 202 54.75 -.75 .75 257.75 54 .75 202 54 @a np 255 50 :M 255 58 :L 263 54 :L 255 50 :L .75 lw eofill -.75 -.75 255.75 58.75 .75 .75 255 50 @b -.75 -.75 255.75 58.75 .75 .75 263 54 @b 255 50.75 -.75 .75 263.75 54 .75 255 50 @a 295 53.75 -.75 .75 344.75 53 .75 295 53 @a np 342 49 :M 342 56 :L 350 53 :L 342 49 :L eofill -.75 -.75 342.75 56.75 .75 .75 342 49 @b -.75 -.75 342.75 56.75 .75 .75 350 53 @b 342 49.75 -.75 .75 350.75 53 .75 342 49 @a -.75 -.75 281.75 85.75 .75 .75 281 74 @b np 278 76 :M 285 76 :L 281 68 :L 278 76 :L eofill 278 76.75 -.75 .75 285.75 76 .75 278 76 @a 281 68.75 -.75 .75 285.75 76 .75 281 68 @a -.75 -.75 278.75 76.75 .75 .75 281 68 @b -.75 -.75 367.75 83.75 .75 .75 367 70 @b np 363 72 :M 370 72 :L 367 64 :L 363 72 :L eofill 363 72.75 -.75 .75 370.75 72 .75 363 72 @a 367 64.75 -.75 .75 370.75 72 .75 367 64 @a -.75 -.75 363.75 72.75 .75 .75 367 64 @b 90 180 80 26 323.5 108.5 @n 0 90 78 26 322.5 108.5 @n 176.5 43.5 23 19 rS 269.5 45.5 23 19 rS 355.5 45.5 23 19 rS gR gS 0 0 552 730 rC 173 144 :M f2_12 sf (Figure )S 210 144 :M (2. SEM S)S 258 147 :M f2_7 sf (2)S 262 144 :M f2_12 sf ( with)S 288 144 :M ( correlated errors)S 60 186 :M f0_12 sf .939 .094(The typical path diagram that would be given for S)J 318 189 :M f0_7 sf (2)S 322 186 :M f0_12 sf 1.116 .112( is shown in )J 389 186 :M .907 .091(Figure )J 425 186 :M 1.012 .101(2. This is not)J 60 204 :M 1.372 .137(strictly a directed graph because of the curved line between error variables )J f1_12 sf .395(e)A f0_7 sf 0 3 rm .328 .033(2 )J 0 -3 rm f0_12 sf 1.089 .109(and )J f1_12 sf .395(e)A f0_7 sf 0 3 rm (3)S 0 -3 rm 489 204 :M f0_12 sf (,)S 60 222 :M .353 .035(which indicates that )J 161 222 :M f1_12 sf .295(e)A f0_7 sf 0 3 rm .267 .027(2 )J 0 -3 rm 172 222 :M f0_12 sf .294 .029(and )J f1_12 sf .107(e)A f0_7 sf 0 3 rm (3)S 0 -3 rm 201 222 :M f0_12 sf .361 .036( are correlated. It is generally accepted that correlation is to)J 60 240 :M .152 .015(be explained by some form of causal connection \(see section 4.1 below\), and accordingly)J 60 258 :M (we assume that curved lines are an ambiguous representation of a causal connection.)S 60 282 :M .76 .076(Interpreted causally, the error variable for a variable X represents all causes of X other)J 60 300 :M .357 .036(than the substantive variables explicitly included in the model. Thus the error variable is)J 60 318 :M 1.632 .163(really an additive amalgamation of perhaps hundreds of other variables that are left)J 60 336 :M .295 .03(unmeasured. To say that two error variable )J f1_12 sf .084(e)A f0_7 sf 0 3 rm .076 .008(2 )J 0 -3 rm 283 336 :M f0_12 sf .219 .022(and )J f1_12 sf .079(e)A f0_7 sf 0 3 rm (3)S 0 -3 rm 312 336 :M f0_12 sf .28 .028( are correlated and have an unknown)J 60 354 :M 1.668 .167(causal connection is to say that one or more of the variables V)J 389 357 :M f0_7 sf (i)S 391 354 :M f0_12 sf 1.66 .166( that make up )J f1_12 sf .694(e)A f0_7 sf 0 3 rm (2)S 0 -3 rm 478 354 :M f0_12 sf 2.108 .211( is)J 60 372 :M .39 .039(identical with, or causally connected to one or more of the variables W)J f0_7 sf 0 3 rm (j)S 0 -3 rm 409 372 :M f0_12 sf .472 .047( that make up )J 480 372 :M f1_12 sf (e)S f0_7 sf 0 3 rm (2)S 0 -3 rm 489 372 :M f0_12 sf (.)S 60 390 :M .13 .013(Whether V)J f0_7 sf 0 3 rm (i)S 0 -3 rm 115 390 :M f0_12 sf .198 .02( is a cause of W)J 193 393 :M f0_7 sf (j)S 195 390 :M f0_12 sf .186 .019(, or W)J f0_7 sf 0 3 rm (j)S 0 -3 rm 227 390 :M f0_12 sf .198 .02( is a cause of V)J 302 393 :M f0_7 sf (i)S 304 390 :M f0_12 sf .181 .018(, or there is a latent variable C that is a)J 60 408 :M .488 .049(cause of both V)J 138 411 :M f0_7 sf (i)S 140 408 :M f0_12 sf .573 .057( and W)J 177 411 :M f0_7 sf (j)S 179 408 :M f0_12 sf .477 .048( \(Figure 3\), or some combination of these, in all cases there is a)J 60 426 :M .766 .077(latent common cause of the substantive variables X)J f0_7 sf 0 3 rm (2)S 0 -3 rm 320 426 :M f0_12 sf 1.065 .107( and X)J 355 429 :M f0_7 sf (3)S 359 426 :M f0_12 sf 1.088 .109( for which )J f1_12 sf .449(e)A f0_7 sf 0 3 rm .373 .037(2 )J 0 -3 rm f0_12 sf 1.238 .124(and )J f1_12 sf .449(e)A f0_7 sf 0 3 rm .407 .041(3 )J 0 -3 rm 459 426 :M f0_12 sf .832 .083(are the)J 60 444 :M 1.065 .106(error variables. Thus we will convert SEMs with correlated errors into SEMs without)J 60 462 :M 2.42 .242(correlated errors by adding a latent common cause \(which possibly represents an)J 60 480 :M 1.57 .157(amalgam of many latents\) of the appropriate substantive variables and replacing the)J 60 498 :M .909 .091(previously correlated error variables with uncorrelated ones. \(This method of handling)J 60 516 :M (correlated errors is also used in Koster 1995.\) We illustrate this process in )S 418 516 :M (Figure 3.)S 60 540 :M .161 .016(Let )J f1_12 sf .063(e)A f0_7 sf 0 3 rm (2)S 0 -3 rm 88 540 :M f0_12 sf .223 .022( be a linear combination of the set of variables )J 317 540 :M f2_12 sf (V)S 326 540 :M f0_12 sf .191 .019(, and )J f1_12 sf .099(e)A f0_7 sf 0 3 rm (3)S 0 -3 rm 361 540 :M f0_12 sf .217 .022( be a linear combination of)J 60 558 :M .941 .094(the set of variables )J f2_12 sf .739(W)A f0_12 sf 1.004 .1(. \(We do not preclude the possibility that )J 383 558 :M f2_12 sf (V)S 392 558 :M f0_12 sf .618 .062( and )J f2_12 sf .794(W)A f0_12 sf 1.22 .122( overlap.\) In)J 60 576 :M .191 .019(Figure 3, we indicate that )J 187 576 :M f1_12 sf (e)S f0_7 sf 0 3 rm (2)S 0 -3 rm 196 576 :M f0_12 sf .204 .02( is a function of the variables in )J 353 576 :M f2_12 sf (V)S 362 576 :M f0_12 sf .176 .018( with braces. The variables)J 60 594 :M .424 .042(in )J 73 594 :M f2_12 sf (V)S 82 594 :M f0_12 sf .181 .018( and )J f2_12 sf .233(W)A f0_12 sf .329 .033( can be re-partitioned into three disjoint subsets \(one each for )J f1_12 sf .102(e)A f0_7 sf 0 3 rm (2)S 0 -3 rm 429 594 :M f0_12 sf .424 .042J 440 594 :M f1_12 sf (e)S f0_7 sf 0 3 rm (3)S 0 -3 rm 449 594 :M f0_12 sf .363 .036(\325, and T\))J 60 612 :M .364 .036(such that no variable in any of the subsets is causally connected to a variable in either of)J 60 630 :M .145 .015(the other subsets. By the Causal Independence assumption \(introduced in section )J 454 630 :M .119 .012(4.1\) )J f1_12 sf (e)S f0_7 sf 0 3 rm (2)S 0 -3 rm 485 630 :M f0_12 sf S 60 648 :M f1_12 sf (e)S f0_7 sf 0 3 rm (3)S 0 -3 rm 69 648 :M f0_12 sf (\325, and T are independent.)S endp %%Page: 8 8 %%BeginPageSetup initializepage (peter; page: 8 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 540 26 :M f0_12 sf (8)S gR gS 167 230 24 19 rC 167 239 :M f0_12 sf ( )S 170 239 :M ( )S 173 239 :M (C)S gR .75 lw gS 101 55 386 283 rC 13 15 177 236 @f gR 1 lw gS 0 0 552 730 rC gS 1.333 1.08 scale 291.261 133.074 :T 90 rotate -291.261 -133.074 :T 281.261 149.074 :M f11_48 sf (})S gR 392 232 24 19 rC 392 241 :M f0_12 sf ( )S 395 241 :M ( )S 398 241 :M (C)S gR .75 lw gS 101 55 386 283 rC 13 15 402 238 @f 201 58 19 14 rC 201 67 :M f0_12 sf (X)S gR gS 210 63 14 12 rC 210 73 :M f0_10 sf (3)S gR gS 101 55 386 283 rC 131 68.75 -.75 .75 142.75 68 .75 131 68 @a np 140 65 :M 140 72 :L 148 68 :L 140 65 :L eofill -.75 -.75 140.75 72.75 .75 .75 140 65 @b -.75 -.75 140.75 72.75 .75 .75 148 68 @b 140 65.75 -.75 .75 148.75 68 .75 140 65 @a -.75 -.75 121.75 116.75 .75 .75 144 84 @b np 140 84 :M 146 88 :L 148 79 :L 140 84 :L eofill 140 84.75 -.75 .75 146.75 88 .75 140 84 @a -.75 -.75 146.75 88.75 .75 .75 148 79 @b -.75 -.75 140.75 84.75 .75 .75 148 79 @b 186 111 69 40 rC 186 132 :M f0_12 sf (W)S 197 135 :M f0_7 sf (1)S 201 132 :M f0_12 sf ( )S 204 132 :M (,)S 208 132 :M (.)S 212 132 :M (,)S 216 132 :M (W)S 227 135 :M f0_7 sf (i)S 229 135 :M .25 .025( )J 231 135 :M .25 .025( )J 233 135 :M (,)S 235 132 :M f0_12 sf (W)S 246 135 :M f0_7 sf (n)S gR gS 101 55 386 283 rC 174 127.75 -.75 .75 184.75 127 .75 174 127 @a np 175 131 :M 174 124 :L 171 127 :L 175 131 :L eofill 174 124.75 -.75 .75 175.75 131 .75 174 124 @a -.75 -.75 171.75 127.75 .75 .75 174 124 @b 171 127.75 -.75 .75 175.75 131 .75 171 127 @a -.75 -.75 139.75 115.75 .75 .75 154 85 @b np 150 86 :M 156 89 :L 157 80 :L 150 86 :L eofill 150 86.75 -.75 .75 156.75 89 .75 150 86 @a -.75 -.75 156.75 89.75 .75 .75 157 80 @b -.75 -.75 150.75 86.75 .75 .75 157 80 @b -.75 -.75 158.75 118.75 .75 .75 163 85 @b np 159 87 :M 166 88 :L 164 79 :L 159 87 :L eofill 159 87.75 -.75 .75 166.75 88 .75 159 87 @a 164 79.75 -.75 .75 166.75 88 .75 164 79 @a -.75 -.75 159.75 87.75 .75 .75 164 79 @b -.75 -.75 193.75 119.75 .75 .75 203 84 @b np 199 85 :M 206 87 :L 205 78 :L 199 85 :L eofill 199 85.75 -.75 .75 206.75 87 .75 199 85 @a 205 78.75 -.75 .75 206.75 87 .75 205 78 @a -.75 -.75 199.75 85.75 .75 .75 205 78 @b 212 86.75 -.75 .75 219.75 119 .75 212 86 @a np 209 89 :M 216 87 :L 211 80 :L 209 89 :L eofill -.75 -.75 209.75 89.75 .75 .75 216 87 @b 211 80.75 -.75 .75 216.75 87 .75 211 80 @a -.75 -.75 209.75 89.75 .75 .75 211 80 @b 219 84.75 -.75 .75 238.75 121 .75 219 84 @a np 216 88 :M 223 84 :L 216 79 :L 216 88 :L eofill -.75 -.75 216.75 88.75 .75 .75 223 84 @b 216 79.75 -.75 .75 223.75 84 .75 216 79 @a -.75 -.75 216.75 88.75 .75 .75 216 79 @b 110 61 19 14 rC 110 70 :M f0_12 sf (X)S gR gS 119 66 14 12 rC 119 75 :M f0_12 sf (1)S gR gS 101 55 386 283 rC 106.5 57.5 23 21 rS 153 58 19 14 rC 153 67 :M f0_12 sf (X)S gR gS 162 63 14 12 rC 162 73 :M f0_10 sf (2)S gR gS 101 55 386 283 rC 149.5 55.5 23 21 rS 196.5 56.5 23 21 rS 173 66.75 -.75 .75 192.75 66 .75 173 66 @a np 190 63 :M 190 70 :L 198 66 :L 190 63 :L eofill -.75 -.75 190.75 70.75 .75 .75 190 63 @b -.75 -.75 190.75 70.75 .75 .75 198 66 @b 190 63.75 -.75 .75 198.75 66 .75 190 63 @a 203 187 19 14 rC 203 196 :M f0_12 sf (X)S gR gS 212 192 14 13 rC 212 202 :M f0_10 sf (3)S gR gS 101 55 386 283 rC 133 196.75 -.75 .75 144.75 196 .75 133 196 @a np 142 193 :M 142 200 :L 150 196 :L 142 193 :L eofill -.75 -.75 142.75 200.75 .75 .75 142 193 @b -.75 -.75 142.75 200.75 .75 .75 150 196 @b 142 193.75 -.75 .75 150.75 196 .75 142 193 @a -.75 -.75 123.75 245.75 .75 .75 146 213 @b np 142 213 :M 148 217 :L 150 208 :L 142 213 :L eofill 142 213.75 -.75 .75 148.75 217 .75 142 213 @a -.75 -.75 148.75 217.75 .75 .75 150 208 @b -.75 -.75 142.75 213.75 .75 .75 150 208 @b 105 234 65 34 rC 105 255 :M f0_12 sf ( )S 108 255 :M (V)S 116 258 :M f0_7 sf (1)S 120 255 :M f0_12 sf (,)S 124 255 :M (.)S 128 255 :M (.)S 132 255 :M (,)S 136 255 :M (V)S 144 258 :M f0_7 sf (i)S 146 258 :M .25 .025( )J 148 258 :M .25 .025( )J 150 258 :M .25 .025( )J 152 258 :M .25 .025( )J 154 255 :M f0_12 sf ( )S 157 255 :M (V)S 165 258 :M f0_7 sf (n)S gR gS 192 236 60 32 rC 192 257 :M f0_12 sf (W)S 203 260 :M f0_7 sf (1)S 207 257 :M f0_12 sf ( )S 210 257 :M (,)S 214 257 :M (.)S 218 257 :M (,)S 222 257 :M (V)S 230 260 :M f0_7 sf (i)S 232 260 :M .25 .025( )J 234 260 :M .25 .025( )J 236 260 :M (,)S 238 257 :M f0_12 sf (V)S 246 260 :M f0_7 sf (n)S gR gS 101 55 386 283 rC -.75 -.75 141.75 244.75 .75 .75 156 215 @b np 152 215 :M 158 218 :L 159 209 :L 152 215 :L eofill 152 215.75 -.75 .75 158.75 218 .75 152 215 @a -.75 -.75 158.75 218.75 .75 .75 159 209 @b -.75 -.75 152.75 215.75 .75 .75 159 209 @b -.75 -.75 160.75 248.75 .75 .75 165 214 @b np 161 216 :M 168 217 :L 166 208 :L 161 216 :L eofill 161 216.75 -.75 .75 168.75 217 .75 161 216 @a 166 208.75 -.75 .75 168.75 217 .75 166 208 @a -.75 -.75 161.75 216.75 .75 .75 166 208 @b -.75 -.75 196.75 243.75 .75 .75 205 213 @b np 201 214 :M 208 216 :L 207 207 :L 201 214 :L eofill 201 214.75 -.75 .75 208.75 216 .75 201 214 @a 207 207.75 -.75 .75 208.75 216 .75 207 207 @a -.75 -.75 201.75 214.75 .75 .75 207 207 @b 215 215.75 -.75 .75 223.75 245 .75 215 215 @a np 212 218 :M 219 216 :L 213 209 :L 212 218 :L eofill -.75 -.75 212.75 218.75 .75 .75 219 216 @b 213 209.75 -.75 .75 219.75 216 .75 213 209 @a -.75 -.75 212.75 218.75 .75 .75 213 209 @b 221 213.75 -.75 .75 239.75 246 .75 221 213 @a np 219 217 :M 225 213 :L 218 208 :L 219 217 :L eofill -.75 -.75 219.75 217.75 .75 .75 225 213 @b 218 208.75 -.75 .75 225.75 213 .75 218 208 @a 218 208.75 -.75 .75 219.75 217 .75 218 208 @a 112 190 19 14 rC 112 199 :M f0_12 sf (X)S gR gS 121 195 14 12 rC 121 205 :M f0_10 sf (1)S gR gS 101 55 386 283 rC 108.5 186.5 23 21 rS 155 187 19 14 rC 155 196 :M f0_12 sf (X)S gR gS 164 192 14 13 rC 164 202 :M f0_10 sf (2)S gR gS 101 55 386 283 rC 151.5 185.5 23 21 rS 198.5 186.5 23 21 rS 175 196.75 -.75 .75 194.75 196 .75 175 196 @a np 192 192 :M 192 199 :L 200 196 :L 192 192 :L eofill -.75 -.75 192.75 199.75 .75 .75 192 192 @b -.75 -.75 192.75 199.75 .75 .75 200 196 @b 192 192.75 -.75 .75 200.75 196 .75 192 192 @a 133 153 12 18 rC 133 166 :M f1_12 sf (e)S 138 169 :M f0_7 sf (2)S gR gS 216 153 13 19 rC 216 166 :M f1_12 sf (e)S 221 169 :M f0_7 sf (3)S gR gS 132 284 13 18 rC 132 297 :M f1_12 sf (e)S 137 300 :M f0_7 sf (2)S gR gS 216 283 12 19 rC 216 296 :M f1_12 sf (e)S 221 299 :M f0_7 sf (3)S gR gS 101 55 386 283 rC 144 127.75 -.75 .75 152.75 127 .75 144 127 @a np 151 124 :M 151 131 :L 154 127 :L 151 124 :L eofill -.75 -.75 151.75 131.75 .75 .75 151 124 @b -.75 -.75 151.75 131.75 .75 .75 154 127 @b 151 124.75 -.75 .75 154.75 127 .75 151 124 @a 423 58 19 15 rC 423 67 :M f0_12 sf (X)S gR gS 432 64 14 12 rC 432 74 :M f0_10 sf (3)S gR gS 101 55 386 283 rC 352 68.75 -.75 .75 363.75 68 .75 352 68 @a np 361 64 :M 361 71 :L 369 68 :L 361 64 :L eofill -.75 -.75 361.75 71.75 .75 .75 361 64 @b -.75 -.75 361.75 71.75 .75 .75 369 68 @b 361 64.75 -.75 .75 369.75 68 .75 361 64 @a -.75 -.75 342.75 117.75 .75 .75 366 85 @b np 362 84 :M 367 88 :L 369 80 :L 362 84 :L eofill 362 84.75 -.75 .75 367.75 88 .75 362 84 @a -.75 -.75 367.75 88.75 .75 .75 369 80 @b -.75 -.75 362.75 84.75 .75 .75 369 80 @b 408 111 68 40 rC 408 132 :M f0_12 sf (W)S 419 135 :M f0_7 sf (1)S 423 132 :M f0_12 sf ( )S 426 132 :M (,)S 430 132 :M (.)S 434 132 :M (,)S 438 132 :M (W)S 449 135 :M f0_7 sf (i)S 451 135 :M .25 .025( )J 453 135 :M .25 .025( )J 455 135 :M (,)S 457 132 :M f0_12 sf (W)S 468 135 :M f0_7 sf (n)S gR gS 101 55 386 283 rC 395 128.75 -.75 .75 406.75 128 .75 395 128 @a np 396 131 :M 396 124 :L 392 128 :L 396 131 :L eofill -.75 -.75 396.75 131.75 .75 .75 396 124 @b -.75 -.75 392.75 128.75 .75 .75 396 124 @b 392 128.75 -.75 .75 396.75 131 .75 392 128 @a -.75 -.75 360.75 116.75 .75 .75 375 86 @b np 371 86 :M 377 89 :L 378 81 :L 371 86 :L eofill 371 86.75 -.75 .75 377.75 89 .75 371 86 @a -.75 -.75 377.75 89.75 .75 .75 378 81 @b -.75 -.75 371.75 86.75 .75 .75 378 81 @b -.75 -.75 379.75 119.75 .75 .75 384 86 @b np 381 87 :M 387 88 :L 385 80 :L 381 87 :L eofill 381 87.75 -.75 .75 387.75 88 .75 381 87 @a 385 80.75 -.75 .75 387.75 88 .75 385 80 @a -.75 -.75 381.75 87.75 .75 .75 385 80 @b -.75 -.75 415.75 121.75 .75 .75 425 85 @b np 421 86 :M 427 88 :L 426 79 :L 421 86 :L eofill 421 86.75 -.75 .75 427.75 88 .75 421 86 @a 426 79.75 -.75 .75 427.75 88 .75 426 79 @a -.75 -.75 421.75 86.75 .75 .75 426 79 @b 434 87.75 -.75 .75 442.75 121 .75 434 87 @a np 431 89 :M 438 88 :L 432 81 :L 431 89 :L eofill -.75 -.75 431.75 89.75 .75 .75 438 88 @b 432 81.75 -.75 .75 438.75 88 .75 432 81 @a -.75 -.75 431.75 89.75 .75 .75 432 81 @b 440 85.75 -.75 .75 461.75 121 .75 440 85 @a np 438 88 :M 444 85 :L 437 79 :L 438 88 :L eofill -.75 -.75 438.75 88.75 .75 .75 444 85 @b 437 79.75 -.75 .75 444.75 85 .75 437 79 @a 437 79.75 -.75 .75 438.75 88 .75 437 79 @a 331 61 19 14 rC 331 70 :M f0_12 sf (X)S gR gS 340 66 14 13 rC 340 76 :M f0_10 sf (1)S gR gS 101 55 386 283 rC 327.5 58.5 23 21 rS 374 58 19 15 rC 374 67 :M f0_12 sf (X)S gR gS 383 64 14 12 rC 383 74 :M f0_10 sf (2)S gR gS 101 55 386 283 rC 370.5 56.5 23 21 rS 418.5 57.5 23 21 rS 395 67.75 -.75 .75 413.75 67 .75 395 67 @a np 411 64 :M 411 71 :L 419 67 :L 411 64 :L eofill -.75 -.75 411.75 71.75 .75 .75 411 64 @b -.75 -.75 411.75 71.75 .75 .75 419 67 @b 411 64.75 -.75 .75 419.75 67 .75 411 64 @a 330 157 16 18 rC 330 170 :M f1_12 sf (e)S 335 173 :M f0_7 sf (2)S 339 170 :M f0_12 sf S gR gS 443 154 17 18 rC 443 167 :M f1_12 sf (e)S 448 170 :M f0_7 sf (3)S 452 167 :M f0_12 sf S gR gS 101 55 386 283 rC 365 128.75 -.75 .75 373.75 128 .75 365 128 @a np 372 124 :M 372 131 :L 376 128 :L 372 124 :L eofill -.75 -.75 372.75 131.75 .75 .75 372 124 @b -.75 -.75 372.75 131.75 .75 .75 376 128 @b 372 124.75 -.75 .75 376.75 128 .75 372 124 @a 384 156 17 18 rC 384 169 :M f1_12 sf (T)S gR gS 425 187 19 14 rC 425 196 :M f0_12 sf (X)S gR gS 434 192 14 13 rC 434 202 :M f0_10 sf (3)S gR gS 101 55 386 283 rC 355 196.75 -.75 .75 366.75 196 .75 355 196 @a np 364 193 :M 364 200 :L 372 196 :L 364 193 :L eofill -.75 -.75 364.75 200.75 .75 .75 364 193 @b -.75 -.75 364.75 200.75 .75 .75 372 196 @b 364 193.75 -.75 .75 372.75 196 .75 364 193 @a -.75 -.75 339.75 249.75 .75 .75 368 213 @b np 364 212 :M 369 217 :L 372 208 :L 364 212 :L eofill 364 212.75 -.75 .75 369.75 217 .75 364 212 @a -.75 -.75 369.75 217.75 .75 .75 372 208 @b -.75 -.75 364.75 212.75 .75 .75 372 208 @b 327 239 65 34 rC 327 260 :M f0_12 sf ( )S 330 260 :M (V)S 338 263 :M f0_7 sf (1)S 342 260 :M f0_12 sf (,)S 346 260 :M (.)S 350 260 :M (.)S 354 260 :M (,)S 358 260 :M (V)S 366 263 :M f0_7 sf (i)S 368 263 :M .25 .025( )J 370 263 :M .25 .025( )J 372 263 :M .25 .025( )J 374 263 :M .25 .025( )J 376 260 :M f0_12 sf ( )S 379 260 :M (V)S 387 263 :M f0_7 sf (n)S gR gS 411 238 60 32 rC 411 259 :M f0_12 sf (W)S 422 262 :M f0_7 sf (1)S 426 259 :M f0_12 sf ( )S 429 259 :M (,)S 433 259 :M (.)S 437 259 :M (,)S 441 259 :M (V)S 449 262 :M f0_7 sf (i)S 451 262 :M .25 .025( )J 453 262 :M .25 .025( )J 455 262 :M (,)S 457 259 :M f0_12 sf (V)S 465 262 :M f0_7 sf (n)S gR gS 101 55 386 283 rC -.75 -.75 364.75 248.75 .75 .75 378 215 @b np 374 215 :M 381 218 :L 381 209 :L 374 215 :L eofill 374 215.75 -.75 .75 381.75 218 .75 374 215 @a -.75 -.75 381.75 218.75 .75 .75 381 209 @b -.75 -.75 374.75 215.75 .75 .75 381 209 @b -.75 -.75 382.75 248.75 .75 .75 387 214 @b np 383 216 :M 390 217 :L 388 208 :L 383 216 :L eofill 383 216.75 -.75 .75 390.75 217 .75 383 216 @a 388 208.75 -.75 .75 390.75 217 .75 388 208 @a -.75 -.75 383.75 216.75 .75 .75 388 208 @b -.75 -.75 419.75 247.75 .75 .75 427 213 @b np 423 214 :M 430 216 :L 429 207 :L 423 214 :L eofill 423 214.75 -.75 .75 430.75 216 .75 423 214 @a 429 207.75 -.75 .75 430.75 216 .75 429 207 @a -.75 -.75 423.75 214.75 .75 .75 429 207 @b 436 215.75 -.75 .75 440.75 246 .75 436 215 @a np 432 218 :M 439 217 :L 435 209 :L 432 218 :L eofill -.75 -.75 432.75 218.75 .75 .75 439 217 @b 435 209.75 -.75 .75 439.75 217 .75 435 209 @a -.75 -.75 432.75 218.75 .75 .75 435 209 @b 442 213.75 -.75 .75 458.75 246 .75 442 213 @a np 440 217 :M 446 214 :L 440 208 :L 440 217 :L eofill -.75 -.75 440.75 217.75 .75 .75 446 214 @b 440 208.75 -.75 .75 446.75 214 .75 440 208 @a -.75 -.75 440.75 217.75 .75 .75 440 208 @b 334 190 19 14 rC 334 199 :M f0_12 sf (X)S gR gS 343 195 14 12 rC 343 205 :M f0_10 sf (1)S gR gS 101 55 386 283 rC 330.5 186.5 23 21 rS 377 187 19 14 rC 377 196 :M f0_12 sf (X)S gR gS 386 192 14 13 rC 386 202 :M f0_10 sf (2)S gR gS 101 55 386 283 rC 373.5 185.5 23 21 rS 420.5 186.5 23 21 rS 397 196.75 -.75 .75 416.75 196 .75 397 196 @a np 414 192 :M 414 199 :L 422 196 :L 414 192 :L eofill -.75 -.75 414.75 199.75 .75 .75 414 192 @b -.75 -.75 414.75 199.75 .75 .75 422 196 @b 414 192.75 -.75 .75 422.75 196 .75 414 192 @a 345 285 16 18 rC 345 298 :M f1_12 sf (e)S 350 301 :M f0_7 sf (2)S 354 298 :M f0_12 sf S gR gS 448 287 17 19 rC 448 300 :M f1_12 sf (e)S 453 303 :M f0_7 sf (3)S 457 300 :M f0_12 sf S gR gS 101 55 386 283 rC 408 243.75 -.75 .75 410.75 245 .75 408 243 @a np 411 241 :M 407 247 :L 412 246 :L 411 241 :L eofill -.75 -.75 407.75 247.75 .75 .75 411 241 @b -.75 -.75 407.75 247.75 .75 .75 412 246 @b 411 241.75 -.75 .75 412.75 246 .75 411 241 @a -.75 -.75 388.75 248.75 .75 .75 394 241 @b np 391 250 :M 386 245 :L 386 250 :L 391 250 :L eofill 386 245.75 -.75 .75 391.75 250 .75 386 245 @a -.75 -.75 386.75 250.75 .75 .75 386 245 @b 386 250.75 -.75 .75 391.75 250 .75 386 250 @a 395 287 17 19 rC 395 300 :M f1_12 sf (T)S gR gS 101 55 386 283 rC 183 241.75 -.75 .75 187.75 246 .75 183 241 @a np 189 243 :M 184 247 :L 189 248 :L 189 243 :L eofill -.75 -.75 184.75 247.75 .75 .75 189 243 @b 184 247.75 -.75 .75 189.75 248 .75 184 247 @a -.75 -.75 189.75 248.75 .75 .75 189 243 @b 252 56 61 19 rC 252 65 :M f0_12 sf (C)S 260 65 :M (o)S 266 65 :M (n)S 272 65 :M (v)S 278 65 :M (e)S 283 65 :M (r)S 287 65 :M (s)S 292 65 :M (i)S 295 65 :M (o)S 301 65 :M (n)S gR gS 320 301 166 36 rC 345 314 :M f1_12 sf (e)S 350 317 :M f0_7 sf (2)S 354 314 :M f0_12 sf S 358 317 :M f0_7 sf .25 .025( )J 360 314 :M f0_12 sf (a)S 365 314 :M (n)S 371 314 :M (d)S 377 314 :M ( )S 380 314 :M f1_12 sf (e)S 385 317 :M f0_7 sf (3)S 389 314 :M f0_12 sf S 393 317 :M f0_7 sf .25 .025( )J 395 314 :M f0_12 sf (U)S 404 314 :M (n)S 410 314 :M (c)S 415 314 :M (o)S 421 314 :M (r)S 425 314 :M (r)S 429 314 :M (e)S 434 314 :M (l)S 437 314 :M (a)S 442 314 :M (t)S 445 314 :M (e)S 450 314 :M (d)S 456 314 :M (,)S 323 327 :M (T)S 330 327 :M ( )S 333 327 :M (a)S 338 327 :M ( )S 341 327 :M (C)S 349 327 :M (o)S 355 327 :M (m)S 364 327 :M (m)S 373 327 :M (o)S 379 327 :M (n)S 385 327 :M ( )S 388 327 :M (C)S 396 327 :M (a)S 401 327 :M (u)S 407 327 :M (s)S 412 327 :M (e)S 417 327 :M ( )S 420 327 :M (o)S 426 327 :M (f)S 430 327 :M ( )S 433 327 :M (X)S 442 330 :M f0_7 sf (2)S 446 327 :M f0_12 sf ( )S 449 327 :M (a)S 454 327 :M (n)S 460 327 :M (d)S 466 327 :M ( )S 469 327 :M (X)S 478 330 :M f0_7 sf (3)S gR gS 118 303 97 18 rC 118 316 :M f1_12 sf (e)S 123 319 :M f0_7 sf (2)S 127 319 :M .25 .025( )J 129 316 :M f0_12 sf (a)S 134 316 :M (n)S 140 316 :M (d)S 146 316 :M ( )S 149 316 :M f1_12 sf (e)S 154 319 :M f0_7 sf (3)S 158 319 :M .25 .025( )J 160 316 :M f0_12 sf (C)S 168 316 :M (o)S 174 316 :M (r)S 178 316 :M (r)S 182 316 :M (e)S 187 316 :M (l)S 190 316 :M (a)S 195 316 :M (t)S 198 316 :M (e)S 203 316 :M (d)S gR gS 101 55 386 283 rC 255 98 -4 4 306 94 4 255 94 @a np 302 86 :M 302 106 :L 312 96 :L 302 86 :L 4 lw eofill -1 -1 303 107 1 1 302 86 @b -1 -1 303 107 1 1 312 96 @b 302 87 -1 1 313 96 1 302 86 @a 255 224 -4 4 306 220 4 255 220 @a np 302 212 :M 302 232 :L 312 222 :L 302 212 :L eofill -1 -1 303 233 1 1 302 212 @b -1 -1 303 233 1 1 312 222 @b 302 213 -1 1 313 222 1 302 212 @a gR 1 lw gS 0 0 552 730 rC gS .985 1.24 scale 143.015 116.257 :T 90 rotate -143.015 -116.257 :T 133.015 132.257 :M f11_48 sf (})S gR gS .924 .92 scale 239.381 160.09 :T 90 rotate -239.381 -160.09 :T 229.381 176.09 :M f11_48 sf (})S gR gS .591 .6 scale 571.844 244.011 :T 90 rotate -571.844 -244.011 :T 561.844 260.011 :M f11_48 sf (})S gR gS .879 .88 scale 248.967 312.413 :T 90 rotate -248.967 -312.413 :T 238.967 328.413 :M f11_48 sf (})S gR gS .591 .6 scale 590.459 459.019 :T 90 rotate -590.459 -459.019 :T 580.459 475.019 :M f11_48 sf (})S gR gS .591 .6 scale 759.689 459.019 :T 90 rotate -759.689 -459.019 :T 749.689 475.019 :M f11_48 sf (})S gR gS .833 .8 scale 167.204 342.755 :T 90 rotate -167.204 -342.755 :T 157.204 358.755 :M f11_48 sf (})S gR gS .727 .6 scale 617.77 239.01 :T 90 rotate -617.77 -239.01 :T 607.77 255.01 :M f11_48 sf (})S gR gS .879 .96 scale 456.071 290.253 :T 90 rotate -456.071 -290.253 :T 446.071 306.253 :M f11_48 sf (})S gR 102 111 79 34 rC 102 132 :M f0_12 sf ( )S 105 132 :M (V)S 113 135 :M f0_7 sf (1)S 117 132 :M f0_12 sf (,)S 121 132 :M (.)S 125 132 :M (.)S 129 132 :M (,)S 133 132 :M (V)S 141 135 :M f0_7 sf (i)S 143 135 :M .25 .025( )J 145 135 :M .25 .025( )J 147 135 :M .25 .025( )J 149 135 :M .25 .025( )J 151 132 :M f0_12 sf ( )S 154 132 :M ( )S 157 132 :M ( )S 160 132 :M (V)S 168 135 :M f0_7 sf (n)S gR gS 321 111 78 34 rC 321 132 :M f0_12 sf ( )S 324 132 :M (V)S 332 135 :M f0_7 sf (1)S 336 132 :M f0_12 sf (,)S 340 132 :M (.)S 344 132 :M (.)S 348 132 :M (,)S 352 132 :M (V)S 360 135 :M f0_7 sf (i)S 362 135 :M .25 .025( )J 364 135 :M .25 .025( )J 366 135 :M .25 .025( )J 368 135 :M .25 .025( )J 370 132 :M f0_12 sf ( )S 373 132 :M ( )S 376 132 :M ( )S 379 132 :M (V)S 387 135 :M f0_7 sf (n)S gR gS 101 55 386 283 rC -.75 -.75 166.75 246.75 .75 .75 170 242 @b np 169 248 :M 164 243 :L 164 248 :L 169 248 :L .75 lw eofill 164 243.75 -.75 .75 169.75 248 .75 164 243 @a -.75 -.75 164.75 248.75 .75 .75 164 243 @b 164 248.75 -.75 .75 169.75 248 .75 164 248 @a gR gS 0 0 552 730 rC 162 359 :M f2_12 sf (Figure )S 199 359 :M (3. Correlated to Uncorrelated Errors)S 60 401 :M f0_12 sf .66 .066(Applying this procedure to S)J f0_7 sf 0 3 rm (2)S 0 -3 rm 207 401 :M f0_12 sf .822 .082( \(Figure )J 251 401 :M .763 .076(2\) results in a model S)J f0_7 sf 0 3 rm (2)S 0 -3 rm 368 401 :M f0_12 sf .731 .073(\325 which we can associate)J 60 419 :M .597 .06(with a directed graph \()J 173 419 :M .597 .06(Figure 4\). S)J 232 422 :M f0_7 sf (2)S 236 419 :M f0_12 sf .74 .074( and S)J 268 422 :M f0_7 sf (2)S 272 419 :M f0_12 sf .563 .056(\325 represent the same causal relations, and for)J 60 437 :M .753 .075(every covariance matrix )J 183 437 :M f3_12 sf (S)S f2_7 sf 0 3 rm (S2)S 0 -3 rm f0_12 sf <28>S 201 437 :M f3_12 sf .329(q)A f0_12 sf .841 .084(\), there is a covariance matrix )J 361 437 :M f3_12 sf (S)S f2_7 sf 0 3 rm <5332D5>S 0 -3 rm 378 437 :M f0_12 sf <28>S 382 437 :M f3_12 sf .366(q)A f0_12 sf .486 .049(\325\) = )J f3_12 sf .416(S)A f2_7 sf 0 3 rm .216(S2)A 0 -3 rm f0_12 sf <28>S 429 437 :M f3_12 sf .275(q)A f0_12 sf .77 .077(\), and vice-)J 60 455 :M .997 .1(versa. Henceforth, we will consider only SEMs with uncorrelated errors. However the)J 60 473 :M 1.399 .14(theorems in this section can be applied to SEMs with correlated errors by using this)J 60 491 :M .165 .016(transformation technique. A SEM is said to be )J 288 491 :M f2_12 sf .04(recursive)A f0_12 sf .13 .013( \(an RSEM\) if its directed graph)J 60 509 :M (is acyclic; otherwise it is )S 182 509 :M f2_12 sf (non-recursive)S 253 509 :M f0_12 sf (.)S 256 506 :M f0_8 sf (6)S 60 664 :M f0_12 sf ( )S 60 661.48 -.48 .48 204.48 661 .48 60 661 @a 60 673 :M f0_8 sf (6)S 64 676 :M f0_10 sf .177 .018( Note that this use of cyclic directed graphs to represent feedback processes represents an extension of the)J 60 687 :M (causal interpretation of directed graphs.)S endp %%Page: 9 9 %%BeginPageSetup initializepage (peter; page: 9 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 540 26 :M f0_12 sf (9)S gR gS 164 41 224 98 rC 178.5 58.5 27 26 rS 184 63 13 15 rC 184 75 :M f12_12 sf (X)S gR gS 193 69 9 11 rC 193 78 :M f12_10 sf (1)S gR gS 164 41 224 98 rC 269.5 58.5 27 26 rS 276 63 13 15 rC 276 72 :M f0_12 sf (X)S gR gS 285 69 9 11 rC 285 78 :M f0_12 sf (2)S gR gS 164 41 224 98 rC 355.5 58.5 27 26 rS 362 63 13 15 rC 362 72 :M f0_12 sf (X)S gR gS 371 69 9 11 rC 371 78 :M f0_12 sf (3)S gR gS 164 41 224 98 rC np 262 71 :M 250 76 :L 250 76 :L 250 75 :L 250 75 :L 250 75 :L 250 75 :L 250 75 :L 250 74 :L 250 74 :L 249 74 :L 249 74 :L 249 73 :L 249 73 :L 249 73 :L 249 73 :L 249 73 :L 249 72 :L 249 72 :L 249 72 :L 249 72 :L 249 71 :L 249 71 :L 249 71 :L 249 71 :L 249 71 :L 249 70 :L 249 70 :L 249 70 :L 249 70 :L 249 69 :L 249 69 :L 249 69 :L 249 69 :L 249 69 :L 249 68 :L 249 68 :L 249 68 :L 249 68 :L 249 67 :L 249 67 :L 250 67 :L 250 67 :L 250 67 :L 250 66 :L 250 66 :L 250 66 :L 250 66 :L 250 66 :L 262 71 :L 262 71 :L eofill 210 72 -1 1 258 71 1 210 71 @a np 351 70 :M 339 75 :L 339 75 :L 339 74 :L 339 74 :L 339 74 :L 338 74 :L 338 74 :L 338 73 :L 338 73 :L 338 73 :L 338 73 :L 338 73 :L 338 72 :L 338 72 :L 338 72 :L 338 72 :L 338 71 :L 338 71 :L 338 71 :L 338 71 :L 338 70 :L 338 70 :L 338 70 :L 338 70 :L 338 70 :L 338 69 :L 338 69 :L 338 69 :L 338 69 :L 338 69 :L 338 68 :L 338 68 :L 338 68 :L 338 68 :L 338 67 :L 338 67 :L 338 67 :L 338 67 :L 338 67 :L 338 66 :L 338 66 :L 338 66 :L 338 66 :L 338 65 :L 339 65 :L 339 65 :L 339 65 :L 339 65 :L 351 70 :L 351 70 :L eofill 302 71 -1 1 346 70 1 302 70 @a 185 117 9 18 rC 185 130 :M f1_12 sf (e)S gR gS 191 125 9 12 rC 191 134 :M f0_12 sf (1)S gR gS 279 117 9 18 rC 279 130 :M f1_12 sf (e)S gR gS 285 125 9 12 rC 285 134 :M f12_10 sf (2)S gR gS 363 116 9 18 rC 363 129 :M f1_12 sf (e)S gR gS 369 124 9 12 rC 369 133 :M f12_10 sf (3)S gR gS 164 41 224 98 rC np 189 93 :M 195 105 :L 194 105 :L 194 105 :L 194 105 :L 194 105 :L 193 105 :L 193 106 :L 193 106 :L 193 106 :L 193 106 :L 192 106 :L 192 106 :L 192 106 :L 192 106 :L 191 106 :L 191 106 :L 191 106 :L 191 106 :L 191 106 :L 190 106 :L 190 106 :L 190 106 :L 190 106 :L 189 106 :L 189 106 :L 189 106 :L 189 106 :L 189 106 :L 188 106 :L 188 106 :L 188 106 :L 188 106 :L 187 106 :L 187 106 :L 187 106 :L 187 106 :L 187 106 :L 186 106 :L 186 106 :L 186 106 :L 186 106 :L 186 106 :L 185 106 :L 185 105 :L 185 105 :L 185 105 :L 184 105 :L 184 105 :L 189 93 :L 189 93 :L eofill -1 -1 191 116 1 1 190 100 @b np 370 90 :M 375 102 :L 375 102 :L 374 102 :L 374 102 :L 374 103 :L 374 103 :L 374 103 :L 373 103 :L 373 103 :L 373 103 :L 373 103 :L 372 103 :L 372 103 :L 372 103 :L 372 103 :L 372 103 :L 371 103 :L 371 103 :L 371 103 :L 371 103 :L 370 103 :L 370 103 :L 370 103 :L 370 103 :L 370 103 :L 369 103 :L 369 103 :L 369 103 :L 369 103 :L 368 103 :L 368 103 :L 368 103 :L 368 103 :L 368 103 :L 367 103 :L 367 103 :L 367 103 :L 367 103 :L 366 103 :L 366 103 :L 366 103 :L 366 103 :L 366 103 :L 365 103 :L 365 103 :L 365 102 :L 365 102 :L 364 102 :L 370 90 :L 370 90 :L eofill -1 -1 371 113 1 1 370 97 @b np 283 91 :M 288 103 :L 288 103 :L 288 103 :L 288 103 :L 287 104 :L 287 104 :L 287 104 :L 287 104 :L 286 104 :L 286 104 :L 286 104 :L 286 104 :L 286 104 :L 285 104 :L 285 104 :L 285 104 :L 285 104 :L 284 104 :L 284 104 :L 284 104 :L 284 104 :L 284 104 :L 283 104 :L 283 104 :L 283 104 :L 283 104 :L 282 104 :L 282 104 :L 282 104 :L 282 104 :L 282 104 :L 281 104 :L 281 104 :L 281 104 :L 281 104 :L 280 104 :L 280 104 :L 280 104 :L 280 104 :L 280 104 :L 279 104 :L 279 104 :L 279 104 :L 279 104 :L 279 104 :L 278 103 :L 278 103 :L 278 103 :L 283 91 :L 283 91 :L eofill -1 -1 284 114 1 1 283 98 @b 284 120 6 15 rC 284 129 :M f0_12 sf (')S gR gS 368 118 6 15 rC 368 127 :M f0_12 sf (')S gR gS 326 119 12 15 rC 326 128 :M f0_12 sf (T)S gR gS 164 41 224 98 rC 19 20 329 125.5 @f np 294 86 :M 306 91 :L 306 91 :L 306 92 :L 306 92 :L 305 92 :L 305 92 :L 305 93 :L 305 93 :L 305 93 :L 305 93 :L 305 93 :L 305 93 :L 305 94 :L 304 94 :L 304 94 :L 304 94 :L 304 94 :L 304 95 :L 304 95 :L 304 95 :L 303 95 :L 303 95 :L 303 95 :L 303 96 :L 303 96 :L 303 96 :L 303 96 :L 302 96 :L 302 96 :L 302 96 :L 302 97 :L 302 97 :L 301 97 :L 301 97 :L 301 97 :L 301 97 :L 301 97 :L 301 98 :L 300 98 :L 300 98 :L 300 98 :L 300 98 :L 300 98 :L 299 98 :L 299 98 :L 299 98 :L 299 98 :L 298 98 :L 294 86 :L 294 86 :L eofill 298 92 -1 1 322 115 1 298 91 @a np 357 86 :M 353 99 :L 353 99 :L 352 99 :L 352 99 :L 352 98 :L 352 98 :L 352 98 :L 351 98 :L 351 98 :L 351 98 :L 351 98 :L 351 98 :L 350 98 :L 350 98 :L 350 98 :L 350 97 :L 350 97 :L 349 97 :L 349 97 :L 349 97 :L 349 97 :L 349 97 :L 349 96 :L 348 96 :L 348 96 :L 348 96 :L 348 96 :L 348 96 :L 347 96 :L 347 95 :L 347 95 :L 347 95 :L 347 95 :L 347 95 :L 347 95 :L 346 94 :L 346 94 :L 346 94 :L 346 94 :L 346 94 :L 346 93 :L 346 93 :L 346 93 :L 345 93 :L 345 93 :L 345 93 :L 345 92 :L 345 92 :L 357 86 :L 357 86 :L eofill -1 -1 335 115 1 1 353 92 @b gR gS 0 0 552 730 rC 99 160 :M f2_12 sf (Figure )S 136 160 :M (4. S)S 155 163 :M f2_7 sf (2)S 159 160 :M f2_12 sf S 170 160 :M (Correlated Errors Replaced)S 314 160 :M ( b)S 324 160 :M (y Latent Common Cause)S 60 198 :M f0_12 sf .275 .028(A linear SEM \(or its corresponding DAG\) containing disjoint sets of variables )J 444 198 :M f2_12 sf (X)S 453 198 :M f0_12 sf .411 .041(, )J 460 198 :M f2_12 sf (Y)S 469 198 :M f0_12 sf .323 .032(, and)J 60 216 :M f2_12 sf (Z)S f0_12 sf ( )S f2_12 sf (linearly entails)S 147 216 :M f0_12 sf ( that )S 171 216 :M f2_12 sf (X)S 180 216 :M f0_12 sf ( is independent of )S 269 216 :M f2_12 sf (Y)S 278 216 :M f0_12 sf ( given )S 311 216 :M f2_12 sf (Z)S f0_12 sf ( if and only if )S f2_12 sf (X)S 396 216 :M f0_12 sf ( is independent of )S 485 216 :M f2_12 sf (Y)S 60 234 :M f0_12 sf .548 .055(given )J 91 234 :M f2_12 sf .254(Z)A f0_12 sf .514 .051( for all values of the linear coefficients not fixed at zero, and all distributions of)J 60 252 :M .585 .059(the exogenous variables in which they are jointly independent. Let )J 392 252 :M f5_12 sf (r)S 399 254 :M f0_10 sf (XY.)S 416 254 :M f2_10 sf (Z)S 423 252 :M f0_12 sf .663 .066( be the partial)J 60 270 :M .22 .022(correlation of X and Y given )J f2_12 sf .111(Z)A f0_12 sf .266 .027(. A linear SEM \(or its corresponding DAG\) containing X,)J 60 288 :M .354 .035(Y, and )J f2_12 sf .219(Z)A f0_12 sf .341 .034(, where X )J f1_12 sf S 162 288 :M f0_12 sf .41 .041(\312Y and X and Y)J 240 288 :M f4_12 sf .544 .054( )J 244 288 :M f0_12 sf .408 .041(are not in )J 294 288 :M f2_12 sf .097(Z)A f0_12 sf .061 .006(, )J f2_12 sf .367 .037(linearly entails)J 385 288 :M f0_12 sf .328 .033( that )J f5_12 sf (r)S 417 290 :M f0_10 sf (XY.)S 434 290 :M f2_10 sf (Z)S 441 288 :M f0_12 sf .435 .044( = 0 if and)J 60 306 :M 1.958 .196(only if )J 100 306 :M f5_12 sf (r)S 107 308 :M f0_10 sf (XY.)S 124 308 :M f2_10 sf (Z)S 131 306 :M f0_12 sf 1.794 .179( = 0 for all values of the linear coefficients not fixed at zero and all)J 60 324 :M .265 .026(distributions of the exogenous variables in which they are jointly independent. It follows)J 60 342 :M .093 .009(from Kiiveri and Speed \(1982\) that if the error variables are jointly independent, then any)J 60 360 :M .247 .025(distribution that forms a linear, recursive SEM with a directed graph )J 396 360 :M f4_12 sf (G)S 405 360 :M f0_12 sf .246 .025( satisfies the local)J 60 378 :M .662 .066(directed Markov property for )J f4_12 sf (G)S 217 378 :M f0_12 sf .797 .08(. One can therefore apply d)J 354 378 :M .796 .08(-separation to the DAG in a)J 60 396 :M 2.281 .228(linear, recursive SEM to compute the conditional independencies and zero partial)J 60 414 :M 1.124 .112(correlations it linearly entails. The d-separation relation provides a polynomial \(in the)J 60 432 :M 2.114 .211(number of vertices\) time algorithm for deciding whether a given vanishing partial)J 60 450 :M (correlation is linearly entailed by a SEM with a given DAG.)S 60 474 :M .213 .021(Linear non-recursive structural equation models \(linear SEMs\) are commonly used in the)J 60 492 :M .8 .08(econometrics literature to represent feedback processes that have reached equilibrium.)J 488 489 :M f0_8 sf (7)S 60 510 :M f0_12 sf 2.204 .22(Corresponding to a set of non-recursive linear equations is a cyclic graph, as the)J 60 528 :M (following example from Whittaker \(1990\) illustrates.)S 254 544 :M (X)S 263 546 :M f0_10 sf (1)S f0_12 sf 0 -2 rm ( = )S 0 2 rm 281 544 :M f1_12 sf (e)S f0_10 sf 0 2 rm (X1)S 0 -2 rm 254 561 :M f0_12 sf (X)S 263 563 :M f0_10 sf (2)S f0_12 sf 0 -2 rm ( = )S 0 2 rm 281 561 :M f1_12 sf (e)S f0_10 sf 0 2 rm (X2)S 0 -2 rm 210 578 :M f0_12 sf (X)S 219 580 :M f0_10 sf (3)S f0_12 sf 0 -2 rm ( = )S 0 2 rm 237 578 :M f1_12 sf (b)S 244 580 :M f0_10 sf (31)S f0_12 sf 0 -2 rm (X)S 0 2 rm 263 580 :M f0_10 sf (1)S f0_12 sf 0 -2 rm ( + )S 0 2 rm 281 578 :M f1_12 sf (b)S 288 580 :M f0_10 sf (34)S f0_12 sf 0 -2 rm (X)S 0 2 rm 307 580 :M f0_10 sf (4)S f0_12 sf 0 -2 rm ( + )S 0 2 rm 325 578 :M f1_12 sf (e)S f0_10 sf 0 2 rm (X3)S 0 -2 rm 210 595 :M f0_12 sf (X)S 219 597 :M f0_10 sf (4)S f0_12 sf 0 -2 rm ( = )S 0 2 rm 237 595 :M f1_12 sf (b)S 244 597 :M f0_10 sf (42)S f0_12 sf 0 -2 rm (X)S 0 2 rm 263 597 :M f0_10 sf (2)S f0_12 sf 0 -2 rm ( + )S 0 2 rm 281 595 :M f1_12 sf (b)S 288 597 :M f0_10 sf (43)S f0_12 sf 0 -2 rm (X)S 0 2 rm 307 597 :M f0_10 sf (3)S f0_12 sf 0 -2 rm ( + )S 0 2 rm 325 595 :M f1_12 sf (e)S f0_10 sf 0 2 rm (X4)S 0 -2 rm 115 612 :M f5_12 sf (e)S f0_10 sf 0 2 rm (X1)S 0 -2 rm f0_12 sf (, )S f5_12 sf (e)S f0_10 sf 0 2 rm (X2)S 0 -2 rm f0_12 sf (, )S f5_12 sf (e)S f0_10 sf 0 2 rm (X3)S 0 -2 rm f0_12 sf (, )S f5_12 sf (e)S f0_10 sf 0 2 rm (X4)S 0 -2 rm f0_12 sf ( are jointly independent and normally distributed)S 60 642 :M ( )S 60 639.48 -.48 .48 204.48 639 .48 60 639 @a 60 651 :M f0_8 sf (7)S 64 654 :M f0_10 sf .715 .071(Cox and Wermuth \(1993\), Wermuth and Lauritzen\(1990\) and \(indirectly\) Frydenberg\(1990\) consider a)J 60 665 :M .191 .019(class of linear models they call )J 188 665 :M f4_10 sf .261 .026(block recursive)J f0_10 sf .162 .016(. The block recursive models overlap the class of SEMs, but)J 60 676 :M 1.537 .154(they are neither properly included in that class, nor properly include it. Frydenberg \(1990\) presents)J 60 687 :M (necessary and sufficient conditions for the equivalence of two block recursive models.)S endp %%Page: 10 10 %%BeginPageSetup initializepage (peter; page: 10 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (10)S gR gS 217 56 18 13 rC 217 65 :M f0_12 sf (X)S gR gS 228 60 9 12 rC 228 70 :M f0_10 sf (1)S gR gS 220 114 18 13 rC 220 123 :M f0_12 sf (X)S gR gS 231 118 9 12 rC 231 128 :M f0_10 sf (2)S gR gS 308 115 18 13 rC 308 124 :M f0_12 sf (X)S gR gS 320 118 9 12 rC 320 128 :M f0_10 sf (4)S gR gS 307 56 18 13 rC 307 65 :M f0_12 sf (X)S gR gS 319 59 9 12 rC 319 69 :M f0_10 sf (3)S gR gS 213 53 125 80 rC np 303 62 :M 294 66 :L 294 66 :L 294 66 :L 294 66 :L 294 66 :L 294 66 :L 294 65 :L 294 65 :L 294 65 :L 294 65 :L 294 65 :L 294 65 :L 294 64 :L 294 64 :L 294 64 :L 294 64 :L 293 64 :L 293 64 :L 293 63 :L 293 63 :L 293 63 :L 293 63 :L 293 63 :L 293 62 :L 293 62 :L 293 62 :L 293 62 :L 293 62 :L 293 62 :L 293 61 :L 293 61 :L 293 61 :L 293 61 :L 294 61 :L 294 61 :L 294 60 :L 294 60 :L 294 60 :L 294 60 :L 294 60 :L 294 60 :L 294 59 :L 294 59 :L 294 59 :L 294 59 :L 294 59 :L 294 59 :L 294 58 :L 303 62 :L 303 62 :L eofill 237 64 -1 1 296 63 1 237 63 @a np 302 121 :M 293 125 :L 293 124 :L 293 124 :L 293 124 :L 293 124 :L 293 124 :L 293 124 :L 293 123 :L 293 123 :L 293 123 :L 293 123 :L 293 123 :L 293 123 :L 293 122 :L 293 122 :L 293 122 :L 292 122 :L 292 122 :L 292 122 :L 292 121 :L 292 121 :L 292 121 :L 292 121 :L 292 121 :L 292 121 :L 292 120 :L 292 120 :L 292 120 :L 292 120 :L 292 120 :L 292 120 :L 292 119 :L 292 119 :L 293 119 :L 293 119 :L 293 119 :L 293 119 :L 293 118 :L 293 118 :L 293 118 :L 293 118 :L 293 118 :L 293 118 :L 293 117 :L 293 117 :L 293 117 :L 293 117 :L 293 117 :L 302 121 :L 302 121 :L eofill 241 122 -1 1 295 121 1 241 121 @a np 326 70 :M 330 79 :L 330 79 :L 330 79 :L 330 79 :L 329 80 :L 329 80 :L 329 80 :L 329 80 :L 329 80 :L 329 80 :L 328 80 :L 328 80 :L 328 80 :L 328 80 :L 328 80 :L 328 80 :L 327 80 :L 327 80 :L 327 80 :L 327 80 :L 327 80 :L 327 80 :L 326 80 :L 326 80 :L 326 80 :L 326 80 :L 326 80 :L 326 80 :L 325 80 :L 325 80 :L 325 80 :L 325 80 :L 325 80 :L 325 80 :L 324 80 :L 324 80 :L 324 80 :L 324 80 :L 324 80 :L 324 80 :L 323 80 :L 323 80 :L 323 80 :L 323 80 :L 323 80 :L 323 79 :L 322 79 :L 322 79 :L 326 70 :L 326 70 :L eofill -1 -1 327 114 1 1 326 79 @b np 310 113 :M 306 104 :L 306 104 :L 307 104 :L 307 104 :L 307 103 :L 307 103 :L 307 103 :L 307 103 :L 308 103 :L 308 103 :L 308 103 :L 308 103 :L 308 103 :L 308 103 :L 309 103 :L 309 103 :L 309 103 :L 309 103 :L 309 103 :L 309 103 :L 310 103 :L 310 103 :L 310 103 :L 310 103 :L 310 103 :L 310 103 :L 311 103 :L 311 103 :L 311 103 :L 311 103 :L 311 103 :L 311 103 :L 312 103 :L 312 103 :L 312 103 :L 312 103 :L 312 103 :L 312 103 :L 313 103 :L 313 103 :L 313 103 :L 313 103 :L 313 103 :L 313 103 :L 314 103 :L 314 104 :L 314 104 :L 314 104 :L 310 113 :L 310 113 :L eofill -1 -1 311 106 1 1 310 74 @b gR gS 0 0 552 730 rC 169 148 :M f2_12 sf (Figure )S 206 148 :M (5: Example of Non-recursive SEM)S 60 178 :M f0_12 sf 1.278 .128(Theorem 4, )J 122 178 :M 1.171 .117(Theorem 5, and Theorem 6 state that the set of conditional independence)J 60 196 :M 2.548 .255(relations and zero partial correlations entailed by a SEM can be read off of the)J 60 214 :M .105 .01(d-separation relations in the associated graph, even in the case of cyclic graphs. \(Theorem)J 60 232 :M (4 was independently proved by J. Koster in Koster \(1995\).\))S 60 256 :M f2_12 sf .263 .026(Theorem 4:)J 120 256 :M f0_12 sf .32 .032( The probability measure P over the substantive variables of a linear SEM )J 485 256 :M f4_12 sf (L)S 60 274 :M f0_12 sf .695 .069(\(recursive or non-recursive\) with jointly independent error variables satisfies the global)J 60 292 :M .616 .062(directed Markov property for the directed \(cyclic or acyclic\) graph )J f4_12 sf (G)S 402 292 :M f0_12 sf .625 .062( of )J f4_12 sf (L)S 427 292 :M f0_12 sf .635 .063(, i.e. if )J f2_12 sf (X)S 473 292 :M f0_12 sf .362 .036(, )J f2_12 sf (Y)S 489 292 :M f0_12 sf (,)S 60 310 :M (and )S f2_12 sf (Z)S f0_12 sf ( are disjoint sets of variables in )S f4_12 sf (G)S 249 310 :M f0_12 sf ( and )S f2_12 sf (X)S 281 310 :M f0_12 sf ( is d-separated from )S 379 310 :M f2_12 sf (Y)S 388 310 :M f0_12 sf ( given )S 421 310 :M f2_12 sf (Z)S f0_12 sf ( in )S f4_12 sf (G)S 453 310 :M f0_12 sf (, then )S 483 310 :M f2_12 sf (X)S 60 328 :M f0_12 sf (and )S f2_12 sf (Y)S 89 328 :M f0_12 sf ( are independent given )S 201 328 :M f2_12 sf (Z)S f0_12 sf ( in P.)S 60 352 :M f2_12 sf 1.262 .126(Theorem )J 112 352 :M (5:)S 122 352 :M f0_12 sf 1.481 .148( In a linear SEM )J f4_12 sf (L)S 222 352 :M f0_12 sf 1.221 .122( with jointly independent error variables and directed)J 60 370 :M .61 .061(\(cyclic or acyclic\) graph )J f4_12 sf (G)S 193 370 :M f0_12 sf .623 .062( containing disjoint sets of variables )J f2_12 sf (X)S 384 370 :M f0_12 sf .98 .098(, )J 392 370 :M f2_12 sf (Y)S 401 370 :M f0_12 sf .715 .071( and )J f2_12 sf .613(Z)A f0_12 sf .5 .05(, if )J f2_12 sf (X)S 461 370 :M f0_12 sf .83 .083( is not)J 60 388 :M .249 .025(d-separated from )J 145 388 :M f2_12 sf (Y)S 154 388 :M f0_12 sf .346 .035( given )J 188 388 :M f2_12 sf .225(Z)A f0_12 sf .179 .018( in )J f4_12 sf (G)S 221 388 :M f0_12 sf .359 .036( then )J 249 388 :M f4_12 sf (L)S 256 388 :M f0_12 sf .3 .03( does not linearly entail that )J 396 388 :M f2_12 sf (X)S 405 388 :M f0_12 sf .287 .029( is independent of)J 60 406 :M f2_12 sf (Y)S 69 406 :M f0_12 sf ( given )S 102 406 :M f2_12 sf (Z)S f0_12 sf (.)S 60 430 :M .202 .02(Applying Theorem 4)J 161 430 :M .312 .031( and )J 184 430 :M .208 .021(Theorem 5)J 237 430 :M .262 .026( to a linear SEM with the directed graph in )J 449 430 :M .208 .021(Figure 5,)J 60 448 :M .32 .032(the conditional independence relations linearly entailed are just X)J 380 450 :M f0_10 sf .106(1)A f0_12 sf 0 -2 rm .343 .034( is independent of X)J 0 2 rm f0_10 sf .106(2)A f0_12 sf 0 -2 rm (,)S 0 2 rm 60 466 :M (and X)S 89 468 :M f0_10 sf (1)S f0_12 sf 0 -2 rm ( is independent of X)S 0 2 rm f0_10 sf (2)S f0_12 sf 0 -2 rm ( given X)S 0 2 rm f0_10 sf (3)S f0_12 sf 0 -2 rm ( and X)S 0 2 rm 274 468 :M f0_10 sf (4)S f0_12 sf 0 -2 rm (.)S 0 2 rm 60 490 :M f2_12 sf .877 .088(Theorem )J 112 490 :M (6:)S 122 490 :M f0_12 sf 1.03 .103( In a linear SEM )J f4_12 sf (L)S 218 490 :M f0_12 sf .884 .088( with jointly independent error variables and \(cyclic or)J 60 508 :M .221 .022(acyclic\) directed graph )J f4_12 sf (G)S 183 508 :M f0_12 sf .267 .027( containing substantive variables X, Y and )J 393 508 :M f2_12 sf .177(Z)A f0_12 sf .284 .028(, where X )J 453 508 :M f1_12 sf S 460 508 :M f0_12 sf .285 .029(\312Y and)J 60 526 :M f2_12 sf .099(Z)A f0_12 sf .156 .016( does not contain X or Y, X is d)J 223 526 :M .141 .014(-separated from Y given )J 343 526 :M f2_12 sf .152(Z)A f0_12 sf .126 .013( in )J 367 526 :M f4_12 sf (G)S 376 526 :M f0_12 sf .153 .015( if and only if )J f4_12 sf (L)S 452 526 :M f0_12 sf .123 .012( linearly)J 60 544 :M (entails that )S f5_12 sf (r)S 122 546 :M f0_10 sf (XY.)S 139 546 :M f2_10 sf (Z)S 146 544 :M f0_12 sf ( = 0.)S 60 568 :M .84 .084(As in the acyclic case, d-separation provides a polynomial time procedure for deciding)J 60 586 :M .5 .05(whether a linear SEM with a cyclic graph linearly entails a conditional independence or)J 60 604 :M (vanishing partial correlation.)S 60 628 :M .201 .02(In DAGs the global directed Markov property entails the local directed Markov property,)J 60 646 :M 1.938 .194(because a variable V is d)J 195 646 :M 1.471 .147(-separated from its non-parental non-descendants given its)J 60 664 :M .426 .043(parents. However, this is not always the case in cyclic graphs. For example, in )J 448 664 :M .392 .039(Figure 5)J 489 664 :M (,)S endp %%Page: 11 11 %%BeginPageSetup initializepage (peter; page: 11 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (11)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf (X)S 69 58 :M f0_10 sf .172(4)A f0_12 sf 0 -2 rm .701 .07( is not d-separated from its non-parental non-descendant X)J 0 2 rm f0_10 sf .172(1)A f0_12 sf 0 -2 rm .543 .054( given its parents X)J 0 2 rm 466 58 :M f0_10 sf .218(2)A f0_12 sf 0 -2 rm .68 .068( and)J 0 2 rm 60 74 :M f4_12 sf (X)S f0_10 sf 0 2 rm (3)S 0 -2 rm (, )S f0_12 sf (so the local directed Markov property does not hold.)S 329 71 :M f0_8 sf (8)S 60 101 :M f2_14 sf (4)S 67 101 :M (.)S 70 101 :M ( )S 96 101 :M (The Discovery Problem)S 60 128 :M f0_12 sf 1.304 .13(Suppose that we are given data sampled from a population whose causal structure is)J 60 146 :M .408 .041(correctly described by some non-recursive structural equation model )J 399 146 :M f2_12 sf .376(M)A f0_12 sf .452 .045(. Is it possible to)J 60 164 :M .942 .094(discover the causal graph of )J 206 164 :M f2_12 sf .679(M)A f0_12 sf .939 .094( from the data, or at least recover some features of the)J 60 182 :M .024 .002(causal graph from the data? In Spirtes )J 246 182 :M f4_12 sf (et al.)S f0_12 sf .024 .002( \(1995\) the problem of discovering features of)J 60 200 :M .468 .047(the causal graph is considered under the assumption that it is acyclic, but that there may)J 60 218 :M .421 .042(be latent variables \(i.e. there may be unmeasured variables that are the direct cause of at)J 60 236 :M .263 .026(least two measured variables.\) Here we will consider the problem of discovering features)J 60 254 :M .955 .096(of the causal graph under the assumption that it may be cyclic, but there are no latent)J 60 272 :M .145 .014(variables. Future research is needed on the problem of discovering the causal graph when)J 60 290 :M (it may be cyclic )S f4_12 sf (and)S f0_12 sf ( there may be latent variables.)S 60 314 :M 1.808 .181(In order to make inferences about causal relations from a sample distribution, it is)J 60 332 :M 2.424 .242(necessary to introduce some axioms that relate probability distributions to causal)J 60 350 :M 1.65 .165(relations. The two assumptions that we will make are the Causal Independence and)J 60 368 :M (Causal Faithfulness Assumptions, described in the next two subsections.)S 60 392 :M f2_12 sf (4)S 66 392 :M (.)S 69 392 :M (1)S 75 392 :M (.)S 78 392 :M ( )S 96 392 :M (The Causal Independence Assumption)S 60 419 :M f0_12 sf .22 .022(The most fundamental assumption relating causality and probability that we will make is)J 60 437 :M (the following:)S 60 461 :M f2_12 sf .131 .013(Causal Independence Assumption:)J 239 461 :M f0_12 sf .237 .024( If A does not cause B, and B does not cause A, and)J 60 479 :M (there is no third variable that causes both A and B, then A and B are uncorrelated.)S 60 503 :M .792 .079(This assumption allows us to draw a )J 247 503 :M f4_12 sf .115(causal)A f0_12 sf .394 .039( conclusion from )J f4_12 sf .102(statistical)A 412 503 :M f0_12 sf .849 .085( data and lies at)J 60 521 :M .227 .023(the foundation of the theory of randomized experiments. If the value of A is randomized,)J 60 539 :M .963 .096(the experimenter knows that the randomizing device is the sole cause of A. Hence the)J 60 557 :M .589 .059(experimenter knows B did not cause A, and that there is no third variable which causes)J 60 575 :M .031 .003(both A and B. This leaves only two alternatives: either A causes B or it does not. If A and)J 60 627 :M ( )S 60 624.48 -.48 .48 204.48 624 .48 60 624 @a 60 636 :M f0_8 sf (8)S 64 639 :M f0_10 sf 1.006 .101( We are indebted to C. Glymour for pointing out that the local Markov condition fails in Whittaker's)J 60 650 :M .548 .055(model. Indeed, there is )J 158 650 :M f4_10 sf .16(no)A f0_10 sf .499 .05( acyclic graph \(even with additional variables\) that linearly entails all and only)J 60 661 :M .978 .098(conditional independence relations linearly entailed by )J 293 661 :M 1.103 .11(Figure 5)J 329 661 :M 1.148 .115(, although the directed cyclic graph of)J 60 672 :M .08 .008(Figure 5)J 94 672 :M .094 .009( is equivalent to one in which the edges from X)J 285 674 :M (1)S 0 -2 rm .077 .008( to X)J 0 2 rm (3)S 0 -2 rm .103 .01( and X)J 0 2 rm 342 674 :M (2)S 0 -2 rm .044 .004( to X)J 0 2 rm (4)S 0 -2 rm .094 .009( are replaced, respectively, by)J 0 2 rm 60 685 :M (edges from X)S 0 2 rm (1)S 0 -2 rm ( to X)S 139 687 :M (4)S 0 -2 rm ( and from X)S 0 2 rm 193 687 :M (2)S 0 -2 rm ( to X)S 0 2 rm 218 687 :M (3)S 0 -2 rm (.)S 0 2 rm endp %%Page: 12 12 %%BeginPageSetup initializepage (peter; page: 12 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (12)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf .492 .049(B are correlated in the experimental population, the experimenter concludes that A does)J 60 74 :M (cause B, which is an application of the Causal Independence Assumption.)S 60 98 :M .226 .023(The Causal Independence Assumption entails that if two error variables, such as )J f1_12 sf .061(e)A f0_7 sf 0 3 rm .056 .006(2 )J 0 -3 rm 463 98 :M f0_12 sf .178 .018(and )J f1_12 sf .065(e)A f0_7 sf 0 3 rm (3)S 0 -3 rm 60 116 :M f0_12 sf .245 .025(in Figure 2 are correlated there is a latent common cause of X)J 360 119 :M f0_7 sf .337 .034(2 )J 366 116 :M f0_12 sf .22 .022(and X)J f0_7 sf 0 3 rm (3)S 0 -3 rm 399 116 :M f0_12 sf .227 .023( responsible for the)J 60 134 :M .789 .079(correlation. In other words, when X)J 239 137 :M f0_7 sf .339 .034(2 )J f0_12 sf 0 -3 rm 1.602 .16(and X)J 0 3 rm 276 137 :M f0_7 sf .196 .02(3 )J f0_12 sf 0 -3 rm .895 .089(have correlated errors, we assume that the)J 0 3 rm 60 152 :M .661 .066(distribution over X)J 154 155 :M f0_7 sf .417 .042(2 )J f0_12 sf 0 -3 rm 1.844 .184(and X)J 0 3 rm f0_7 sf .455 .046(3 )J 197 152 :M f0_12 sf .754 .075(is the marginal of some other distribution including a finite)J 60 170 :M .481 .048(number of latent causes of X)J 202 173 :M f0_7 sf .674 .067(2 )J 208 170 :M f0_12 sf .441 .044(and X)J f0_7 sf 0 3 rm (3)S 0 -3 rm 241 170 :M f0_12 sf .47 .047( in which the error variables are uncorrelated, as in)J 60 188 :M .077 .008(Figure 4. Since we are making the assumption that there are no latent variables, it follows)J 60 206 :M 1.733 .173(that the error variables of the causal graph are uncorrelated. The correctness of the)J 60 224 :M .894 .089(d-separation criterion for deciding which partial correlations are linearly entailed to be)J 60 242 :M (zero by a SEM with an associated graph )S 256 242 :M f4_12 sf (G)S 265 242 :M f0_12 sf ( then follows necessarily from Theorem 6.)S 60 266 :M f2_12 sf (4)S 66 266 :M (.)S 69 266 :M (2)S 75 266 :M (.)S 78 266 :M ( )S 96 266 :M (The Faithfulness Assumption)S 60 293 :M f0_12 sf .028 .003(In addition to the zero partial correlations that are entailed for )J 360 293 :M f4_12 sf (all)S 373 293 :M f0_12 sf .02 .002( linear parameterizations)J 60 311 :M 1.449 .145(of a graph, there may be zero partial correlations that hold only for some )J 443 311 :M f4_12 sf (particular)S 60 329 :M f0_12 sf .789 .079(parameterizations of a graph. For example, suppose )J 321 329 :M .91 .091(Figure 6 is the directed graph of a)J 60 347 :M (SEM)S 85 347 :M ( that describes the relations among Tax Rate, the Economy, and Tax Revenues.)S 294 383 26 30 rC 294 392 :M (T)S 301 392 :M (a)S 306 392 :M (x)S 294 404 :M (R)S 302 404 :M (a)S 307 404 :M (t)S 310 404 :M (e)S gR gS 203 447 50 15 rC 203 456 :M f0_12 sf (E)S 210 456 :M (c)S 215 456 :M (o)S 221 456 :M (n)S 227 456 :M (o)S 233 456 :M (m)S 242 456 :M (y)S gR gS 322 474 50 30 rC 322 483 :M f0_12 sf (T)S 329 483 :M (a)S 334 483 :M (x)S 322 495 :M (R)S 330 495 :M (e)S 335 495 :M (v)S 341 495 :M (e)S 346 495 :M (n)S 352 495 :M (u)S 358 495 :M (e)S 363 495 :M (s)S gR gS 178 377 195 128 rC np 250 438 :M 257 427 :L 257 427 :L 257 427 :L 257 428 :L 257 428 :L 258 428 :L 258 428 :L 258 428 :L 258 428 :L 258 428 :L 258 429 :L 259 429 :L 259 429 :L 259 429 :L 259 429 :L 259 429 :L 259 430 :L 260 430 :L 260 430 :L 260 430 :L 260 430 :L 260 430 :L 260 431 :L 260 431 :L 260 431 :L 261 431 :L 261 431 :L 261 432 :L 261 432 :L 261 432 :L 261 432 :L 261 432 :L 261 433 :L 261 433 :L 262 433 :L 262 433 :L 262 433 :L 262 434 :L 262 434 :L 262 434 :L 262 434 :L 262 434 :L 262 435 :L 262 435 :L 262 435 :L 262 435 :L 262 436 :L 262 436 :L 250 438 :L 250 438 :L eofill -1 -1 256 437 1 1 286 415 @b np 310 480 :M 297 480 :L 297 480 :L 297 480 :L 297 480 :L 297 480 :L 297 479 :L 297 479 :L 297 479 :L 297 479 :L 297 478 :L 297 478 :L 298 478 :L 298 478 :L 298 478 :L 298 477 :L 298 477 :L 298 477 :L 298 477 :L 298 476 :L 298 476 :L 298 476 :L 298 476 :L 298 476 :L 298 475 :L 298 475 :L 298 475 :L 298 475 :L 299 475 :L 299 474 :L 299 474 :L 299 474 :L 299 474 :L 299 473 :L 299 473 :L 299 473 :L 299 473 :L 299 473 :L 300 473 :L 300 472 :L 300 472 :L 300 472 :L 300 472 :L 300 472 :L 300 471 :L 301 471 :L 301 471 :L 301 471 :L 301 471 :L 310 480 :L 310 480 :L eofill 259 461 -1 1 306 478 1 259 460 @a np 335 465 :M 325 457 :L 325 457 :L 325 456 :L 326 456 :L 326 456 :L 326 456 :L 326 456 :L 326 456 :L 326 455 :L 327 455 :L 327 455 :L 327 455 :L 327 455 :L 327 455 :L 327 455 :L 328 454 :L 328 454 :L 328 454 :L 328 454 :L 328 454 :L 329 454 :L 329 454 :L 329 454 :L 329 453 :L 329 453 :L 330 453 :L 330 453 :L 330 453 :L 330 453 :L 330 453 :L 331 453 :L 331 453 :L 331 453 :L 331 453 :L 331 453 :L 332 452 :L 332 452 :L 332 452 :L 332 452 :L 333 452 :L 333 452 :L 333 452 :L 333 452 :L 333 452 :L 334 452 :L 334 452 :L 334 452 :L 334 452 :L 335 465 :L 335 465 :L eofill 313 421 -1 1 335 460 1 313 420 @a 325 425 11 18 rC 325 438 :M f1_12 sf (b)S gR gS 331 430 10 18 rC 331 443 :M f1_12 sf (1)S gR gS 263 404 10 18 rC 263 417 :M f1_12 sf (b)S gR gS 269 409 9 18 rC 269 422 :M f1_12 sf (2)S gR gS 275 449 11 18 rC 275 462 :M f1_12 sf (b)S gR gS 281 454 10 18 rC 281 467 :M f1_12 sf (3)S gR gS 0 0 552 730 rC 208 526 :M f2_12 sf (Figure )S 245 526 :M (6. Economic Model)S 60 562 :M f0_12 sf .277 .028(In this case there are no vanishing partial correlation constraints entailed for all values of)J 60 580 :M 1.85 .185(the free parameters. But if )J 204 580 :M f1_12 sf (b)S 211 583 :M f0_7 sf (1)S 215 580 :M f0_12 sf 1.634 .163( = )J f1_12 sf (-)S 240 580 :M f0_12 sf <28>S 244 580 :M f1_12 sf (b)S 251 583 :M f0_7 sf (2)S 255 580 :M f0_12 sf 2.849 .285( )J 261 580 :M f1_12 sf S 268 580 :M f0_12 sf 2.849 .285( )J 274 580 :M f1_12 sf (b)S 281 583 :M f0_7 sf (3)S 285 580 :M f0_12 sf 1.881 .188(\), then Tax Rate and Tax Revenues are)J 60 598 :M .877 .088(uncorrelated. The SEM postulates a direct effect of Tax Rate on Revenue \()J 436 598 :M f1_12 sf (b)S 443 601 :M f0_7 sf (1)S 447 598 :M f0_12 sf <29>S 451 598 :M 1.276 .128(, )J 459 598 :M .936 .094(and an)J 60 616 :M .899 .09(indirect effect through the Economy \()J 249 616 :M f1_12 sf (b)S 256 619 :M f0_7 sf (2)S 260 616 :M f0_12 sf 1.547 .155( )J 265 616 :M f1_12 sf S 272 616 :M f0_12 sf .44 .044( )J f1_12 sf (b)S 283 619 :M f0_7 sf (3)S 287 616 :M f0_12 sf .85 .085(\). The parameter constraint indicates that)J 60 634 :M .313 .031(these effects )J f4_12 sf .096(exactly)A 158 634 :M f0_12 sf .521 .052( offset each other, leaving no total effect whatsoever. In such a case)J 60 652 :M .024 .002(we say that the population is )J 201 652 :M f2_12 sf .006(unfaithful)A f0_12 sf .019 .002( to the graph of the causal structure that generated)J endp %%Page: 13 13 %%BeginPageSetup initializepage (peter; page: 13 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (13)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf .112 .011(it. A distribution is )J 155 56 :M f2_12 sf (faithful)S 193 56 :M f0_12 sf .121 .012( to a directed graph )J 290 56 :M f4_12 sf (G)S 299 56 :M f0_12 sf .111 .011( if each partial correlation that is zero in)J 60 74 :M (the distribution is entailed to be zero by )S f4_12 sf (G)S 262 74 :M f0_12 sf (.)S 60 98 :M f2_12 sf 1.342 .134(Causal Faithfulness Assumption:)J 235 98 :M f0_12 sf 2.299 .23( If a distribution P is generated by a SEM with)J 60 116 :M (associated graph )S 143 116 :M f4_12 sf (G)S 152 116 :M f0_12 sf (, then )S 182 116 :M f2_12 sf (A)S 191 116 :M f0_12 sf ( is independent of )S 280 116 :M f2_12 sf (B)S f0_12 sf ( given )S 321 116 :M f2_12 sf (C)S 330 116 :M f0_12 sf ( in P only if )S 390 116 :M f4_12 sf (G)S 399 116 :M f0_12 sf ( linearly entails that)S 60 134 :M f2_12 sf (A)S 69 134 :M f0_12 sf ( is independent of )S 158 134 :M f2_12 sf (B)S f0_12 sf ( given )S 199 134 :M f2_12 sf (C)S 208 134 :M f0_12 sf (.)S 60 158 :M 1.163 .116(The faithfulness assumption limits the SEMs considered to those in which population)J 60 176 :M .993 .099(constraints are entailed by structure, not by particular values of the parameters)J 455 176 :M 1.271 .127(. If one)J 60 194 :M 1.833 .183(assumes faithfulness, then if A and B are )J 283 194 :M f4_12 sf .4(not)A f0_12 sf 1.483 .148( d-separated given )J 396 194 :M f2_12 sf (C)S 405 194 :M f0_12 sf 1.68 .168(, then )J f5_12 sf (r)S 447 197 :M f0_7 sf .458(A,B.)A f2_7 sf .7(C)A f0_7 sf .242 .024( )J 470 194 :M f0_12 sf cF f1_12 sf .213A sf 2.133 .213( 0,)J 60 212 :M 1.051 .105(\(because it is not linearly entailed to equal zero for all values of the free parameters.\))J 60 230 :M 1.201 .12(Faithfulness should not be assumed when there are deterministic relationships among)J 60 248 :M .675 .068(variables, or equality constraints upon free parameters, since either of these can lead to)J 60 266 :M 1.186 .119(violations of the assumption. )J 211 266 :M (S)S 218 266 :M 1.296 .13(ome form of the assumption of faithfulness is used in)J 60 284 :M .901 .09(every science, and amounts to no more that the belief that an improbable and unstable)J 60 302 :M 1.052 .105(cancellation of parameters does not hide real causal influences. When a theory cannot)J 60 320 :M 2.343 .234(explain an empirical regularity save by invoking a special parameterization, most)J 60 338 :M (scientists are uneasy with the theory and look for an alternative.)S 60 362 :M .298 .03(It is also possible to give a personalist Bayesian argument for assuming faithfulness. For)J 60 380 :M .212 .021(any graph, the set of linear parameterizations of the graph that lead to violations of linear)J 60 398 :M 1.974 .197(faithfulness are Lebesgue measure zero. Hence any Bayesian whose prior over the)J 60 416 :M 2.519 .252(parameters is absolutely continuous with Lebesgue measure, assigns a zero prior)J 60 434 :M .341 .034(probability to violations of faithfulness. Of course, this argument is not relevant to those)J 60 452 :M .722 .072(Bayesians who place a prior over the parameters that is not absolutely continuous with)J 60 470 :M (Lebesgue measure and assigns a non-zero probability to violations of faithfulness.)S 60 494 :M 1.869 .187(The assumption of faithfulness guarantees the asymptotic correctness of the Cyclic)J 60 512 :M .027 .003(Causal Discovery \(CCD\) algorithm described in Section )J 335 512 :M .024 .002(4.4. It does )J f4_12 sf (not)S f0_12 sf .033 .003( guarantee that on)J 60 530 :M (samples of finite size this algorithm is reliable.)S 60 554 :M 1.449 .145(Given the Causal Independence Assumption, an assumption of no latent variables, a)J 60 572 :M 2.352 .235(linearity assumption, and the Causal Faithfulness assumption, it follows that in a)J 60 590 :M .109 .011(distribution P generated by a causal structure represented by a directed graph )J f4_12 sf (G)S 443 590 :M f0_12 sf .168 .017(, )J 450 590 :M f5_12 sf (r)S 457 593 :M f0_7 sf .043(XY.Z)A f0_12 sf 0 -3 rm .087 .009( = 0)J 0 3 rm 60 608 :M .259 .026(if and only if X is d-separated from Y given Z in )J 300 608 :M f4_12 sf (G)S 309 608 :M f0_12 sf .234 .023(. So if we can perform statistical tests)J 60 626 :M .809 .081(of zero partial correlations then we can use that information to draw conclusions about)J 60 644 :M .714 .071(the d)J 85 644 :M .555 .056(-separation relations in )J 200 644 :M f4_12 sf (G)S 209 644 :M f0_12 sf .638 .064(, and then to reconstruct as much information about )J 469 644 :M f4_12 sf (G)S 478 644 :M f0_12 sf .833 .083( as)J 60 662 :M 1.309 .131(possible. Henceforth we will speak of reconstructing features of )J 391 662 :M f4_12 sf (G)S 400 662 :M f0_12 sf 1.6 .16( from d-s)J 447 662 :M (eparation)S 60 680 :M .29 .029(relations, and from zero partial correlation interchangeably, since given our assumptions,)J endp %%Page: 14 14 %%BeginPageSetup initializepage (peter; page: 14 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (14)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf 2.923 .292(these are equivalent. \(The algorithm is correct for any distribution for which a)J 60 74 :M .396 .04(d-separation oracle is available, but the only case we know of in which such an oracle is)J 60 92 :M (available is the linear case.\))S 60 116 :M ( )S 63 116 :M 1.481 .148(Of course the number of distinct d)J 242 116 :M 1.159 .116(-separation relations grows exponentially with the)J 60 134 :M .535 .053(number of variables in the graph. Therefore it is important to discover the features of )J f4_12 sf (G)S 60 152 :M f0_12 sf .122 .012(from a subset of the set of all d)J 211 152 :M .095 .009(-separation relations. The CCD algorithm that we describe)J 60 170 :M .418 .042(below chooses the subset of d-separation relations that it needs to reconstruct features of)J 60 188 :M f4_12 sf (G)S 69 188 :M f0_12 sf .476 .048( as it goes along. Therefore we assume that it has access to a )J 372 188 :M f2_12 sf (d)S 379 188 :M .4 .04(-separation oracle)J f0_12 sf .182 .018( that)J 60 206 :M .539 .054(correctly answers questions about d-separation relations in )J 350 206 :M f4_12 sf (G)S 359 206 :M f0_12 sf .637 .064(. In practice, of course, the)J 60 224 :M .249 .025(oracle is some kind of statistical test of the hypothesis that a particular partial correlation)J 60 242 :M (is zero in a population that satisfies the global Markov and faithfulness properties for )S f4_12 sf (G)S 480 242 :M f0_12 sf (.)S 60 266 :M f2_12 sf (4)S 66 266 :M (.)S 69 266 :M (3)S 75 266 :M (.)S 78 266 :M ( )S 96 266 :M (Output Representation \320 Partial Ancestral Graphs \(PAGs\))S 60 293 :M f0_12 sf .056 .006(In general, it is not possible to reconstruct a unique graph )J 340 293 :M f4_12 sf (G)S 349 293 :M f0_12 sf .051 .005( given information only about)J 60 311 :M .144 .014(its d-separation relations, because there may be more than one graph that contains exactly)J 60 329 :M 2.583 .258(the same set of d)J 156 329 :M 1.866 .187(-separation relations. Two directed graphs )J f4_12 sf (G)S 387 329 :M f0_12 sf 3.288 .329(, )J 397 329 :M f4_12 sf (G)S 406 326 :M f14_11 sf (*)S 411 329 :M f0_12 sf 2.679 .268( are said to be)J 60 347 :M .407 .041(d-separation )J f2_12 sf .08(equivalent)A f4_12 sf ( )S 179 347 :M f0_12 sf .626 .063(if they both have the same set of d)J 351 347 :M .454 .045(-separation relations. The set)J 60 365 :M .8 .08(of directed graphs d)J 160 365 :M .792 .079(-separation equivalent to a given graph )J 357 365 :M f4_12 sf (G)S 366 365 :M f0_12 sf .661 .066( is denoted by )J f2_12 sf .398(Equiv)A 472 365 :M f0_12 sf <28>S 476 365 :M f4_12 sf (G)S 485 365 :M f0_12 sf (\).)S 60 383 :M 2.326 .233(Richardson\(1994b\) and Richardson\(1995\) present a polynomial-time algorithm for)J 60 401 :M 2.267 .227(determining when two graphs are d)J 249 401 :M 2.105 .21(-separation equivalent to each other; a simpler)J 60 419 :M .755 .075(algorithm is presented in Section 5. \(Note that there is a stronger sense of equivalence,)J 60 437 :M .809 .081(which we will call linear statistical equivalence between two graphs which holds when)J 60 455 :M 2.047 .205(every distribution described by a linear parameterization of one graph can also be)J 60 473 :M .646 .065(described by a linear parameterization of the other graph, and vice-versa. In the acyclic)J 60 491 :M 3.57 .357(case it is known that d)J 193 491 :M 2.614 .261(-separation equivalence is equivalent to linear statistical)J 60 509 :M (equivalence, but it is not known if this is so for cyclic graphs.\))S 60 533 :M .598 .06(The members of )J 145 533 :M f2_12 sf (Equiv)S 176 533 :M f0_12 sf <28>S 180 533 :M f4_12 sf (G)S 189 533 :M f0_12 sf .537 .054(\) always have certain features in common. We now introduce)J 60 551 :M .536 .054(the formalism with which we will represent features common to all graphs in )J 444 551 :M f2_12 sf (Equiv)S 475 551 :M f0_12 sf <28>S 479 551 :M f4_12 sf (G)S 488 551 :M f0_12 sf <29>S 60 569 :M .524 .052(for some fixed )J f4_12 sf (G)S 144 569 :M f0_12 sf .555 .056(. A partial ancestral graph \(PAG\) is an extended graph, consisting of a)J 60 587 :M .813 .081(set of vertices )J 134 587 :M f2_12 sf (V)S 143 587 :M f0_12 sf .792 .079(, a set of edges between vertices, and a set of edge-endpoints, two for)J 60 605 :M 1.406 .141(each edge, drawn from the set {)J f0_10 sf .419(o)A f0_12 sf 1.349 .135(, \320, >}. In addition pairs of edge endpoints may be)J 60 623 :M .292 .029(connected by underlining, or dotted underlining. A partial ancestral graph for )J 440 623 :M f4_12 sf (G)S 449 623 :M f0_12 sf .277 .028( contains)J 60 641 :M .105 .01(partial information about the ancestor relations in )J f4_12 sf (G)S 310 641 :M f0_12 sf .109 .011(, namely only those ancestor relations)J 60 659 :M 1.395 .139(common to all members of )J f2_12 sf .606(Equiv)A 236 659 :M f0_12 sf <28>S 240 659 :M f4_12 sf (G)S 249 659 :M f0_12 sf 1.542 .154(\). In the following definition, which provides a)J endp %%Page: 15 15 %%BeginPageSetup initializepage (peter; page: 15 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (15)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf .879 .088(semantics for PAGs we use '*' as a meta-symbol indicating the presence of any one of)J 60 74 :M ({)S 66 74 :M f0_10 sf (o)S f0_12 sf (,\312\320, >}.)S 60 106 :M f2_12 sf (Partial Ancestral Graphs)S f0_12 sf ( \()S 196 106 :M f2_12 sf (PAGs)S 226 106 :M f0_12 sf <29>S 230 103 :M f0_8 sf (9)S 60 122 :M f3_12 sf (Y)S 70 122 :M f0_11 sf ( )S 73 122 :M f0_12 sf (is a PAG for Directed Cyclic Graph )S 248 122 :M f4_12 sf (G)S 257 122 :M f0_12 sf ( with vertex set )S 334 122 :M f2_12 sf (V)S 343 122 :M f0_12 sf (, if and only if)S 78 138 :M .357 .036(\(i\) There is an edge between A and B in )J 278 138 :M f4_12 sf (G)S 287 138 :M f0_12 sf .352 .035( if and only if A and B are d-connected in)J 78 154 :M f3_12 sf (Y )S 91 154 :M f0_12 sf (given any subset )S 174 154 :M f2_12 sf (W)S f0_12 sf ( )S f1_12 sf S 198 154 :M f0_12 sf ( )S f2_12 sf (V)S 210 154 :M f0_12 sf (\\{A,B}.)S 78 170 :M .229 .023(\(ii\) If there is an edge in )J 200 170 :M f3_12 sf .288 .029(Y )J 213 170 :M f0_12 sf .238 .024(out of A \(not necessarily into B\), i.e. A\320)J f1_12 sf .09(*)A f0_12 sf .172 .017( B, then A is an)J 78 186 :M (ancestor of B in every graph in )S 230 186 :M f2_12 sf (Equiv)S 261 186 :M f0_12 sf <28>S 265 186 :M f4_12 sf (G)S 274 186 :M f0_12 sf (\).)S 78 202 :M .116 .012(\(iii\) If there is an edge in )J f3_12 sf .107 .011(Y )J 214 202 :M f0_12 sf .11 .011(into B, i.e. A)J 277 202 :M f1_12 sf (*)S f0_12 sf .085 .008(\320>B, then in every graph in )J f2_12 sf .042(Equiv)A 450 202 :M f0_12 sf <28>S 454 202 :M f4_12 sf (G)S 463 202 :M f0_12 sf .121 .012(\), B is)J 78 218 :M f2_12 sf (not)S 95 218 :M f0_12 sf ( an ancestor of A.)S 78 234 :M .662 .066(\(iv\) If there is an underlining A*\321)J 254 0 6 730 rC 254 234 :M 12 f6_1 :p 4.037 :m 1.037 .104( )J 256 234 :M 8.075 :m .988 .099( )J gR gS 0 0 552 730 rC 254 234 :M f0_12 sf 12 f6_1 :p 20.004 :m (*B*)S 268 0 6 730 rC 268 234 :M 4.037 :m 1.037 .104( )J 270 234 :M 8.075 :m .988 .099( )J gR gS 0 0 552 730 rC 274 234 :M f0_12 sf .624 .062(\321*C in )J f3_12 sf (Y)S 327 234 :M f0_12 sf .728 .073( then B is an ancestor of \(at least)J 78 250 :M (one of\) A or C in every graph in )S 236 250 :M f2_12 sf (Equiv)S 267 250 :M f0_12 sf <28>S 271 250 :M f4_12 sf (G)S 280 250 :M f0_12 sf (\).)S 78 266 :M .295 .029(\(v\) If there is an edge from A to B, and from C to B, \(A\321> B)J f1_12 sf (<)S 388 266 :M f0_12 sf .281 .028(\321 C\), then the arrow)J 78 282 :M .226 .023(heads at B are joined by dotted underlining, thus A\321)J 338 0 7 730 rC 338 282 :M .338 .034( )J 342 282 :M .338 .034( )J gR gS 0 0 552 730 rC 338 282 :M f0_12 sf (>B<)S 353 0 7 730 rC 353 282 :M .338 .034( )J 357 282 :M .338 .034( )J gR gS 0 0 552 730 rC 360 282 :M f0_12 sf .236 .024(\321C, only if in every graph)J 1 G 0 0 1 1 rF 491 282 :M psb /wp$x1 338 def /wp$x2 359 def /wp$y 284 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 78 298 :M 0 G (in )S f2_12 sf (Equiv)S 121 298 :M f0_12 sf <28>S 125 298 :M f4_12 sf (G)S 134 298 :M f0_12 sf (\) B is not a descendant of a common child of A and C.)S 78 314 :M .033 .003(\(vi\) Any edge endpoint not marked in one of the above ways is left with a small circle)J 78 330 :M (thus: )S f0_9 sf (o)S 109 330 :M f0_12 sf S 60 354 :M .765 .076(Observe that condition \(i\) differs from the other five conditions in providing necessary)J 60 372 :M f4_12 sf .082(and)A f0_12 sf .249 .025( sufficient condtions on )J 196 372 :M f2_12 sf (Equiv)S 227 372 :M f0_12 sf <28>S 231 372 :M f4_12 sf (G)S 240 372 :M f0_12 sf .312 .031(\) for a given symbol, in this case an edge, to appear)J 60 390 :M .877 .088(in a PAG. The other five conditions merely state necessary conditions. For this reason)J 60 408 :M .923 .092(there are in fact many different PAGs for a graph )J f4_12 sf (G)S 322 408 :M f0_12 sf .935 .094(. Although they all have the same)J 60 426 :M .232 .023(edges, they do not necessarily have the same endpoints. Some of the PAGs provide more)J 60 444 :M 1.287 .129(information than others about causal structure, e.g. they have fewer ')J 409 444 :M f0_10 sf .584(o)A f0_12 sf 1.476 .148('s at the end of)J 60 462 :M (edges.)S 90 459 :M f0_9 sf (1)S 94 459 :M (0)S 60 486 :M f0_12 sf .076 .008(If )J f3_12 sf (Y)S 81 486 :M f0_11 sf .183 .018( )J 84 486 :M f0_12 sf .116 .012(is a PAG for Directed Cyclic Graph )J f4_12 sf (G)S 269 486 :M f0_12 sf .119 .012(, we also say that )J f3_12 sf (Y)S 365 486 :M f0_11 sf .183 .018( )J 368 486 :M f0_12 sf .092 .009(represents )J 421 486 :M f4_12 sf (G)S 430 486 :M f0_12 sf .115 .011(. Since every)J 60 504 :M -.004(clause in the definition refers only to )A 240 504 :M f2_12 sf (Equiv)S 271 504 :M f0_12 sf <28>S 275 504 :M f4_12 sf (G)S 284 504 :M f0_12 sf -.005(\), it follows that if )A 374 504 :M f3_12 sf (Y)S 384 504 :M f0_11 sf ( )S 387 504 :M f0_12 sf -.005(is a PAG for Directed)A 60 522 :M .257 .026(Cyclic Graph )J f4_12 sf (G)S 137 522 :M f0_12 sf .277 .028(, and )J f4_12 sf (G)S 173 519 :M f14_11 sf (*)S 178 522 :M f1_12 sf S 187 522 :M f2_12 sf (Equiv)S 218 522 :M f0_12 sf <28>S 222 522 :M f4_12 sf (G)S 231 522 :M f0_12 sf .354 .035(\), then )J 266 522 :M f3_12 sf (Y)S 276 522 :M f0_11 sf .478 .048( )J 280 522 :M f0_12 sf .319 .032(is also a PAG for )J f4_12 sf (G)S 377 519 :M f14_11 sf (*)S 382 522 :M f0_12 sf .304 .03(. This is not surprising)J 60 540 :M .233 .023(since, as the output of the discovery algorithm we present below, the PAG is designed to)J 60 572 :M ( )S 60 569.48 -.48 .48 204.48 569 .48 60 569 @a 60 581 :M f0_8 sf (9)S 64 584 :M f0_9 sf .107 .011( )J f0_10 sf .756 .076(The extended graphs which we introduce here - Partial Ancestral Graphs - use a superset of the set of)J 60 595 :M .314 .031(symbols used by Partially Oriented Inducing Path Graphs \(POIPGs\) descaribed in Spirtes )J f4_10 sf .241 .024(et al.)J 447 595 :M f0_10 sf .352 .035( \(1993\) but)J 60 606 :M .589 .059(the )J 76 606 :M f4_10 sf .403 .04(graphical )J 117 606 :M f0_10 sf .462 .046(interpretation of the orientations given to edges is different. However, it has been shown in)J 60 617 :M .172 .017(Sprites )J f4_10 sf .128 .013(et al.)J f0_10 sf .142 .014( \(1996\) that a POIPG can be interpreted directly as a PAG. A direct corollary is that PAGs can)J 60 628 :M .032 .003(be used to represent the d-separation equivalence class for directed )J 331 628 :M f4_10 sf .032 .003(acyclic )J 362 628 :M f0_10 sf .032 .003(graphs with )J f4_10 sf .031 .003(latent )J f0_10 sf .033 .003(variables. It is)J 60 639 :M .202 .02(an open question whether or not the set of symbols is sufficiently rich to allow us to represent d)J 448 639 :M (-separation)S 60 650 :M (classes of cyclic graphs with latent variables.)S 60 660 :M f0_9 sf (1)S 64 660 :M (0)S 68 663 :M f0_10 sf .204 .02(If one PAG for a graph )J f4_10 sf .129(G)A f0_10 sf .216 .022( has a '>' at the end of an edge, then every other PAG for the same graph either)J 60 675 :M .698 .07(has a '>' or a ')J 119 675 :M f0_9 sf (o)S 124 675 :M f0_10 sf .68 .068(' in that location. Similarly if one PAG for a graph )J f4_10 sf .393(G)A f0_10 sf .557 .056( has a '\320' at the end of an edge then)J 60 687 :M (every other PAG for the same graph either has a '\320' or an ')S f0_9 sf (o)S 296 687 :M f0_10 sf (' in that location.)S endp %%Page: 16 16 %%BeginPageSetup initializepage (peter; page: 16 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (16)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf .509 .051(represent features common to all graphs in the d)J 298 56 :M .357 .036(-separation equivalence class. However,)J 60 74 :M 2.126 .213(some PAGs \(providing less information\) may also represent graphs from different)J 60 92 :M .593 .059(d-separation equivalence classes. However, any PAG output by the discovery algorithm)J 60 110 :M .904 .09(we present provides sufficient information so as to ensure that all of the graphs that it)J 60 128 :M 1.994 .199(represents lie in one d-separation equivalence class; hence any PAG output by the)J 60 146 :M .348 .035(discovery algorithm can be thought of as representing a unique d-separation equivalence)J 60 164 :M (class.)S 60 188 :M .021 .002(The set of features described by a PAG is rich enough to enable us to distinguish between)J 60 206 :M .298 .03(any two d-separation equivalence classes, i.e. there is some feature common to all graphs)J 60 224 :M 3.203 .32(in one d)J 108 224 :M 2.702 .27(-separation equivalence class that is not true of all graphs in the other)J 60 242 :M .145 .015(d-separation equivalence class, and this difference can be expressed by a difference in the)J 60 260 :M (PAGs representing those d-separation equivalence classes.)S 60 292 :M f2_12 sf (Example:)S 60 308 :M f0_12 sf (Suppose )S 104 308 :M f4_12 sf (G)S 113 308 :M f0_12 sf ( is as follows:)S 261 321 13 13 rC 261 330 :M (A)S gR gS 262 361 12 12 rC 262 370 :M f0_12 sf (B)S gR gS 313 321 13 13 rC 313 330 :M f0_12 sf (X)S gR gS 314 361 13 12 rC 314 370 :M f0_12 sf (Y)S gR gS 224 319 103 54 rC np 311 326 :M 300 330 :L 300 330 :L 300 329 :L 300 329 :L 300 329 :L 300 329 :L 300 329 :L 300 329 :L 300 328 :L 299 328 :L 299 328 :L 299 328 :L 299 327 :L 299 327 :L 299 327 :L 299 327 :L 299 327 :L 299 326 :L 299 326 :L 299 326 :L 299 326 :L 299 326 :L 299 325 :L 299 325 :L 299 325 :L 299 325 :L 299 325 :L 299 324 :L 299 324 :L 299 324 :L 299 324 :L 299 324 :L 300 323 :L 300 323 :L 300 323 :L 300 323 :L 300 323 :L 300 322 :L 300 322 :L 300 322 :L 311 326 :L 311 326 :L eofill 275 327 -1 1 300 326 1 275 326 @a np 311 365 :M 300 369 :L 300 369 :L 300 369 :L 300 369 :L 300 368 :L 300 368 :L 300 368 :L 300 368 :L 300 368 :L 299 367 :L 299 367 :L 299 367 :L 299 367 :L 299 367 :L 299 366 :L 299 366 :L 299 366 :L 299 366 :L 299 366 :L 299 365 :L 299 365 :L 299 365 :L 299 365 :L 299 365 :L 299 364 :L 299 364 :L 299 364 :L 299 364 :L 299 364 :L 299 363 :L 299 363 :L 299 363 :L 300 363 :L 300 363 :L 300 362 :L 300 362 :L 300 362 :L 300 362 :L 300 362 :L 300 361 :L 311 365 :L 311 365 :L eofill 275 366 -1 1 300 365 1 275 365 @a 1 G 10 27 318 347.5 @j 0 G 11 28 318 347.5 @f np 318 332 :M 325 341 :L 325 341 :L 325 341 :L 325 341 :L 324 342 :L 324 342 :L 324 342 :L 324 342 :L 324 342 :L 324 342 :L 323 342 :L 323 342 :L 323 342 :L 323 342 :L 323 343 :L 323 343 :L 322 343 :L 322 343 :L 322 343 :L 322 343 :L 322 343 :L 321 343 :L 321 343 :L 321 343 :L 321 343 :L 321 343 :L 320 343 :L 320 343 :L 320 343 :L 320 343 :L 320 343 :L 319 343 :L 319 343 :L 319 344 :L 319 344 :L 319 344 :L 318 344 :L 318 344 :L 318 344 :L 318 344 :L 318 332 :L 318 332 :L eofill -1 -1 323 345 1 1 322 343 @b np 316 360 :M 310 350 :L 310 350 :L 310 350 :L 310 350 :L 310 350 :L 311 350 :L 311 350 :L 311 350 :L 311 350 :L 311 349 :L 311 349 :L 312 349 :L 312 349 :L 312 349 :L 312 349 :L 312 349 :L 313 349 :L 313 349 :L 313 349 :L 313 349 :L 313 349 :L 314 349 :L 314 349 :L 314 349 :L 314 348 :L 314 348 :L 315 348 :L 315 348 :L 315 348 :L 315 348 :L 315 348 :L 316 348 :L 316 348 :L 316 348 :L 316 348 :L 316 348 :L 317 348 :L 317 348 :L 317 348 :L 317 348 :L 316 360 :L 316 360 :L eofill -1 -1 314 350 1 1 313 348 @b 225 339 11 16 rC 225 348 :M f4_12 sf (G)S gR gS 0 0 552 730 rC 254 388 :M f2_12 sf (Figure )S 291 388 :M (7)S 60 412 :M f0_12 sf (In this case it can be shown that )S 216 412 :M f2_12 sf (Equiv)S 247 412 :M f0_12 sf <28>S 251 412 :M f4_12 sf (G)S 260 412 :M f0_12 sf (\) contains \(only\) two graphs:)S 1 G 149 429 254 65 rC 210 430 401 492 15 @q 0 G 209.5 429.5 401.5 492.5 15.5 @s 224 435 12 13 rC 224 444 :M (A)S 224 458 :M f14_13 sf (G)S 231 458 :M f0_12 sf <29>S gR gS 225 476 11 13 rC 225 485 :M f0_12 sf (B)S gR gS 276 435 13 13 rC 276 444 :M f0_12 sf (X)S gR gS 277 476 13 13 rC 277 485 :M f0_12 sf (Y)S gR gS 149 429 254 65 rC np 274 440 :M 263 444 :L 263 444 :L 263 444 :L 263 444 :L 263 444 :L 262 443 :L 262 443 :L 262 443 :L 262 443 :L 262 443 :L 262 442 :L 262 442 :L 262 442 :L 262 442 :L 262 441 :L 262 441 :L 262 441 :L 262 441 :L 262 441 :L 262 440 :L 262 440 :L 262 440 :L 262 440 :L 262 440 :L 262 439 :L 262 439 :L 262 439 :L 262 439 :L 262 439 :L 262 438 :L 262 438 :L 262 438 :L 262 438 :L 262 438 :L 262 437 :L 262 437 :L 263 437 :L 263 437 :L 263 437 :L 263 436 :L 274 440 :L 274 440 :L eofill 237 441 -1 1 263 440 1 237 440 @a np 274 480 :M 263 484 :L 263 483 :L 263 483 :L 263 483 :L 263 483 :L 262 483 :L 262 483 :L 262 482 :L 262 482 :L 262 482 :L 262 482 :L 262 481 :L 262 481 :L 262 481 :L 262 481 :L 262 481 :L 262 480 :L 262 480 :L 262 480 :L 262 480 :L 262 480 :L 262 479 :L 262 479 :L 262 479 :L 262 479 :L 262 479 :L 262 478 :L 262 478 :L 262 478 :L 262 478 :L 262 478 :L 262 477 :L 262 477 :L 262 477 :L 262 477 :L 262 477 :L 263 476 :L 263 476 :L 263 476 :L 263 476 :L 274 480 :L 274 480 :L eofill 237 481 -1 1 263 480 1 237 480 @a 1 G 10 27 281 461.5 @j 0 G 11 28 281 461.5 @f np 281 446 :M 288 455 :L 288 456 :L 288 456 :L 288 456 :L 288 456 :L 287 456 :L 287 456 :L 287 456 :L 287 456 :L 287 457 :L 287 457 :L 286 457 :L 286 457 :L 286 457 :L 286 457 :L 286 457 :L 285 457 :L 285 457 :L 285 457 :L 285 457 :L 285 457 :L 284 457 :L 284 458 :L 284 458 :L 284 458 :L 284 458 :L 283 458 :L 283 458 :L 283 458 :L 283 458 :L 283 458 :L 282 458 :L 282 458 :L 282 458 :L 282 458 :L 282 458 :L 281 458 :L 281 458 :L 281 458 :L 281 458 :L 281 446 :L 281 446 :L eofill -1 -1 286 459 1 1 285 457 @b np 279 475 :M 273 465 :L 273 465 :L 273 465 :L 273 465 :L 273 464 :L 274 464 :L 274 464 :L 274 464 :L 274 464 :L 274 464 :L 274 464 :L 275 464 :L 275 464 :L 275 464 :L 275 464 :L 275 463 :L 276 463 :L 276 463 :L 276 463 :L 276 463 :L 276 463 :L 277 463 :L 277 463 :L 277 463 :L 277 463 :L 277 463 :L 278 463 :L 278 463 :L 278 463 :L 278 463 :L 278 463 :L 279 463 :L 279 463 :L 279 463 :L 279 463 :L 279 463 :L 280 463 :L 280 463 :L 280 463 :L 280 463 :L 279 475 :L 279 475 :L eofill -1 -1 277 464 1 1 276 462 @b 322 434 13 13 rC 322 443 :M f0_12 sf (A)S gR gS 323 475 12 13 rC 323 484 :M f0_12 sf (B)S gR gS 375 434 13 13 rC 375 443 :M f0_12 sf (X)S gR gS 376 475 13 13 rC 376 484 :M f0_12 sf (Y)S gR gS 149 429 254 65 rC np 376 480 :M 365 475 :L 365 474 :L 365 474 :L 365 474 :L 365 474 :L 365 474 :L 366 473 :L 366 473 :L 366 473 :L 366 473 :L 366 473 :L 366 473 :L 366 472 :L 366 472 :L 366 472 :L 367 472 :L 367 472 :L 367 472 :L 367 472 :L 367 471 :L 367 471 :L 368 471 :L 368 471 :L 368 471 :L 368 471 :L 368 471 :L 368 470 :L 368 470 :L 369 470 :L 369 470 :L 369 470 :L 369 470 :L 369 470 :L 369 470 :L 370 469 :L 370 469 :L 370 469 :L 370 469 :L 370 469 :L 371 469 :L 376 480 :L 376 480 :L eofill 336 440 -1 1 368 471 1 336 439 @a np 373 442 :M 368 453 :L 367 453 :L 367 453 :L 367 453 :L 367 453 :L 367 452 :L 366 452 :L 366 452 :L 366 452 :L 366 452 :L 366 452 :L 366 452 :L 365 452 :L 365 452 :L 365 451 :L 365 451 :L 365 451 :L 365 451 :L 365 451 :L 364 451 :L 364 451 :L 364 450 :L 364 450 :L 364 450 :L 364 450 :L 363 450 :L 363 450 :L 363 450 :L 363 449 :L 363 449 :L 363 449 :L 363 449 :L 363 449 :L 363 449 :L 362 448 :L 362 448 :L 362 448 :L 362 448 :L 362 448 :L 362 447 :L 373 442 :L 373 442 :L eofill -1 -1 337 480 1 1 364 450 @b 1 G 10 28 380 461 @j 0 G 11 29 380 461 @f np 380 445 :M 387 454 :L 387 455 :L 387 455 :L 386 455 :L 386 455 :L 386 455 :L 386 455 :L 386 455 :L 386 455 :L 385 456 :L 385 456 :L 385 456 :L 385 456 :L 385 456 :L 384 456 :L 384 456 :L 384 456 :L 384 456 :L 384 456 :L 383 456 :L 383 456 :L 383 456 :L 383 457 :L 383 457 :L 382 457 :L 382 457 :L 382 457 :L 382 457 :L 382 457 :L 381 457 :L 381 457 :L 381 457 :L 381 457 :L 381 457 :L 380 457 :L 380 457 :L 380 457 :L 380 457 :L 380 457 :L 379 457 :L 380 445 :L 380 445 :L eofill 383 457 -1 1 385 457 1 383 456 @a np 378 474 :M 371 464 :L 371 464 :L 372 464 :L 372 464 :L 372 463 :L 372 463 :L 372 463 :L 373 463 :L 373 463 :L 373 463 :L 373 463 :L 373 463 :L 373 463 :L 374 463 :L 374 463 :L 374 462 :L 374 462 :L 374 462 :L 375 462 :L 375 462 :L 375 462 :L 375 462 :L 375 462 :L 376 462 :L 376 462 :L 376 462 :L 376 462 :L 376 462 :L 377 462 :L 377 462 :L 377 462 :L 377 462 :L 378 462 :L 378 462 :L 378 462 :L 378 462 :L 378 462 :L 379 462 :L 379 462 :L 379 462 :L 378 474 :L 378 474 :L eofill -1 -1 376 463 1 1 375 461 @b 149 450 61 16 rC 149 459 :M f2_12 sf (E)S 157 459 :M (q)S 164 459 :M (u)S 171 459 :M (i)S 175 459 :M (v)S 182 459 :M f0_12 sf <28>S 186 459 :M f4_12 sf (G)S 195 459 :M f0_12 sf <29>S gR gS 0 0 552 730 rC 254 515 :M f2_12 sf (Figure )S 291 515 :M (8)S 60 549 :M f0_12 sf 2.017 .202(The PAG which the CCD algorithm outputs given as input an oracle for deciding)J 60 565 :M (conditional independence facts in )S 224 565 :M f4_12 sf (G)S 233 565 :M f0_12 sf (, is:)S 1 G 160 576 231 54 rC 0 90 12 22 374 616 @l 0 G np 380 617 :M 380 618 :L 379 618 :L 379 616 :L 380 617 :L eofill np 380 618 :M 379 620 :L 378 619 :L 379 618 :L 380 618 :L eofill np 380 618 :M 380 618 :L 379 618 :L 379 618 :L 380 618 :L eofill np 379 622 :M 379 622 :L 378 622 :L 378 621 :L 379 622 :L eofill np 379 622 :M 378 623 :L 377 623 :L 378 622 :L 379 622 :L eofill np 379 622 :M 379 622 :L 378 622 :L 378 622 :L 379 622 :L eofill np 378 623 :M 378 624 :L 377 624 :L 377 623 :L 378 623 :L eofill np 378 623 :M 378 623 :L 377 623 :L 377 623 :L 378 623 :L eofill np 376 626 :M 376 626 :L 375 625 :L 376 625 :L 376 626 :L eofill np 376 626 :M 375 626 :L 375 625 :L 376 625 :L 376 626 :L eofill np 376 626 :M 376 626 :L 375 625 :L 376 625 :L 376 626 :L eofill 1 G -90 0 14 28 372 592 @l 0 G np 373 578 :M 374 578 :L 374 579 :L 373 579 :L 373 578 :L eofill np 374 578 :M 375 579 :L 375 580 :L 374 579 :L 374 578 :L eofill np 374 578 :M 374 578 :L 374 579 :L 374 579 :L 374 578 :L eofill np 375 579 :M 375 579 :L 375 580 :L 375 579 :L 375 579 :L eofill np 375 579 :M 375 579 :L 375 580 :L 375 579 :L 375 579 :L eofill np 377 581 :M 377 582 :L 376 582 :L 376 581 :L 377 581 :L eofill np 377 582 :M 378 583 :L 377 583 :L 376 582 :L 377 582 :L eofill np 377 582 :M 377 582 :L 376 582 :L 376 582 :L 377 582 :L eofill np 378 585 :M 378 586 :L 377 586 :L 377 585 :L 378 585 :L eofill np 378 586 :M 379 588 :L 378 588 :L 377 586 :L 378 586 :L eofill np 378 586 :M 378 586 :L 377 586 :L 377 586 :L 378 586 :L eofill np 379 589 :M 379 590 :L 378 591 :L 378 589 :L 379 589 :L eofill 325 577 13 13 rC 325 586 :M (A)S 325 600 :M f14_13 sf (G)S 325 613 :M f0_12 sf (g)S 331 613 :M (i)S 325 625 :M (v)S 331 625 :M (e)S 325 637 :M (n)S gR gS 326 617 12 13 rC 326 626 :M f0_12 sf (B)S gR gS 380 577 11 13 rC 380 586 :M f0_12 sf (X)S gR gS 381 617 10 13 rC 381 626 :M f0_12 sf (Y)S gR gS 160 576 231 54 rC np 376 582 :M 364 586 :L 364 586 :L 364 585 :L 364 585 :L 364 585 :L 364 585 :L 364 585 :L 364 585 :L 364 584 :L 364 584 :L 364 584 :L 364 584 :L 364 583 :L 364 583 :L 364 583 :L 364 583 :L 364 583 :L 364 582 :L 364 582 :L 364 582 :L 364 582 :L 364 582 :L 364 581 :L 364 581 :L 364 581 :L 364 581 :L 364 581 :L 364 580 :L 364 580 :L 364 580 :L 364 580 :L 364 580 :L 364 579 :L 364 579 :L 364 579 :L 364 579 :L 364 579 :L 364 578 :L 364 578 :L 364 578 :L 376 582 :L 376 582 :L eofill 339 583 -1 1 365 582 1 339 582 @a np 377 621 :M 365 625 :L 365 625 :L 365 625 :L 365 625 :L 365 624 :L 365 624 :L 365 624 :L 365 624 :L 365 624 :L 365 623 :L 365 623 :L 365 623 :L 365 623 :L 365 623 :L 365 622 :L 365 622 :L 365 622 :L 365 622 :L 365 622 :L 365 621 :L 365 621 :L 365 621 :L 365 621 :L 365 621 :L 365 620 :L 365 620 :L 365 620 :L 365 620 :L 365 620 :L 365 619 :L 365 619 :L 365 619 :L 365 619 :L 365 619 :L 365 618 :L 365 618 :L 365 618 :L 365 618 :L 365 618 :L 365 617 :L 377 621 :L 377 621 :L eofill 340 622 -1 1 366 621 1 340 621 @a np 377 620 :M 366 615 :L 367 615 :L 367 615 :L 367 614 :L 367 614 :L 367 614 :L 367 614 :L 367 614 :L 367 614 :L 367 613 :L 368 613 :L 368 613 :L 368 613 :L 368 613 :L 368 613 :L 368 612 :L 368 612 :L 368 612 :L 369 612 :L 369 612 :L 369 612 :L 369 611 :L 369 611 :L 369 611 :L 370 611 :L 370 611 :L 370 611 :L 370 611 :L 370 611 :L 370 610 :L 370 610 :L 371 610 :L 371 610 :L 371 610 :L 371 610 :L 371 610 :L 372 610 :L 372 610 :L 372 609 :L 372 609 :L 377 620 :L 377 620 :L eofill 340 584 -1 1 370 612 1 340 583 @a np 377 584 :M 372 594 :L 371 594 :L 371 594 :L 371 594 :L 371 594 :L 371 594 :L 370 594 :L 370 594 :L 370 594 :L 370 594 :L 370 593 :L 370 593 :L 369 593 :L 369 593 :L 369 593 :L 369 593 :L 369 593 :L 369 593 :L 368 592 :L 368 592 :L 368 592 :L 368 592 :L 368 592 :L 368 592 :L 368 592 :L 367 591 :L 367 591 :L 367 591 :L 367 591 :L 367 591 :L 367 591 :L 367 590 :L 367 590 :L 366 590 :L 366 590 :L 366 590 :L 366 589 :L 366 589 :L 366 589 :L 366 589 :L 377 584 :L 377 584 :L eofill -1 -1 340 622 1 1 368 592 @b 161 594 149 16 rC 161 603 :M f0_12 sf (P)S 168 603 :M (A)S 176 603 :M (G)S 185 603 :M ( )S 188 603 :M (f)S 192 603 :M (o)S 198 603 :M (r)S 202 603 :M ( )S 205 603 :M f2_12 sf (E)S 213 603 :M (q)S 220 603 :M (u)S 227 603 :M (i)S 231 603 :M (v)S 238 603 :M f0_12 sf <28>S 242 603 :M f4_12 sf (G)S 251 603 :M f0_12 sf <29>S gR gS 160 576 231 54 rC -1 -1 386 619 1 1 385 589 @b gR gS 0 0 552 730 rC 254 645 :M f2_12 sf (Figure )S 291 645 :M (9)S endp %%Page: 17 17 %%BeginPageSetup initializepage (peter; page: 17 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (17)S gR gS 0 0 552 730 rC 60 54 :M f0_12 sf (Observe that the PAG tells us the following facts about )S 328 54 :M f2_12 sf (Equiv)S 359 54 :M f0_12 sf <28>S 363 54 :M f4_12 sf (G)S 372 54 :M f0_12 sf (\):)S 379 51 :M f0_9 sf (1)S 383 51 :M (1)S 78 70 :M f0_12 sf (\(a\) X is an ancestor of Y, and Y is an ancestor of X in every graph in )S 412 70 :M f2_12 sf (Equiv)S 443 70 :M f0_12 sf <28>S 447 70 :M f4_12 sf (G)S 456 70 :M f0_12 sf (\).)S 78 86 :M (\(b\) In no graph in )S 166 86 :M f2_12 sf (Equiv)S 197 86 :M f0_12 sf <28>S 201 86 :M f4_12 sf (G)S 210 86 :M f0_12 sf (\) is X or Y an ancestor of A or B.)S 78 102 :M (\(c\) In every graph in )S 180 102 :M f2_12 sf (Equiv)S 211 102 :M f0_12 sf <28>S 215 102 :M f4_12 sf (G)S 224 102 :M f0_12 sf (\) both A and B are ancestors of X and Y.)S 60 126 :M .126 .013(Note that not every edge in the PAG appears in every graph in )J f2_12 sf .062(Equiv)A 395 126 :M f0_12 sf <28>S 399 126 :M f4_12 sf (G)S 408 126 :M f0_12 sf .131 .013(\). This is because)J 60 144 :M 1.45 .145(an edge in the PAG indicates only that the two variables connected by the edge are)J 60 162 :M 1.97 .197(d-connected given any subset of the other variables. In fact)J 372 162 :M 2.265 .226( it is possible to show)J 60 180 :M .403 .04(something stronger, namely that if there is an edge between two vertices in a PAG, then)J 60 198 :M (there is some graph represented by the PAG in which that edge is present.)S 415 195 :M f0_9 sf (1)S 419 195 :M (2)S 60 222 :M f0_12 sf .085 .009(This example is atypical in that the PAG given by the algorithm contains no ')J f0_10 sf (o)S f0_12 sf .131 .013(' endpoints;)J 60 240 :M .468 .047(however it shows how much information a PAG may provide. Notice that the following)J 60 258 :M (are also PAGs for )S 149 258 :M f4_12 sf (G)S 158 258 :M f0_12 sf ( though they are less informative.)S 135 269 281 55 rC np 318 309 :M 318 311 :L 317 311 :L 317 309 :L 318 309 :L eofill np 318 311 :M 318 312 :L 317 312 :L 317 311 :L 318 311 :L eofill np 318 311 :M 318 311 :L 317 311 :L 317 311 :L 318 311 :L eofill np 318 314 :M 318 314 :L 317 314 :L 317 313 :L 318 314 :L eofill np 318 314 :M 317 315 :L 316 315 :L 317 314 :L 318 314 :L eofill np 318 314 :M 318 314 :L 317 314 :L 317 314 :L 318 314 :L eofill np 317 315 :M 316 316 :L 316 316 :L 316 315 :L 317 315 :L eofill np 317 315 :M 317 315 :L 316 315 :L 316 315 :L 317 315 :L eofill np 315 318 :M 315 318 :L 314 317 :L 314 317 :L 315 318 :L eofill np 314 318 :M 314 318 :L 313 317 :L 314 317 :L 314 318 :L eofill np 314 318 :M 315 318 :L 314 317 :L 314 317 :L 314 318 :L eofill np 312 270 :M 313 270 :L 313 271 :L 311 271 :L 312 270 :L eofill np 313 270 :M 314 271 :L 313 272 :L 312 271 :L 313 270 :L eofill np 313 270 :M 313 270 :L 313 271 :L 312 271 :L 313 270 :L eofill np 314 271 :M 314 272 :L 314 272 :L 313 272 :L 314 271 :L eofill np 314 271 :M 314 271 :L 313 272 :L 313 272 :L 314 271 :L eofill np 316 273 :M 316 274 :L 315 275 :L 315 274 :L 316 273 :L eofill np 316 274 :M 317 276 :L 316 276 :L 315 275 :L 316 274 :L eofill np 316 274 :M 316 274 :L 315 275 :L 315 275 :L 316 274 :L eofill np 317 276 :M 317 276 :L 316 277 :L 316 276 :L 317 276 :L eofill np 317 276 :M 317 276 :L 316 276 :L 316 276 :L 317 276 :L eofill np 317 278 :M 318 280 :L 317 281 :L 316 278 :L 317 278 :L eofill np 318 280 :M 318 281 :L 317 281 :L 317 280 :L 318 280 :L eofill np 318 280 :M 318 280 :L 317 281 :L 317 280 :L 318 280 :L eofill 264 269 13 13 rC 264 278 :M (A)S 264 292 :M f14_13 sf (G)S gR gS 265 309 12 13 rC 265 318 :M f0_12 sf (B)S gR gS 319 269 12 13 rC 319 278 :M f0_12 sf (X)S gR gS 319 311 13 13 rC 319 320 :M f0_12 sf (Y)S gR gS 135 269 281 55 rC np 314 274 :M 303 278 :L 303 270 :L 314 274 :L 314 274 :L eofill 278 275 -1 1 304 274 1 278 274 @a np 315 313 :M 304 317 :L 304 309 :L 315 313 :L 315 313 :L eofill 279 314 -1 1 305 313 1 279 313 @a np 316 312 :M 305 307 :L 311 301 :L 316 312 :L 316 312 :L eofill 279 276 -1 1 309 304 1 279 275 @a np 315 276 :M 310 287 :L 304 281 :L 315 276 :L 315 276 :L eofill -1 -1 279 314 1 1 307 284 @b -1 -1 324 306 1 1 323 284 @b 136 286 89 30 rC 142 295 :M f0_12 sf (O)S 150 295 :M (t)S 153 295 :M (h)S 159 295 :M (e)S 164 295 :M (r)S 168 295 :M ( )S 171 295 :M (P)S 178 295 :M (A)S 186 295 :M (G)S 195 295 :M (s)S 200 295 :M ( )S 203 295 :M (f)S 207 295 :M (o)S 213 295 :M (r)S 155 307 :M f2_12 sf (E)S 163 307 :M (q)S 170 307 :M (u)S 177 307 :M (i)S 181 307 :M (v)S 188 307 :M f0_12 sf <28>S 192 307 :M f4_12 sf (G)S 201 307 :M f0_12 sf <29>S gR gS 351 269 12 13 rC 351 278 :M f0_12 sf (A)S gR gS 352 309 11 13 rC 352 318 :M f0_12 sf (B)S gR gS 405 269 11 13 rC 405 278 :M f0_12 sf (X)S gR gS 406 311 10 13 rC 406 320 :M f0_12 sf (Y)S gR gS 135 269 281 55 rC 364 275 -1 1 399 274 1 364 274 @a 366 315 -1 1 401 314 1 366 314 @a 366 276 -1 1 401 306 1 366 275 @a -1 -1 365 315 1 1 400 281 @b -1 -1 411 306 1 1 410 285 @b 401.5 274.5 2 @e 410.5 284.5 2 @e 410.5 307.5 2 @e 402.5 307.5 2 @e 402.5 280.5 2 @e 403.5 314.5 2 @e 323.5 283.5 2 @e 323.5 308.5 2 @e gR gS 0 0 552 730 rC 251 339 :M f2_12 sf (Figure )S 288 339 :M (10)S 60 369 :M f0_12 sf .751 .075(The CCD algorithm we describe does not always give the most informative PAG for a)J 60 387 :M .921 .092(given graph )J f4_12 sf (G)S 132 387 :M f0_12 sf 1.16 .116( in that there may be features common to all graphs in the d)J 440 387 :M (-separation)S 60 405 :M .569 .057(equivalence class which are not captured by the PAG that the algorithm outputs. In this)J 60 423 :M .294 .029(sense the algorithm is not complete. However, the algorithm is 'd)J 377 423 :M .224 .022(-separation complete' in)J 60 441 :M .686 .069(the sense that if the d)J 167 441 :M .571 .057(-separation oracles for two different graphs cause the algorithm to)J 60 459 :M (produce the same PAG as output then the two graphs are d)S 342 459 :M (-separation equivalent.)S 60 483 :M .399 .04(Two definitions are required to state the algorithm. Two vertices, X and Y in a PAG are)J 60 501 :M f2_12 sf (p)S 67 501 :M 1.541 .154(-adjacent )J 120 501 :M f0_12 sf 1.952 .195(if there is an edge between them, X*\321*Y in the PAG.)J 412 498 :M f0_9 sf (1)S 416 498 :M (3)S 420 501 :M f2_12 sf 2.928 .293( )J 426 501 :M f0_12 sf 2.169 .217(For PAG )J 479 501 :M f3_12 sf (Y)S 489 501 :M f0_12 sf (,)S 60 519 :M f2_12 sf (Adjacencies)S f0_12 sf <28>S 125 519 :M f3_12 sf (Y)S 135 519 :M f0_12 sf .039 .004(,X\) is a function giving the set of variables Y s.t. there is an edge X*\321*Y)J 60 537 :M .555 .056(in )J f3_12 sf (Y)S 83 537 :M f0_12 sf .412 .041(. )J f3_12 sf (Y)S 100 537 :M f0_12 sf .8 .08( is a dynamic object in the algorithm that changes as the algorithm progresses,)J 60 555 :M .518 .052(and hence )J f2_12 sf .169(Adjacencies)A f0_12 sf <28>S 179 555 :M f3_12 sf (Y)S 189 555 :M f0_12 sf 1.111 .111(,X\) also changes as the algorithm progresses. A trace of the)J 60 573 :M (algorithm on a simple example is given in section )S 302 573 :M (4.6.)S 60 625 :M ( )S 60 622.48 -.48 .48 204.48 622 .48 60 622 @a 60 636 :M f0_9 sf (1)S 64 636 :M (1)S 68 639 :M f0_10 sf .266 .027(This is not an exhaustive list. For example, the presence of the dotted line connecting the arrowheads on)J 60 650 :M 1.104 .11(the A \321>X, and B\321>X edges, tells us that in no graph in Equiv\()J 344 650 :M f4_10 sf .507(G)A f0_10 sf .992 .099(\) are both of these edges present.)J 60 661 :M (Likewise with the dotted line connecting the arrowheads of the B\321>Y, and A\321>Y edges.)S 60 671 :M f0_9 sf (1)S 64 671 :M (2)S 68 674 :M f0_10 sf (See footnote )S 121 674 :M (10.)S 60 684 :M f0_9 sf (1)S 64 684 :M (3)S 68 687 :M f0_10 sf (Here as elsewhere '*' as a meta-symbol indicating any of the three ends -, )S f0_8 sf (o, >.)S endp %%Page: 18 18 %%BeginPageSetup initializepage (peter; page: 18 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (18)S gR gS 0 0 552 730 rC 60 50 :M f2_12 sf (4)S 66 50 :M (.)S 69 50 :M (4)S 75 50 :M (.)S 78 50 :M ( )S 96 50 :M (The Cyclic Causal Discovery \(CCD\) Algorithm)S 60 73 :M f0_12 sf (The overall strategy for discovery is shown in Figure 11.)S .5 lw 60 108 434 130 rC 102 49 111.5 161 @f 164 162.75 -.75 .75 246.75 162 .75 164 162 @a np 245 158 :M 245 165 :L 252 162 :L 245 158 :L .75 lw eofill -.75 -.75 245.75 165.75 .75 .75 245 158 @b -.75 -.75 245.75 165.75 .75 .75 252 162 @b 245 158.75 -.75 .75 252.75 162 .75 245 158 @a 150 146 142 42 rC gS .941 .942 scale 159.334 163.479 :M ( )S gR gS .941 .942 scale 162.52 163.479 :M ( )S gR gS .941 .942 scale 165.707 163.479 :M ( )S gR gS .941 .942 scale 168.894 163.479 :M ( )S gR gS .941 .942 scale 172.08 163.479 :M ( )S gR gS .941 .942 scale 174.205 163.479 :M ( )S gR gS .941 .942 scale 177.391 163.479 :M ( )S gR gS .941 .942 scale 180.578 163.479 :M ( )S gR gS .941 .942 scale 183.765 163.479 :M (D)S gR gS .941 .942 scale 192.262 163.479 :M (i)S gR gS .941 .942 scale 194.387 163.479 :M (s)S gR gS .941 .942 scale 199.698 163.479 :M (c)S gR gS .941 .942 scale 205.009 163.479 :M (o)S gR gS .941 .942 scale 210.32 163.479 :M (v)S gR gS .941 .942 scale 216.694 163.479 :M (e)S gR gS .941 .942 scale 222.005 163.479 :M (r)S gR gS .941 .942 scale 226.254 163.479 :M (y)S gR gS .941 .942 scale 159.334 175.156 :M ( )S gR gS .941 .942 scale 162.52 175.156 :M ( )S gR gS .941 .942 scale 165.707 175.156 :M ( )S gR gS .941 .942 scale 168.894 175.156 :M ( )S gR gS .941 .942 scale 172.08 175.156 :M ( )S gR gS .941 .942 scale 174.205 175.156 :M ( )S gR gS .941 .942 scale 177.391 175.156 :M ( )S gR gS .941 .942 scale 180.578 175.156 :M ( )S gR gS .941 .942 scale 183.765 175.156 :M ( )S gR gS .941 .942 scale 186.951 175.156 :M ( )S gR gS .941 .942 scale 190.138 175.156 :M ( )S gR gS .941 .942 scale 192.262 175.156 :M ( )S gR gS .941 .942 scale 195.449 175.156 :M ( )S gR gS .941 .942 scale 198.636 175.156 :M ( )S gR gS .941 .942 scale 201.822 175.156 :M ( )S gR gS .941 .942 scale 205.009 175.156 :M ( )S gR gS .941 .942 scale 208.196 175.156 :M ( )S gR gS .941 .942 scale 210.32 175.156 :M ( )S gR gS .941 .942 scale 213.507 175.156 :M ( )S gR gS .941 .942 scale 216.694 175.156 :M ( )S gR gS .941 .942 scale 219.88 175.156 :M ( )S gR gS .941 .942 scale 223.067 175.156 :M ( )S gR gS .941 .942 scale 226.254 175.156 :M ( )S gR gS .941 .942 scale 228.378 175.156 :M ( )S gR gS .941 .942 scale 231.565 175.156 :M ( )S gR gS .941 .942 scale 234.751 175.156 :M ( )S gR gS .941 .942 scale 237.938 175.156 :M ( )S gR gS .941 .942 scale 241.125 175.156 :M ( )S gR gS .941 .942 scale 243.249 175.156 :M ( )S gR gS .941 .942 scale 246.436 175.156 :M ( )S gR gS .941 .942 scale 249.623 175.156 :M ( )S gR gS .941 .942 scale 252.809 175.156 :M ( )S gR gS .941 .942 scale 255.996 175.156 :M ( )S gR gS .941 .942 scale 259.183 175.156 :M ( )S gR gS .941 .942 scale 261.307 175.156 :M ( )S gR gS .941 .942 scale 264.494 175.156 :M ( )S gR gS .941 .942 scale 267.68 175.156 :M ( )S gR gS .941 .942 scale 270.867 175.156 :M (P)S gR gS .941 .942 scale 277.24 175.156 :M (A)S gR gS .941 .942 scale 285.738 175.156 :M (G)S gR gS .941 .942 scale 159.334 187.895 :M ( )S gR gS .941 .942 scale 162.52 187.895 :M ( )S gR gS .941 .942 scale 165.707 187.895 :M ( )S gR gS .941 .942 scale 168.894 187.895 :M ( )S gR gS .941 .942 scale 172.08 187.895 :M ( )S gR gS .941 .942 scale 174.205 187.895 :M ( )S gR gS .941 .942 scale 177.391 187.895 :M ( )S gR gS .941 .942 scale 180.578 187.895 :M ( )S gR gS .941 .942 scale 183.765 187.895 :M (A)S gR gS .941 .942 scale 192.262 187.895 :M (l)S gR gS .941 .942 scale 194.387 187.895 :M (g)S gR gS .941 .942 scale 200.76 187.895 :M (o)S gR gS .941 .942 scale 207.134 187.895 :M (r)S gR gS .941 .942 scale 210.32 187.895 :M (i)S gR gS .941 .942 scale 213.507 187.895 :M (t)S gR gS .941 .942 scale 216.694 187.895 :M (h)S gR gS .941 .942 scale 223.067 187.895 :M (m)S gR gR gS 60 108 434 130 rC 286 165.75 -.75 .75 365.75 199 .75 286 165 @a np 364 196 :M 361 201 :L 370 201 :L 364 196 :L .75 lw eofill -.75 -.75 361.75 201.75 .75 .75 364 196 @b 361 201.75 -.75 .75 370.75 201 .75 361 201 @a 364 196.75 -.75 .75 370.75 201 .75 364 196 @a 287 165.75 -.75 .75 368.75 217 .75 287 165 @a np 368 214 :M 364 219 :L 373 220 :L 368 214 :L eofill -.75 -.75 364.75 219.75 .75 .75 368 214 @b 364 219.75 -.75 .75 373.75 220 .75 364 219 @a 368 214.75 -.75 .75 373.75 220 .75 368 214 @a 292 145 96 41 rC gS .941 .942 scale 335.663 162.418 :M f0_12 sf (r)S gR gS .941 .942 scale 339.911 162.418 :M f0_12 sf (e)S gR gS .941 .942 scale 345.223 162.418 :M f0_12 sf (p)S gR gS .941 .942 scale 350.534 162.418 :M f0_12 sf (r)S gR gS .941 .942 scale 354.783 162.418 :M f0_12 sf (e)S gR gS .941 .942 scale 360.094 162.418 :M f0_12 sf (s)S gR gS .941 .942 scale 364.343 162.418 :M f0_12 sf (e)S gR gS .941 .942 scale 369.654 162.418 :M f0_12 sf (n)S gR gS .941 .942 scale 376.027 162.418 :M f0_12 sf (t)S gR gS .941 .942 scale 379.214 162.418 :M f0_12 sf (s)S gR gS .941 .942 scale 342.036 175.156 :M f0_12 sf (f)S gR gS .941 .942 scale 345.223 175.156 :M f0_12 sf (e)S gR gS .941 .942 scale 350.534 175.156 :M f0_12 sf (a)S gR gS .941 .942 scale 355.845 175.156 :M f0_12 sf (t)S gR gS .941 .942 scale 359.032 175.156 :M f0_12 sf (u)S gR gS .941 .942 scale 364.343 175.156 :M f0_12 sf (r)S gR gS .941 .942 scale 368.592 175.156 :M f0_12 sf (e)S gR gS .941 .942 scale 373.903 175.156 :M f0_12 sf (s)S gR gS .941 .942 scale 333.538 186.833 :M f0_12 sf (c)S gR gS .941 .942 scale 338.849 186.833 :M f0_12 sf (o)S gR gS .941 .942 scale 345.223 186.833 :M f0_12 sf (m)S gR gS .941 .942 scale 353.72 186.833 :M f0_12 sf (m)S gR gS .941 .942 scale 362.218 186.833 :M f0_12 sf (o)S gR gS .941 .942 scale 368.592 186.833 :M f0_12 sf (n)S gR gS .941 .942 scale 374.965 186.833 :M f0_12 sf ( )S gR gS .941 .942 scale 378.152 186.833 :M f0_12 sf (t)S gR gS .941 .942 scale 380.276 186.833 :M f0_12 sf (o)S gR gR .75 lw gS 60 108 434 130 rC 45 128 390 172.5 @f 389 111 16 118 rC gS .941 .942 scale 413.205 126.325 :M f0_12 sf (G)S gR gS .941 .942 scale 421.703 129.509 :M f0_7 sf (1)S gR gS .941 .942 scale 413.205 139.063 :M f0_12 sf ( )S gR gS .941 .942 scale 415.329 139.063 :M f0_12 sf (.)S gR gS .941 .942 scale 413.205 150.74 :M f0_12 sf ( )S gR gS .941 .942 scale 415.329 150.74 :M f0_12 sf (.)S gR gS .941 .942 scale 413.205 162.418 :M f0_12 sf ( )S gR gS .941 .942 scale 415.329 162.418 :M f0_12 sf (.)S gR gS .941 .942 scale 413.205 175.156 :M f0_12 sf ( )S gR gS .941 .942 scale 415.329 175.156 :M f0_12 sf (.)S gR gS .941 .942 scale 413.205 186.833 :M f0_12 sf ( )S gR gS .941 .942 scale 415.329 186.833 :M f0_12 sf (.)S gR gS .941 .942 scale 413.205 198.51 :M f0_12 sf ( )S gR gS .941 .942 scale 415.329 198.51 :M f0_12 sf (.)S gR gS .941 .942 scale 413.205 210.187 :M f0_12 sf ( )S gR gS .941 .942 scale 415.329 210.187 :M f0_12 sf (.)S gR gS .941 .942 scale 413.205 222.926 :M f0_12 sf (G)S gR gS .941 .942 scale 421.703 226.111 :M f0_7 sf (n)S gR gS .941 .942 scale 413.205 234.603 :M f0_12 sf ( )S gR gS .941 .942 scale 415.329 234.603 :M f0_12 sf ( )S gR gS .941 .942 scale 418.516 234.603 :M f0_12 sf (.)S gR gS .941 .942 scale 413.205 246.28 :M f0_12 sf ( )S gR gS .941 .942 scale 415.329 246.28 :M f0_12 sf ( )S gR gS .941 .942 scale 418.516 246.28 :M f0_12 sf (.)S gR gR gS 410 145 83 36 rC gS .941 .942 scale 449.321 162.418 :M f0_12 sf (d)S gR gS .941 .942 scale 455.694 162.418 :M f0_12 sf (-)S gR gS .941 .942 scale 459.943 162.418 :M f0_12 sf (s)S gR gS .941 .942 scale 465.254 162.418 :M f0_12 sf (e)S gR gS .941 .942 scale 469.503 162.418 :M f0_12 sf (p)S gR gS .941 .942 scale 475.876 162.418 :M f0_12 sf (a)S gR gS .941 .942 scale 481.187 162.418 :M f0_12 sf (r)S gR gS .941 .942 scale 484.374 162.418 :M f0_12 sf (a)S gR gS .941 .942 scale 489.685 162.418 :M f0_12 sf (t)S gR gS .941 .942 scale 492.872 162.418 :M f0_12 sf (i)S gR gS .941 .942 scale 496.058 162.418 :M f0_12 sf (o)S gR gS .941 .942 scale 501.369 162.418 :M f0_12 sf (n)S gR gS .941 .942 scale 451.445 175.156 :M f0_12 sf (e)S gR gS .941 .942 scale 456.756 175.156 :M f0_12 sf (q)S gR gS .941 .942 scale 463.129 175.156 :M f0_12 sf (u)S gR gS .941 .942 scale 468.441 175.156 :M f0_12 sf (i)S gR gS .941 .942 scale 471.627 175.156 :M f0_12 sf (v)S gR gS .941 .942 scale 478.001 175.156 :M f0_12 sf (a)S gR gS .941 .942 scale 482.249 175.156 :M f0_12 sf (l)S gR gS .941 .942 scale 485.436 175.156 :M f0_12 sf (e)S gR gS .941 .942 scale 490.747 175.156 :M f0_12 sf (n)S gR gS .941 .942 scale 497.121 175.156 :M f0_12 sf (c)S gR gS .941 .942 scale 501.369 175.156 :M f0_12 sf (e)S gR gS .941 .942 scale 467.378 186.833 :M f0_12 sf (c)S gR gS .941 .942 scale 472.689 186.833 :M f0_12 sf (l)S gR gS .941 .942 scale 475.876 186.833 :M f0_12 sf (a)S gR gS .941 .942 scale 481.187 186.833 :M f0_12 sf (s)S gR gS .941 .942 scale 485.436 186.833 :M f0_12 sf (s)S gR gR gS 69 151 84 28 rC gS .941 .942 scale 89.227 169.848 :M f0_12 sf (d)S gR gS .941 .942 scale 94.538 169.848 :M f0_12 sf (-)S gR gS .941 .942 scale 98.787 169.848 :M f0_12 sf (s)S gR gS .941 .942 scale 104.098 169.848 :M f0_12 sf (e)S gR gS .941 .942 scale 108.347 169.848 :M f0_12 sf (p)S gR gS .941 .942 scale 114.72 169.848 :M f0_12 sf (a)S gR gS .941 .942 scale 120.031 169.848 :M f0_12 sf (r)S gR gS .941 .942 scale 123.218 169.848 :M f0_12 sf (a)S gR gS .941 .942 scale 128.529 169.848 :M f0_12 sf (t)S gR gS .941 .942 scale 131.716 169.848 :M f0_12 sf (i)S gR gS .941 .942 scale 134.902 169.848 :M f0_12 sf (o)S gR gS .941 .942 scale 140.214 169.848 :M f0_12 sf (n)S gR gS .941 .942 scale 104.098 181.525 :M f0_12 sf (o)S gR gS .941 .942 scale 109.409 181.525 :M f0_12 sf (r)S gR gS .941 .942 scale 113.658 181.525 :M f0_12 sf (a)S gR gS .941 .942 scale 118.969 181.525 :M f0_12 sf (c)S gR gS .941 .942 scale 123.218 181.525 :M f0_12 sf (l)S gR gS .941 .942 scale 126.405 181.525 :M f0_12 sf (e)S gR gR gS 60 108 434 130 rC -.75 -.75 280.75 162.75 .75 .75 371 120 @b np 367 117 :M 370 124 :L 375 117 :L 367 117 :L eofill 367 117.75 -.75 .75 370.75 124 .75 367 117 @a -.75 -.75 370.75 124.75 .75 .75 375 117 @b 367 117.75 -.75 .75 375.75 117 .75 367 117 @a gR gS 0 0 552 730 rC 251 269 :M f2_12 sf (Figure )S 288 269 :M (11)S 235 295 :M (CCD Algorithm)S 60 319 :M .094 .009(Input: )J 96 319 :M f0_12 sf .096 .01(An oracle for answering questions of the form: "Is X d-separated from Y given set)J 60 335 :M f2_12 sf (Z)S f0_12 sf (, \(X,Y)S f1_12 sf S 107 335 :M f2_12 sf (Z)S f0_12 sf (\) in graph )S 165 335 :M f4_12 sf (G)S 174 335 :M f0_12 sf (?")S 60 359 :M f2_12 sf (Output:)S f0_12 sf ( A PAG for )S 160 359 :M f4_12 sf (G)S 169 359 :M f0_12 sf (.)S 60 379 :M f2_12 sf .213A f0_12 sf .504 .05( a\) Form the complete undirected PAG )J 271 379 :M f3_12 sf (Y)S 281 379 :M f0_12 sf .58 .058(, i.e. for each pair of variables A and B, )J 482 379 :M f3_12 sf (Y)S 60 395 :M f0_12 sf (contains the edge A )S 158 395 :M f0_10 sf (o)S f0_12 sf S f0_10 sf (o)S f0_12 sf ( B.)S 96 407 :M (b\) n = 0.)S 96 419 :M (repeat)S 114 431 :M (repeat)S 150 443 :M 2.816 .282(select an ordered pair of variables X and Y that are)J 150 459 :M 2.496 .25(p-adjacent )J 207 459 :M 2.166 .217(in )J f3_12 sf (Y)S 234 459 :M f0_12 sf 3.328 .333( such that the number of vertices in)J 150 476 :M f2_12 sf (Adjacencies)S f0_12 sf <28>S 215 476 :M f3_12 sf (Y)S 225 476 :M f0_12 sf (,X\)\\{Y} is greater than or equal to n;)S 150 489 :M (repeat)S 168 505 :M 2.524 .252(select a subset )J 252 505 :M f2_12 sf (S)S 259 505 :M f0_12 sf .621 .062( of )J f2_12 sf .519(Adjacencies)A f0_12 sf <28>S 347 505 :M f3_12 sf (Y)S 357 505 :M f0_12 sf 2.295 .229(,X\)\\{Y} with n)J 168 518 :M (vertices;)S 168 530 :M .699 .07(if X and Y are d)J 251 530 :M .455 .045(-separated given )J f2_12 sf (S)S 340 530 :M f0_12 sf .673 .067( delete edge X )J 416 530 :M f0_10 sf (o)S f0_12 sf S f0_10 sf (o)S 168 546 :M f0_12 sf .419 .042(Y from )J f3_12 sf (Y)S 217 546 :M f0_12 sf .559 .056( and set )J 259 546 :M f2_12 sf (Sepset)S 292 546 :M f0_12 sf .524 .052(\(X,Y\) = )J 335 546 :M f2_12 sf (S)S 342 546 :M f0_12 sf .236 .024( and )J f2_12 sf .165(Sepset)A 399 546 :M f0_12 sf .442 .044(\(Y,X\) =)J 168 559 :M f2_12 sf (S)S 175 559 :M f0_12 sf (;)S 150 575 :M 2.545 .255(until every subset )J 251 575 :M f2_12 sf (S)S 258 575 :M f0_12 sf .668 .067( of )J f2_12 sf .558(Adjacencies)A f0_12 sf <28>S 347 575 :M f3_12 sf (Y)S 357 575 :M f0_12 sf 2.468 .247(,X\)\\{Y} with n)J 150 588 :M .229 .023(vertices has been selected or some subset )J 354 588 :M f2_12 sf (S)S 361 588 :M f0_12 sf .248 .025( has been found)J 150 600 :M (for which X and Y are d)S 267 600 :M (-separated given )S 348 600 :M f2_12 sf (S)S 355 600 :M f0_12 sf (;)S 114 612 :M 2.593 .259(until all ordered pairs of p)J 259 612 :M 2.522 .252(-adjacent vertices X and Y such that)J 114 628 :M f2_12 sf (Adjacencies)S f0_12 sf <28>S 179 628 :M f3_12 sf (Y)S 189 628 :M f0_12 sf 1.447 .145(,X\)\\{Y} has greater than or equal to n vertices have)J 114 641 :M (been selected;)S 114 653 :M (n = n + 1;)S 96 669 :M 1.229 .123(until for each ordered pair of p-adjacent vertices X, Y, )J 378 669 :M f2_12 sf (Adjacencies)S f0_12 sf <28>S 443 669 :M f3_12 sf (Y)S 453 669 :M f0_12 sf (,X\)\\{Y})S 96 685 :M (has less than n vertices.)S endp %%Page: 19 19 %%BeginPageSetup initializepage (peter; page: 19 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (19)S gR gS 0 0 552 730 rC 78 50 :M f2_12 sf .074A f0_12 sf .197 .02( For each triple of vertices A,B,C such that the pair A,B and the pair B,C are each)J 78 67 :M (p-adjacent in )S f3_12 sf (Y)S 152 67 :M f0_12 sf ( but the pair A, C are not p-adjacent in )S f3_12 sf (Y)S 348 67 :M f0_12 sf (, orient A)S 394 67 :M f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (B)S f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (C as)S 78 85 :M <41D1>S 99 85 :M f1_12 sf (>)S 106 85 :M f0_12 sf (B)S f1_12 sf (<)S 121 85 :M f0_12 sf (\321C if and only if B is not in )S 262 85 :M f2_12 sf (Sepset)S 295 85 :M f0_12 sf (; orient A)S f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (B)S f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (C as)S 78 103 :M (A)S 87 103 :M f1_12 sf (*)S f0_12 sf S 105 0 6 730 rC 105 103 :M f1_12 sf 12 f7_1 :p 6 :m ( )S 108 103 :M 6 :m ( )S gR gS 0 0 552 730 rC 105 103 :M f1_12 sf 12 f7_1 :p 6 :m (*)S 105 0 6 730 rC 105 103 :M 6 :m ( )S 108 103 :M 6 :m ( )S gR gS 111 0 8 730 rC 111 103 :M f0_12 sf 12 f6_1 :p 6 :m ( )S 116 103 :M 6 :m ( )S gR gS 0 0 552 730 rC 111 103 :M f0_12 sf 12 f6_1 :p 8.004 :m (B)S 111 0 8 730 rC 111 103 :M 6 :m ( )S 116 103 :M 6 :m ( )S gR gS 119 0 6 730 rC 119 103 :M f1_12 sf 12 f7_1 :p 6 :m ( )S 122 103 :M 6 :m ( )S gR gS 0 0 552 730 rC 119 103 :M f1_12 sf 12 f7_1 :p 6 :m (*)S 119 0 6 730 rC 119 103 :M 6 :m ( )S 122 103 :M 6 :m ( )S gR gS 0 0 552 730 rC 125 103 :M f0_12 sf S f1_12 sf (*)S f0_12 sf (C if and only if B is in )S 254 103 :M f2_12 sf (Sepset)S 287 103 :M f0_12 sf (.)S 60 133 :M f2_12 sf (\246C. )S f0_12 sf (For each triple of vertices in )S f3_12 sf (Y)S 278 133 :M f14_13 sf ( )S 281 133 :M f0_12 sf (such that)S 96 151 :M (\(a\) A is not p-adjacent to X or Y in )S 266 151 :M f3_12 sf (Y)S 276 151 :M f1_12 sf (,)S 96 169 :M f0_12 sf (\(b\) X and Y are p-adjacent in )S f3_12 sf (Y)S 248 169 :M f1_12 sf (,)S 96 187 :M f0_12 sf (\(c\) X )S 124 187 :M f1_12 sf S 133 187 :M f0_12 sf ( )S f2_12 sf (Sepset)S 169 187 :M f0_12 sf ()S 78 204 :M (\(i\) If )S 103 194 180 19 rC 283 213 :M psb currentpoint pse 103 194 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5760 div 608 3 -1 roll exch div scale currentpoint translate 64 56 translate -16 264 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Bold f1 (Sepset) show 1140 264 moveto 384 /Symbol f1 (<) show 1463 264 moveto 384 /Times-Roman f1 (A) show 1742 264 moveto (,) show 1875 264 moveto (Y) show 2263 264 moveto 384 /Symbol f1 (>) show 2591 264 moveto 384 /Times-Roman f1 ( ) show 2737 264 moveto 384 /Symbol f1 (\314) show 2770 506 moveto (\271) show 3065 264 moveto 384 /Times-Roman f1 ( ) show 3161 264 moveto 384 /Times-Bold f1 (Sepset) show 4317 264 moveto 384 /Symbol f1 (<) show 4640 264 moveto 384 /Times-Roman f1 (A) show 4919 264 moveto 384 /Times-Roman f1 (,) show 5056 264 moveto 384 /Times-Roman f1 (X) show 5446 264 moveto 384 /Symbol f1 (>) show end pse gR gS 0 0 552 730 rC 283 204 :M f0_12 sf ( then orient X )S f0_10 sf (o)S f0_12 sf S f1_12 sf (*)S f0_12 sf (Y as X<\321Y)S 78 227 :M (\(ii\) Else if )S 130 227 :M f2_12 sf (Sepset)S 163 227 :M f0_12 sf ( is not a subset of )S 284 227 :M f2_12 sf (Sepset)S 317 227 :M f0_12 sf (, then orient X )S f0_10 sf (o)S f0_12 sf S f1_12 sf (*)S f0_12 sf (Y as X<\321)S 78 246 :M (Y if A and X are d-connected given )S f2_12 sf (Sepset)S 284 246 :M f0_12 sf ()S 60 278 :M f2_12 sf .766 .077(\246D. )J f0_12 sf .736 .074(For each vertex V in )J 190 278 :M f3_12 sf (Y)S 200 278 :M f0_12 sf .639 .064( form the following set: X)J f1_12 sf S 339 278 :M f2_12 sf (Local)S 368 278 :M f0_12 sf <28>S 372 278 :M f3_12 sf (Y)S 382 278 :M f0_12 sf .759 .076(,V\) if and only if X is)J 78 294 :M 1.929 .193(p-adjacent to V in )J f3_12 sf (Y)S 188 294 :M f0_12 sf 2.198 .22(, or there is some vertex Y such that X\321)J 414 294 :M f1_12 sf (>)S 421 294 :M f0_12 sf (Y)S 430 294 :M f1_12 sf (<)S 437 294 :M f0_12 sf 1.915 .192(\321V in )J f3_12 sf (Y)S 489 294 :M f0_12 sf (.)S 78 307 :M f0_9 sf <28>S 81 310 :M f0_12 sf (Local\()S f3_12 sf (Y)S 122 310 :M f0_12 sf 2.44 .244(,V\) is calculated once for each vertex V and does not change as the)J 78 326 :M (algorithm progresses.\))S 78 342 :M (m = 1.)S 96 358 :M (repeat)S 132 374 :M (repeat)S 163 390 :M .018 .002(select an ordered triple such that A\321)J f1_12 sf (>)S 396 390 :M f0_12 sf (B)S f1_12 sf (<)S 411 390 :M f0_12 sf .023 .002(\321C but A and C)J 163 406 :M 2.359 .236(are not p)J 212 406 :M 1.943 .194(-adjacent, and )J 288 406 :M f2_12 sf (Local)S 317 406 :M f0_12 sf <28>S 321 406 :M f3_12 sf (Y)S 331 406 :M f0_12 sf (,A\)\\\()S 354 406 :M f2_12 sf (Sepset)S 387 406 :M f0_12 sf .196(\312)A f1_12 sf .299A f0_12 sf .916 .092(\312{B,C}\) has)J 163 422 :M (greater than or equal to m vertices.)S 164 438 :M (repeat)S 186 454 :M 1.128 .113(select a set )J f2_12 sf .883 .088(T )J 258 454 :M f1_12 sf S 267 454 :M f0_12 sf .125 .013( )J f2_12 sf .329(Local)A 300 454 :M f0_12 sf <28>S 304 454 :M f3_12 sf (Y)S 314 454 :M f0_12 sf (,A\)\\\()S 337 454 :M f2_12 sf (Sepset)S 370 454 :M f0_12 sf .147(\312)A f1_12 sf .224A f0_12 sf .547 .055(\312{B,C}\) with m)J 186 470 :M 4.854 .485(vertices, and test if A and C are d-separated given)J 186 486 :M f2_12 sf (T)S f0_12 sf S f1_12 sf S f0_12 sf S f2_12 sf (Sepset)S 242 486 :M f0_12 sf .078(\312)A f1_12 sf .12A f0_12 sf .241 .024(\312{B} then orient the triple A\321)J 437 486 :M f1_12 sf (>)S 444 486 :M f0_12 sf (B)S f1_12 sf (<)S 459 486 :M f0_12 sf .357 .036(\321C as)J 186 502 :M <41D1>S 207 0 7 730 rC 195 502 :M f1_12 sf 14.875 1.488( )J gR gS 0 0 552 730 rC 207 502 :M f1_12 sf (>)S 207 0 7 730 rC 195 502 :M 14.875 1.488( )J gR gS 214 0 8 730 rC 203 502 :M f0_12 sf 14.875 1.488( )J gR gS 0 0 552 730 rC 214 502 :M f0_12 sf (B)S 214 0 8 730 rC 203 502 :M 14.875 1.488( )J gR gS 222 0 7 730 rC 210 502 :M f1_12 sf 14.875 1.488( )J gR gS 0 0 552 730 rC 222 502 :M f1_12 sf (<)S 222 0 7 730 rC 210 502 :M 14.875 1.488( )J gR gS 0 0 552 730 rC 229 502 :M f0_12 sf 10.649 1.065(\321C, and record )J 356 502 :M f2_12 sf (T)S f0_12 sf S f1_12 sf S f0_12 sf S f2_12 sf (Sepset)S 412 502 :M f0_12 sf (\312)S f1_12 sf S f0_12 sf (\312{B}\312in)S 1 G 0 0 1 1 rF 491 502 :M psb /wp$x1 207 def /wp$x2 228 def /wp$y 505 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 186 518 :M 0 G f2_12 sf (SupSepset)S 239 518 :M f0_12 sf (.)S 164 534 :M .014 .001(until every subset )J 252 534 :M f2_12 sf (T )S f1_12 sf S 272 534 :M f0_12 sf ( )S f2_12 sf (Local)S 304 534 :M f0_12 sf <28>S 308 534 :M f3_12 sf (Y)S 318 534 :M f0_12 sf (,A\)\\\()S 341 534 :M f2_12 sf (Sepset)S 374 534 :M f0_12 sf ( )S f1_12 sf S f0_12 sf (\312{B,C}\) with m)S 164 550 :M 1.339 .134(vertices has been selected or a d-separating set for A and C has)J 164 566 :M (been recorded in )S 247 566 :M f2_12 sf (SupSepset)S 300 566 :M f0_12 sf (.)S 132 582 :M .033 .003(until all triples such that A\321)J 273 582 :M f1_12 sf (>)S 280 582 :M f0_12 sf (B)S f1_12 sf (<)S 295 582 :M f0_12 sf .035 .004(\321C, \(i.e. not A\321)J 382 0 7 730 rC 370 582 :M f1_12 sf .05 .005( )J gR gS 0 0 552 730 rC 382 582 :M f1_12 sf (>)S 382 0 7 730 rC 370 582 :M .05 .005( )J gR gS 389 0 8 730 rC 378 582 :M f0_12 sf .05 .005( )J gR gS 0 0 552 730 rC 389 582 :M f0_12 sf (B)S 389 0 8 730 rC 378 582 :M .05 .005( )J gR gS 397 0 7 730 rC 385 582 :M f1_12 sf .05 .005( )J gR gS 0 0 552 730 rC 397 582 :M f1_12 sf (<)S 397 0 7 730 rC 385 582 :M .05 .005( )J gR gS 0 0 552 730 rC 404 582 :M f0_12 sf .038 .004(\321C\), A and C are)J 1 G 0 0 1 1 rF 491 582 :M psb /wp$x1 382 def /wp$x2 403 def /wp$y 585 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 132 598 :M 0 G 3.496 .35(not p)J 160 598 :M 1.987 .199(-adjacent, and )J f2_12 sf .709(Local)A 268 598 :M f0_12 sf <28>S 272 598 :M f3_12 sf (Y)S 282 598 :M f0_12 sf (,A\)\\\()S 305 598 :M f2_12 sf (Sepset)S 338 598 :M f0_12 sf .408(\312)A f1_12 sf .623A f0_12 sf 1.723 .172(\312{B,C}\) have greater)J 132 614 :M (than or equal to m vertices have been selected.)S 132 630 :M (m = m)S f4_12 sf ( )S f0_12 sf (+1.)S 96 646 :M 1.142 .114(until each ordered triple such that A\321)J f1_12 sf (>)S 348 646 :M f0_12 sf (B)S f1_12 sf (<)S 363 646 :M f0_12 sf 1.442 .144(\321C but A and C are not)J 96 662 :M 2.251 .225(p-adjacent, is such that )J 223 662 :M f2_12 sf (Local)S 252 662 :M f0_12 sf <28>S 256 662 :M f3_12 sf (Y)S 266 662 :M f0_12 sf (,A\)\\\()S 289 662 :M f2_12 sf .428(Sepset<)A f0_12 sf .446(A,C>\312)A f1_12 sf .699A f0_12 sf 1.39 .139(\312{B}\) has fewer than m)J 96 678 :M (vertices.)S endp %%Page: 20 20 %%BeginPageSetup initializepage (peter; page: 20 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (20)S gR gS 0 0 552 730 rC 60 50 :M f2_12 sf (\246E. )S f0_12 sf (If there is a quadruple of distinct vertices such that)S 73 67 :M (\(i\) A\321)S 108 0 7 730 rC 96 67 :M f1_12 sf ( )S gR gS 0 0 552 730 rC 108 67 :M f1_12 sf (>)S 108 0 7 730 rC 96 67 :M ( )S gR gS 115 0 8 730 rC 115 67 :M f0_12 sf ( )S 120 67 :M ( )S gR gS 0 0 552 730 rC 115 67 :M f0_12 sf (B)S 115 0 8 730 rC 115 67 :M ( )S 120 67 :M ( )S gR gS 123 0 7 730 rC 111 67 :M f1_12 sf ( )S gR gS 0 0 552 730 rC 123 67 :M f1_12 sf (<)S 123 0 7 730 rC 111 67 :M ( )S gR gS 0 0 552 730 rC 130 67 :M f0_12 sf (\321C in )S f3_12 sf (Y)S 1 G 0 0 1 1 rF 174 67 :M psb /wp$x1 108 def /wp$x2 129 def /wp$y 70 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 73 85 :M 0 G f0_12 sf (\(ii\) A\321)S f1_12 sf (>)S 118 85 :M f0_12 sf (D)S 127 85 :M f1_12 sf (<)S 134 85 :M f0_12 sf (\321C or A\321)S 191 0 7 730 rC 179 85 :M f1_12 sf ( )S gR gS 0 0 552 730 rC 191 85 :M f1_12 sf (>)S 191 0 7 730 rC 179 85 :M ( )S gR gS 198 0 9 730 rC 198 85 :M f0_12 sf ( )S 204 85 :M ( )S gR gS 0 0 552 730 rC 198 85 :M f0_12 sf (D)S 198 0 9 730 rC 198 85 :M ( )S 204 85 :M ( )S gR gS 207 0 7 730 rC 195 85 :M f1_12 sf ( )S gR gS 0 0 552 730 rC 207 85 :M f1_12 sf (<)S 207 0 7 730 rC 195 85 :M ( )S gR gS 0 0 552 730 rC 214 85 :M f0_12 sf (\321C in )S f3_12 sf (Y)S 1 G 0 0 1 1 rF 258 85 :M psb /wp$x1 191 def /wp$x2 213 def /wp$y 88 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 73 103 :M 0 G f0_12 sf (\(iii\) B and D are p)S 161 103 :M (-adjacent in )S f3_12 sf (Y)S 60 121 :M f0_12 sf (then orient B)S 123 121 :M f1_12 sf (*)S f0_12 sf S f0_10 sf (o)S f0_12 sf (D as B\321)S 185 121 :M f1_12 sf (>)S 192 121 :M f0_12 sf (D in )S 216 121 :M f3_12 sf (Y)S 226 121 :M f14_13 sf ( )S 229 121 :M f0_12 sf (if D is not in )S 293 121 :M f2_12 sf (SupSepset)S 346 121 :M f0_12 sf ()S 60 139 :M (else orient B)S 121 139 :M f1_12 sf (*)S f0_12 sf S f0_10 sf (o)S f0_12 sf (D as B)S 171 139 :M f1_12 sf (*)S f0_12 sf (\321D in )S 213 139 :M f3_12 sf (Y)S 223 139 :M f0_12 sf ( if D is in )S f2_12 sf (SupSepset)S 324 139 :M f0_12 sf ()S 60 170 :M f2_12 sf (\246F. )S 80 170 :M f0_12 sf (For each quadruple of distinct vertices such that)S 73 187 :M (\(i\) A\321)S 108 0 7 730 rC 96 187 :M f1_12 sf ( )S gR gS 0 0 552 730 rC 108 187 :M f1_12 sf (>)S 108 0 7 730 rC 96 187 :M ( )S gR gS 115 0 8 730 rC 115 187 :M f0_12 sf ( )S 120 187 :M ( )S gR gS 0 0 552 730 rC 115 187 :M f0_12 sf (B)S 115 0 8 730 rC 115 187 :M ( )S 120 187 :M ( )S gR gS 123 0 7 730 rC 111 187 :M f1_12 sf ( )S gR gS 0 0 552 730 rC 123 187 :M f1_12 sf (<)S 123 0 7 730 rC 111 187 :M ( )S gR gS 0 0 552 730 rC 130 187 :M f0_12 sf (\321C in )S f3_12 sf (Y)S 1 G 0 0 1 1 rF 174 187 :M psb /wp$x1 108 def /wp$x2 129 def /wp$y 190 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 73 205 :M 0 G f0_12 sf (\(ii\) D is not p)S 138 205 :M (-adjacent to both A and C in )S 276 205 :M f3_12 sf (Y)S 60 223 :M f0_12 sf .667 .067(if A and D are d-connected given )J 230 223 :M f2_12 sf (SupSepset)S 283 223 :M f0_12 sf .116(\312)A f1_12 sf .181A f0_12 sf .376 .038(\312{D}, then orient B)J 434 223 :M f1_12 sf .209(*)A f0_12 sf .419A f0_10 sf .175(o)A f0_12 sf .456 .046(D as B)J 60 241 :M (\321>D in )S 103 241 :M f3_12 sf (Y)S 60 282 :M f2_12 sf (4)S 66 282 :M (.)S 69 282 :M (5)S 75 282 :M (.)S 78 282 :M ( )S 96 282 :M (Soundness and Completeness)S 60 309 :M .102 .01(Theorem )J 110 309 :M (7:)S 120 309 :M f0_12 sf .105 .01( \(Soundness\) Given as input an oracle for d-separation relations in the \(cyclic)J 60 327 :M (or acyclic\) directed graph )S 186 327 :M f4_12 sf (G)S 195 327 :M f0_12 sf (, then the output is a PAG )S f3_12 sf (Y)S 332 327 :M f14_13 sf ( )S 335 327 :M f0_12 sf (for )S 352 327 :M f4_12 sf (G)S 361 327 :M f0_12 sf (.)S 60 351 :M 1.981 .198(Theorem 7 is proved by showing that each section of the algorithm makes correct)J 60 369 :M (inferences from the answers of the d)S 235 369 :M (-separation oracle applied to )S f4_12 sf (G)S 382 369 :M f0_12 sf (.)S 60 393 :M .314 .031(In practice, an approximation to a d-separation oracle can be implemented as a statistical)J 60 411 :M 1.563 .156(test that the corresponding partial correlation vanishes. As the sample size increases)J 60 429 :M .097 .01(without limit, if the significance level of the statistical test is systematically lowered, then)J 60 447 :M .662 .066(the probabilities of both Type I and Type II error for the test approach zero, so that the)J 60 465 :M .453 .045(statistical test is correct with probability one. Of course, this does not guarantee that the)J 60 483 :M .237 .024(CCD algorithm as implemented is reliable on realistic sample sizes. The reliability of the)J 60 501 :M (algorithm depends upon the following factors:)S 78 519 :M (1)S 84 519 :M (.)S 87 519 :M 3 .3( )J 96 519 :M 1.802 .18(Whether the Causal Independence Assumption holds \(i.e. there are no latent)J 78 531 :M (variables\).)S 78 549 :M (2)S 84 549 :M (.)S 87 549 :M ( )S 96 549 :M (Whether the Causal Faithfulness Assumption holds.)S 78 567 :M (3)S 84 567 :M (.)S 87 567 :M ( )S 96 567 :M (Whether the distributional assumptions made by the statistical tests hold.)S 78 585 :M (4)S 84 585 :M (.)S 87 585 :M ( )S 96 585 :M (The power of the statistical tests against alternatives.)S 78 603 :M (5)S 84 603 :M (.)S 87 603 :M ( )S 96 603 :M (The significance level used in the statistical tests.)S 60 627 :M .887 .089(In the future, we will test the sensitivity of the algorithm to these factors on simulated)J 60 645 :M (data.)S endp %%Page: 21 21 %%BeginPageSetup initializepage (peter; page: 21 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (21)S gR gS 0 0 552 730 rC 60 56 :M f2_12 sf 1.278 .128(Theorem )J 112 56 :M .945 .094(8: )J f0_12 sf .945(\(d)A 137 56 :M 1.253 .125(-separation Completeness\) If the CCD algorithm, when given as input)J 60 74 :M .089 .009(d-separation oracles for the graphs )J 229 74 :M f4_12 sf (G)S 238 76 :M f0_9 sf (1)S 243 74 :M f0_12 sf .052 .005(, )J f4_12 sf (G)S 258 76 :M f0_9 sf (2)S 263 74 :M f0_12 sf .102 .01( produces as output PAGs )J 392 74 :M f3_12 sf (Y)S 402 76 :M f3_9 sf (1)S 407 74 :M f0_12 sf .049 .005(, )J f3_12 sf (Y)S 423 76 :M f3_9 sf (2)S 428 74 :M f0_12 sf .066 .007( respectively,)J 60 92 :M 1.56 .156(then )J 86 92 :M f3_12 sf (Y)S 96 94 :M f3_9 sf (1)S 101 92 :M f0_12 sf 1.56 .156( is identical to )J 181 92 :M f3_12 sf (Y)S 191 94 :M f3_9 sf (2)S 196 92 :M f0_12 sf 1.469 .147( if and only if )J f4_12 sf (G)S 284 94 :M f0_9 sf (1)S 289 92 :M f0_12 sf 1.274 .127( and )J f4_12 sf (G)S 325 94 :M f0_9 sf (2)S 330 92 :M f0_12 sf 1.747 .175( are d)J 362 92 :M .929 .093(-separation equivalent, i.e.)J 60 110 :M f4_12 sf (G)S 69 112 :M f0_9 sf (2)S 74 110 :M f0_12 sf S f1_12 sf S 86 110 :M f2_12 sf (Equiv)S 117 110 :M f0_12 sf <28>S 121 110 :M f4_12 sf (G)S 130 112 :M f0_9 sf (1)S 135 110 :M f0_12 sf (\) and vice versa.)S 60 134 :M 1.477 .148(The proof is based on the characterization of d)J 302 134 :M 1.057 .106(-separation equivalence in Richardson)J 60 152 :M 2.599 .26(\(1994b\). \(It follows directly from Theorem 7 that if )J 346 152 :M f4_12 sf (G)S 355 154 :M f0_9 sf (1)S 360 152 :M f0_12 sf 2.392 .239( and )J f4_12 sf (G)S 400 154 :M f0_9 sf (2)S 405 152 :M f0_12 sf 3.28 .328( are d)J 440 152 :M (-separation)S 60 170 :M (equivalent then )S 137 170 :M f3_12 sf (Y)S 147 172 :M f3_9 sf (1 )S 154 170 :M f0_12 sf (is identical to )S 222 170 :M f3_12 sf (Y)S 232 172 :M f3_9 sf (2)S 237 170 :M f0_12 sf (.\))S 60 210 :M f2_12 sf (4)S 66 210 :M (.)S 69 210 :M (6)S 75 210 :M (.)S 78 210 :M ( )S 96 210 :M (Trace of CCD Algorithm)S 60 237 :M f0_12 sf 1.381 .138(The following illustrates the operation of the algorithm given as input a d-separation)J 60 255 :M (oracle for the following graph:)S 261 259 13 13 rC 261 268 :M (A)S gR gS 262 299 12 13 rC 262 308 :M f0_12 sf (B)S gR gS 313 259 13 13 rC 313 268 :M f0_12 sf (X)S gR gS 314 299 13 13 rC 314 308 :M f0_12 sf (Y)S gR gS 224 258 103 54 rC np 311 264 :M 300 268 :L 300 268 :L 300 267 :L 300 267 :L 300 267 :L 300 267 :L 300 267 :L 300 267 :L 300 266 :L 299 266 :L 299 266 :L 299 266 :L 299 265 :L 299 265 :L 299 265 :L 299 265 :L 299 265 :L 299 264 :L 299 264 :L 299 264 :L 299 264 :L 299 264 :L 299 263 :L 299 263 :L 299 263 :L 299 263 :L 299 263 :L 299 262 :L 299 262 :L 299 262 :L 299 262 :L 299 262 :L 300 261 :L 300 261 :L 300 261 :L 300 261 :L 300 261 :L 300 260 :L 300 260 :L 300 260 :L 311 264 :L 311 264 :L eofill 275 265 -1 1 300 264 1 275 264 @a np 311 303 :M 300 307 :L 300 307 :L 300 307 :L 300 307 :L 300 306 :L 300 306 :L 300 306 :L 300 306 :L 300 306 :L 299 305 :L 299 305 :L 299 305 :L 299 305 :L 299 305 :L 299 304 :L 299 304 :L 299 304 :L 299 304 :L 299 304 :L 299 303 :L 299 303 :L 299 303 :L 299 303 :L 299 303 :L 299 302 :L 299 302 :L 299 302 :L 299 302 :L 299 302 :L 299 301 :L 299 301 :L 299 301 :L 300 301 :L 300 301 :L 300 300 :L 300 300 :L 300 300 :L 300 300 :L 300 300 :L 300 299 :L 311 303 :L 311 303 :L eofill 275 304 -1 1 300 303 1 275 303 @a 1 G 10 27 318 285.5 @j 0 G 11 28 318 285.5 @f np 318 270 :M 325 279 :L 325 279 :L 325 279 :L 325 279 :L 324 280 :L 324 280 :L 324 280 :L 324 280 :L 324 280 :L 324 280 :L 323 280 :L 323 280 :L 323 280 :L 323 280 :L 323 281 :L 323 281 :L 322 281 :L 322 281 :L 322 281 :L 322 281 :L 322 281 :L 321 281 :L 321 281 :L 321 281 :L 321 281 :L 321 281 :L 320 281 :L 320 281 :L 320 281 :L 320 281 :L 320 281 :L 319 281 :L 319 281 :L 319 282 :L 319 282 :L 319 282 :L 318 282 :L 318 282 :L 318 282 :L 318 282 :L 318 270 :L 318 270 :L eofill -1 -1 323 283 1 1 322 281 @b np 316 298 :M 310 288 :L 310 288 :L 310 288 :L 310 288 :L 310 288 :L 311 288 :L 311 288 :L 311 288 :L 311 288 :L 311 287 :L 311 287 :L 312 287 :L 312 287 :L 312 287 :L 312 287 :L 312 287 :L 313 287 :L 313 287 :L 313 287 :L 313 287 :L 313 287 :L 314 287 :L 314 287 :L 314 287 :L 314 286 :L 314 286 :L 315 286 :L 315 286 :L 315 286 :L 315 286 :L 315 286 :L 316 286 :L 316 286 :L 316 286 :L 316 286 :L 316 286 :L 317 286 :L 317 286 :L 317 286 :L 317 286 :L 316 298 :L 316 298 :L eofill -1 -1 314 288 1 1 313 286 @b 225 277 12 16 rC 225 286 :M f4_12 sf (G)S gR gS 0 0 552 730 rC 251 327 :M f2_12 sf (Figure )S 288 327 :M (12)S 60 349 :M (Initial PAG )S 123 349 :M f3_12 sf (Y)S 133 349 :M f0_12 sf (:)S 250 352 13 13 rC 250 361 :M (A)S gR gS 250 390 12 12 rC 250 399 :M f0_12 sf (B)S gR gS 290 352 12 13 rC 290 361 :M f0_12 sf (X)S gR gS 290 390 12 12 rC 290 399 :M f0_12 sf (Y)S gR 1 G gS 250 352 52 50 rC 265 365 2 @i 0 G 265 365 2.5 @e 1 G 288 389 2 @i 0 G 288 389 2.5 @e 267 368 -1 1 287 387 1 267 367 @a 1 G 287 358 2 @i 0 G 287 358 2.5 @e 1 G 263 358 2 @i 0 G 263 358 2.5 @e 264 358 -1 1 285 357 1 264 357 @a 1 G 288 396 2 @i 0 G 288 396 2.5 @e 1 G 264 396 2 @i 0 G 264 396 2.5 @e 265 397 -1 1 286 396 1 265 396 @a 1 G 266 389 2 @i 0 G 266 389 2.5 @e 1 G 290 365 2 @i 0 G 290 365 2.5 @e -1 -1 269 387 1 1 286 367 @b 1 G 295 388 2 @i 0 G 295 388 2.5 @e 1 G 295 365 2 @i 0 G 295 365 2.5 @e -1 -1 295 387 1 1 294 366 @b 1 G 260 392 2 @i 0 G 260 392 2.5 @e 1 G 260 363 2 @i 0 G 260 363 2.5 @e -1 -1 260 388 1 1 259 367 @b gR gS 0 0 552 730 rC 251 417 :M 0 G f2_12 sf (Figure )S 288 417 :M (13)S 60 439 :M (Section \246A:)S 60 463 :M f0_12 sf 1.555 .156(Since A and B are d-separated given the empty set, the algorithm removes the edge)J 60 481 :M .196 .02(between A and B and records )J 207 481 :M f2_12 sf (Sepset)S 240 481 :M f0_12 sf (\312=\312)S 286 481 :M f2_12 sf (Sepset)S 319 481 :M f0_12 sf .167 .017( = )J f1_12 sf S 375 481 :M f0_12 sf .2 .02(. This is the only pair of)J 60 499 :M .274 .027(vertices that are d-separated given a subset of the other variables. Hence after \246A )J 457 499 :M f3_12 sf (Y)S 467 499 :M f0_12 sf .35 .035( is as)J 60 517 :M (follows:)S 250 528 13 13 rC 250 537 :M (A)S gR gS 250 566 12 12 rC 250 575 :M 0 G f0_12 sf (B)S gR gS 290 528 12 13 rC 290 537 :M 0 G f0_12 sf (X)S gR gS 290 566 12 12 rC 290 575 :M 0 G f0_12 sf (Y)S gR gS 250 528 52 50 rC 262 540 2 @i 0 G 262 540 2.5 @e 1 G 288 565 2 @i 0 G 288 565 2.5 @e 263 541 -1 1 287 564 1 263 540 @a 1 G 287 534 2 @i 0 G 287 534 2.5 @e 1 G 263 534 2 @i 0 G 263 534 2.5 @e 264 534 -1 1 285 533 1 264 533 @a 1 G 288 572 2 @i 0 G 288 572 2.5 @e 1 G 264 572 2 @i 0 G 264 572 2.5 @e 265 573 -1 1 286 572 1 265 572 @a 1 G 264 566 2 @i 0 G 264 566 2.5 @e 1 G 290 541 2 @i 0 G 290 541 2.5 @e -1 -1 266 566 1 1 288 541 @b 1 G 295 564 2 @i 0 G 295 564 2.5 @e 1 G 295 541 2 @i 0 G 295 541 2.5 @e -1 -1 295 563 1 1 294 542 @b gR gS 0 0 552 730 rC 251 593 :M 0 G f2_12 sf (Figure )S 288 593 :M (14)S 60 615 :M (Section \246B)S 60 639 :M f0_12 sf .358 .036(Since X)J f1_12 sf S 108 639 :M f2_12 sf (Sepset)S 141 639 :M f0_12 sf .425 .043( and Y)J f1_12 sf S 216 639 :M f2_12 sf (Sepset)S 249 639 :M f0_12 sf .439 .044(, A)J 298 639 :M f0_10 sf (o)S f0_12 sf S f0_10 sf (o)S f0_12 sf <586FD0>S 335 639 :M f0_10 sf .166(o)A f0_12 sf .509 .051(B and A)J 382 639 :M f0_10 sf (o)S f0_12 sf S f0_10 sf (o)S f0_12 sf (Y)S 407 639 :M f0_10 sf .089(o)A f0_12 sf .107A f0_10 sf .089(o)A f0_12 sf .366 .037(B are oriented)J 60 657 :M .765 .077(respectively as A\320>X<\320B and A\320>Y<\320B. The state of )J 335 657 :M f3_12 sf (Y)S 345 657 :M f0_12 sf .914 .091( at the end of \246B is shown in)J 60 675 :M (Figure 15.)S endp %%Page: 22 22 %%BeginPageSetup initializepage (peter; page: 22 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (22)S gR gS 250 41 12 13 rC 250 50 :M f0_12 sf (A)S gR gS 250 79 12 12 rC 250 88 :M f0_12 sf (B)S gR gS 289 41 13 13 rC 289 50 :M f0_12 sf (X)S gR gS 289 79 13 12 rC 289 88 :M f0_12 sf (Y)S gR gS 250 41 52 50 rC np 290 83 :M 279 76 :L 283 72 :L 290 83 :L 290 83 :L eofill 259 52 -1 1 283 75 1 259 51 @a np 289 45 :M 277 48 :L 277 42 :L 289 45 :L 289 45 :L eofill 260 46 -1 1 278 45 1 260 45 @a np 291 84 :M 279 86 :L 279 81 :L 291 84 :L 291 84 :L eofill 261 85 -1 1 280 84 1 261 84 @a np 288 50 :M 281 60 :L 277 56 :L 288 50 :L 288 50 :L eofill -1 -1 260 78 1 1 279 58 @b 1 G 292 76 2 @i 0 G 292 76 2.5 @e 1 G 292 53 2 @i 0 G 292 53 2.5 @e -1 -1 293 75 1 1 292 54 @b gR gS 0 0 552 730 rC 251 106 :M f2_12 sf (Figure )S 288 106 :M (15)S 60 136 :M (Section \246C)S f0_12 sf ( No orientations are performed in this case.)S 60 160 :M f2_12 sf (Section \246D)S 60 184 :M f0_12 sf .222 .022(Since A and B are d)J 158 184 :M .139 .014(-separated given {X,Y}, the algorithm records )J f2_12 sf .043(SupSepset)A 437 184 :M f0_12 sf .168 .017( =)J 60 202 :M f2_12 sf (SupSepset)S 113 202 :M f0_12 sf .597 .06( = {X,Y}, and it orients A\320>X<\320B as A\320)J 363 0 7 730 rC 363 202 :M .926 .093( )J 366 202 :M .882 .088( )J gR gS 0 0 552 730 rC 363 202 :M f0_12 sf (>X<)S 378 0 7 730 rC 378 202 :M .926 .093( )J 381 202 :M .882 .088( )J gR gS 0 0 552 730 rC 385 202 :M f0_12 sf .591 .059(\320B, and A\320>Y<\320B as)J 1 G 0 0 1 1 rF 491 202 :M psb /wp$x1 363 def /wp$x2 384 def /wp$y 204 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 220 :M 0 G <41D0>S 75 0 7 730 rC 75 220 :M ( )S 78 220 :M ( )S gR gS 0 0 552 730 rC 75 220 :M f0_12 sf (>Y<)S 90 0 7 730 rC 90 220 :M ( )S 93 220 :M ( )S gR gS 0 0 552 730 rC 97 220 :M f0_12 sf (\320B. The state of PAG )S 204 220 :M f3_12 sf (Y)S 214 220 :M f0_12 sf ( after \246D is shown in )S 317 220 :M (Figure 16.)S 1 G 0 0 1 1 rF 366 220 :M psb /wp$x1 75 def /wp$x2 96 def /wp$y 222 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 244 282 :M 0 G ( )S 1 G 256 231 52 51 rC -90 0 16 24 287 243 @l 0 G np 288 231 :M 289 231 :L 289 232 :L 287 232 :L 288 231 :L eofill np 289 231 :M 290 232 :L 290 233 :L 289 232 :L 289 231 :L eofill np 289 231 :M 289 231 :L 289 232 :L 289 232 :L 289 231 :L eofill np 290 232 :M 291 232 :L 290 233 :L 290 233 :L 290 232 :L eofill np 290 232 :M 290 232 :L 290 233 :L 290 233 :L 290 232 :L eofill np 292 233 :M 293 234 :L 292 235 :L 291 234 :L 292 233 :L eofill np 293 234 :M 293 236 :L 293 236 :L 291 235 :L 293 234 :L eofill np 293 234 :M 293 234 :L 292 235 :L 291 235 :L 293 234 :L eofill np 293 236 :M 293 236 :L 292 237 :L 292 236 :L 293 236 :L eofill np 293 236 :M 293 236 :L 292 236 :L 292 236 :L 293 236 :L eofill np 294 238 :M 294 239 :L 293 239 :L 293 238 :L 294 238 :L eofill np 294 239 :M 294 241 :L 293 241 :L 293 239 :L 294 239 :L eofill np 294 239 :M 294 239 :L 293 239 :L 293 239 :L 294 239 :L eofill 1 G 0 90 16 22 289 270 @l 0 G np 297 271 :M 296 273 :L 295 273 :L 296 271 :L 297 271 :L eofill np 296 273 :M 296 274 :L 295 273 :L 296 272 :L 296 273 :L eofill np 296 273 :M 296 273 :L 295 272 :L 296 272 :L 296 273 :L eofill np 296 276 :M 295 276 :L 295 276 :L 295 275 :L 296 276 :L eofill np 295 276 :M 295 278 :L 294 277 :L 295 276 :L 295 276 :L eofill np 295 276 :M 295 276 :L 295 276 :L 295 276 :L 295 276 :L eofill np 295 278 :M 294 278 :L 293 278 :L 294 277 :L 295 278 :L eofill np 295 278 :M 295 278 :L 294 277 :L 294 277 :L 295 278 :L eofill np 293 280 :M 293 280 :L 292 279 :L 292 279 :L 293 280 :L eofill np 293 280 :M 291 281 :L 291 280 :L 292 279 :L 293 280 :L eofill np 293 280 :M 293 280 :L 292 279 :L 292 279 :L 293 280 :L eofill np 291 281 :M 290 281 :L 290 280 :L 291 280 :L 291 281 :L eofill np 291 281 :M 291 281 :L 291 280 :L 291 280 :L 291 281 :L eofill 256 231 13 13 rC 256 240 :M (A)S gR gS 256 268 12 13 rC 256 277 :M f0_12 sf (B)S gR gS 296 231 12 13 rC 296 240 :M f0_12 sf (X)S gR gS 296 268 12 13 rC 296 277 :M f0_12 sf (Y)S gR gS 256 231 52 51 rC np 294 273 :M 284 267 :L 288 263 :L 294 273 :L 294 273 :L eofill 263 243 -1 1 287 265 1 263 242 @a np 293 236 :M 281 239 :L 281 233 :L 293 236 :L 293 236 :L eofill 264 237 -1 1 282 236 1 264 236 @a np 295 274 :M 283 277 :L 283 271 :L 295 274 :L 295 274 :L eofill 265 275 -1 1 284 274 1 265 274 @a np 293 237 :M 287 247 :L 282 244 :L 293 237 :L 293 237 :L eofill -1 -1 264 268 1 1 285 246 @b 1 G 300 267 2 @i 0 G 300 267 2.5 @e 1 G 300 244 2 @i 0 G 300 244 2.5 @e -1 -1 301 266 1 1 300 245 @b gR gS 0 0 552 730 rC 251 297 :M f2_12 sf (Figure )S 288 297 :M (16)S 60 327 :M (Section \246E)S 60 351 :M f0_12 sf .861 .086(The quadruple is such that \(i\) A\320)J 287 0 7 730 rC 287 351 :M 1.378 .138( )J 290 351 :M 1.312 .131( )J gR gS 0 0 552 730 rC 287 351 :M f0_12 sf (>X<)S 302 0 7 730 rC 302 351 :M 1.378 .138( )J 305 351 :M 1.312 .131( )J gR gS 0 0 552 730 rC 309 351 :M f0_12 sf .919 .092(\320B, \(ii\) A\320)J 364 0 7 730 rC 364 351 :M 1.378 .138( )J 367 351 :M 1.312 .131( )J gR gS 0 0 552 730 rC 364 351 :M f0_12 sf (>Y<)S 379 0 7 730 rC 379 351 :M 1.378 .138( )J 382 351 :M 1.312 .131( )J gR 1 G gS 0 0 552 730 rC 0 0 1 1 rF 390 351 :M psb /wp$x1 287 def /wp$x2 308 def /wp$y 353 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 386 351 :M 0 G f0_12 sf .984 .098(\320B, \(iii\) X and Y are)J 1 G 0 0 1 1 rF 492 351 :M psb /wp$x1 364 def /wp$x2 385 def /wp$y 353 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 369 :M 0 G .681 .068(p-adjacent, thus it satisfies the conditions in section \246E. Since Y)J f1_12 sf S 387 369 :M f0_12 sf .057 .006( )J f2_12 sf .138(SupSepset)A 444 369 :M f0_12 sf (,)S 60 387 :M .12 .012(the edge X)J f0_10 sf (o)S f0_12 sf S f0_10 sf (o)S f0_12 sf .117 .012(Y is oriented as Y\321)J 228 387 :M f0_10 sf (o)S f0_12 sf .102 .01(X. Since X)J f1_12 sf S 295 387 :M f0_12 sf ( )S f2_12 sf .023(SupSepset)A 351 387 :M f0_12 sf .113 .011(, this edge is further)J 60 405 :M (oriented as Y\321X.)S 60 429 :M f2_12 sf (Section )S f0_12 sf (\246F \320 Performs no orientations in this case, hence the PAG that is output is:)S 1 G 221 440 109 51 rC -90 0 16 24 308 452 @l 0 G np 309 440 :M 310 440 :L 310 441 :L 309 441 :L 309 440 :L eofill np 310 440 :M 312 441 :L 311 442 :L 310 441 :L 310 440 :L eofill np 310 440 :M 310 440 :L 310 441 :L 310 441 :L 310 440 :L eofill np 312 441 :M 312 441 :L 311 442 :L 311 442 :L 312 441 :L eofill np 312 441 :M 312 441 :L 311 442 :L 311 442 :L 312 441 :L eofill np 313 442 :M 314 443 :L 313 444 :L 313 443 :L 313 442 :L eofill np 314 443 :M 315 445 :L 314 445 :L 313 444 :L 314 443 :L eofill np 314 443 :M 314 443 :L 313 444 :L 313 444 :L 314 443 :L eofill np 315 445 :M 315 445 :L 314 446 :L 314 445 :L 315 445 :L eofill np 315 445 :M 315 445 :L 314 445 :L 314 445 :L 315 445 :L eofill np 316 447 :M 316 448 :L 315 448 :L 315 447 :L 316 447 :L eofill np 316 448 :M 316 450 :L 315 450 :L 315 448 :L 316 448 :L eofill np 316 448 :M 316 448 :L 315 448 :L 315 448 :L 316 448 :L eofill 1 G 0 90 14 22 311 479 @l 0 G np 318 480 :M 318 482 :L 317 482 :L 317 480 :L 318 480 :L eofill np 318 482 :M 318 483 :L 317 482 :L 317 481 :L 318 482 :L eofill np 318 482 :M 318 482 :L 317 481 :L 317 481 :L 318 482 :L eofill np 317 485 :M 317 485 :L 316 485 :L 316 484 :L 317 485 :L eofill np 317 485 :M 316 487 :L 315 486 :L 316 485 :L 317 485 :L eofill np 317 485 :M 317 485 :L 316 485 :L 316 485 :L 317 485 :L eofill np 316 487 :M 316 487 :L 315 487 :L 315 486 :L 316 487 :L eofill np 316 487 :M 316 487 :L 315 486 :L 315 486 :L 316 487 :L eofill np 314 489 :M 314 489 :L 313 488 :L 314 488 :L 314 489 :L eofill np 314 489 :M 313 490 :L 312 489 :L 313 488 :L 314 489 :L eofill np 314 489 :M 314 489 :L 313 488 :L 313 488 :L 314 489 :L eofill np 312 490 :M 311 490 :L 311 489 :L 312 489 :L 312 490 :L eofill np 312 490 :M 313 490 :L 312 489 :L 312 489 :L 312 490 :L eofill 277 440 13 13 rC 277 449 :M (A)S 277 465 :M f3_12 sf (Y)S gR gS 277 477 12 13 rC 277 486 :M 0 G f0_12 sf (B)S gR gS 317 440 13 13 rC 317 449 :M 0 G f0_12 sf (X)S gR gS 317 477 13 13 rC 317 486 :M 0 G f0_12 sf (Y)S gR 0 G gS 221 440 109 51 rC np 315 482 :M 305 476 :L 309 472 :L 315 482 :L 315 482 :L eofill 284 452 -1 1 308 474 1 284 451 @a np 314 445 :M 302 448 :L 302 442 :L 314 445 :L 314 445 :L eofill 285 446 -1 1 303 445 1 285 445 @a np 316 483 :M 304 486 :L 304 480 :L 316 483 :L 316 483 :L eofill 286 484 -1 1 305 483 1 286 483 @a np 314 446 :M 308 456 :L 304 453 :L 314 446 :L 314 446 :L eofill -1 -1 285 477 1 1 306 455 @b -1 -1 322 475 1 1 321 451 @b 221 457 40 18 rC 221 470 :M f0_12 sf (P)S 228 470 :M (A)S 236 470 :M (G)S 245 470 :M ( )S 248 470 :M f3_12 sf (Y)S gR gS 0 0 552 730 rC 251 506 :M f2_12 sf (Figure )S 288 506 :M (17)S 60 536 :M (4)S 66 536 :M (.)S 69 536 :M (7)S 75 536 :M (.)S 78 536 :M ( )S 96 536 :M (Complexity of CCD Algorithm)S 60 548 295 18 rC 355 566 :M psb currentpoint pse 60 545 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9440 div 672 3 -1 roll exch div scale currentpoint translate 64 39 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 4219 0 moveto 0 492 rlineto stroke 9223 0 moveto 0 492 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (Let MaxDegree\() -4 345 sh (Max) 3488 345 sh (X) 4427 345 sh (|) 4960 345 sh ( Y) 5138 345 sh (X,) 6104 345 sh ( or X) 6522 345 sh (Y in) 7901 345 sh 224 ns (Y) 3598 572 sh (V) 3918 572 sh 384 /Times-Italic f1 (G) 2641 345 sh 416 ns (G) 8736 345 sh 384 /Times-Roman f1 (\)) 2928 345 sh ( ) 4762 345 sh (,) 9252 345 sh 416 ns ( ) 8632 345 sh 384 /Symbol f1 (=) 3161 345 sh (\254) 5619 345 sh (\254) 7420 345 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1304 /Symbol f3 ({) 4250 369 sh (}) 9026 369 sh 224 /Symbol f1 (\316) 3777 572 sh end MTsave restore pse gR gS 60 566 344 18 rC 404 584 :M psb currentpoint pse 60 563 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11008 div 672 3 -1 roll exch div scale currentpoint translate 64 37 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3626 0 moveto 0 496 rlineto stroke 10874 0 moveto 0 496 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (and MaxAdj\() -14 347 sh (\)) 2335 347 sh (Max) 2895 347 sh (X ) 3834 347 sh (|) 4309 347 sh ( X is ) 4487 347 sh (p) 5371 347 sh (adjacent to Y in any PAG for) 5863 347 sh 224 ns (Y) 3005 574 sh (V) 3325 574 sh 384 /Times-Italic f1 (G) 2048 347 sh (G) 10409 347 sh 384 /Symbol f1 (=) 2568 347 sh (-) 5576 347 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1315 /Symbol f3 ({) 3657 372 sh (}) 10677 372 sh 224 /Symbol f1 (\316) 3184 574 sh end MTsave restore pse gR gS 0 0 552 730 rC 60 599 :M f0_12 sf .57 .057(The number of d-separation tests performed by Step \246A of the CCD algorithm will, in a)J 60 617 :M (worst case, be bounded as follows:)S 120 620 312 32 rC 432 652 :M psb currentpoint pse 120 620 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9984 div 1024 3 -1 roll exch div scale currentpoint translate 64 38 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 6963 471 moveto 2786 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 352 /Times-Roman f1 (Total number of) 668 296 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 352 /MT_Times-Roman f1 (oracle consultations in \266) -10 824 sh (A) 3395 824 sh 384 ns ( ) 3707 570 sh ( ) 4249 570 sh ( ) 6803 570 sh 384 /Symbol f1 (\243) 3920 570 sh (\327) 4615 570 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 2270 /Symbol f3 (\() 4755 681 sh (\)) 5046 681 sh 384 1000 2143 /Symbol f3 (\() 5617 711 sh (\)) 6251 711 sh 384 /Symbol f1 (\243) 6474 570 sh (+) 7385 323 sh (-) 8708 323 sh 288 ns (-) 5919 370 sh 224 ns (=) 5290 943 sh (+) 9469 153 sh 576 ns (\345) 5173 657 sh 384 /MT_Times-Roman f1 (2) 4363 570 sh (1) 7645 323 sh (2) 8998 323 sh 224 ns (2) 4907 781 sh (0) 5437 943 sh (2) 8150 153 sh (1) 9597 153 sh 288 ns (2) 6097 370 sh 288 /Times-Italic f1 (n) 4893 349 sh (n) 5755 370 sh (i) 5954 802 sh 224 ns (i) 5202 943 sh (k) 5324 168 sh (k) 9336 153 sh 384 ns (k) 7116 323 sh (n) 7944 323 sh (n) 8436 323 sh (k) 8228 871 sh 384 /MT_Times-Roman f1 (\() 6977 323 sh (\)) 7814 323 sh (\() 8299 323 sh (\)) 9198 323 sh (!) 8392 871 sh (.) 9787 570 sh end MTsave restore pse endp %%Page: 23 23 %%BeginPageSetup initializepage (peter; page: 23 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (23)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf 5.595 .56(where n\312=\312number of vertices in )J 262 56 :M f4_12 sf (G)S 271 56 :M f0_12 sf 4.12 .412(, k\312=\312MaxAdj\()J 349 56 :M f4_12 sf (G)S 358 56 :M f0_12 sf 5.18 .518(\). Since MaxAdj\()J 460 56 :M f4_12 sf (G)S 469 56 :M f0_12 sf 7.554 .755(\) )J cF f1_12 sf 7.554 .755J sf 60 74 :M (\(MaxDegree\()S 125 74 :M f4_12 sf (G)S 134 74 :M f0_12 sf <2929>S 142 71 :M f0_10 sf (2)S f0_12 sf 0 3 rm -.003(, with MaxDegree\()A 0 -3 rm 238 74 :M f4_12 sf (G)S 247 74 :M f0_12 sf -.01(\) = r this step is )A 324 61 43 17 rC 367 78 :M psb currentpoint pse 324 61 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 544 3 -1 roll exch div scale currentpoint translate 64 38 translate -12 378 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (O) show 266 378 moveto 384 /Times-Roman f1 (\() show 411 378 moveto 384 /Times-Roman f1 (n) show 631 207 moveto 224 ns (r) show 734 108 moveto 160 /Times-Roman f1 (2) show 872 207 moveto 224 /Symbol f1 (+) show 1015 207 moveto 224 /Times-Roman f1 (3) show 1164 378 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 367 74 :M f0_12 sf -.005(. It should be stressed that)A 60 92 :M .374 .037(even as a worst case complexity bound this is a very loose one; the bound presumes that)J 60 110 :M 1.816 .182(there is a graph in which every pair of non-adjacent vertices in the graph are only)J 60 128 :M (d-separated given all vertices adjacent to one of them.)S 60 152 :M (Step \246B performs no additional tests of d-separation.)S 60 176 :M .911 .091(Step \246C performs at most one d)J 220 176 :M .745 .074(-separation test for each triple satisfying the conditions)J 60 194 :M (given. Thus this step is O\(n)S 192 191 :M f0_10 sf (3)S f0_12 sf 0 3 rm (\).)S 0 -3 rm 60 218 :M (In a worst case the number of tests of d-separation that Step \246D performs is bounded by)S 125 229 301 32 rC 426 261 :M psb currentpoint pse 125 229 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9632 div 1024 3 -1 roll exch div scale currentpoint translate 64 39 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 6634 470 moveto 2899 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 352 /Times-Roman f1 (Total number of) 664 295 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 352 /MT_Times-Roman f1 (oracle consultations in \266) -10 823 sh (D) 3393 823 sh 384 ns ( ) 3699 569 sh ( ) 4241 569 sh ( ) 5932 569 sh ( ) 6474 569 sh 384 /Symbol f1 (\243) 3912 569 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 2270 /Symbol f3 (\() 4346 680 sh (\)) 4637 680 sh 384 1000 2143 /Symbol f3 (\() 5208 710 sh (\)) 5826 710 sh 384 /Symbol f1 (\243) 6145 569 sh (+) 7150 322 sh (-) 8460 322 sh 288 ns (-) 5510 369 sh 224 ns (=) 4881 942 sh (+) 9253 152 sh 576 ns (\345) 4764 656 sh 288 /Times-Italic f1 (n) 4484 348 sh (n) 5346 369 sh (i) 5537 801 sh 224 ns (i) 4793 942 sh (m) 4889 167 sh (m) 9067 152 sh 384 ns (m) 6789 322 sh (n) 7709 322 sh (n) 8188 322 sh (m) 7910 870 sh 288 /MT_Times-Roman f1 (3) 4486 780 sh (3) 5684 369 sh 224 ns (0) 5028 942 sh (3) 7912 152 sh (1) 9381 152 sh 384 ns (1) 7410 322 sh (3) 8745 322 sh 384 /MT_Times-Roman f1 (\() 6648 322 sh (\)) 7579 322 sh (\() 8051 322 sh (\)) 8928 322 sh (!) 8166 870 sh end MTsave restore pse gR gS 0 0 552 730 rC 60 282 :M f0_12 sf 1.476 .148(where m\312=\312)J 117 272 25 19 rC 142 291 :M psb currentpoint pse 117 272 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 800 div 608 3 -1 roll exch div scale currentpoint translate -3781 -271 translate 3838 591 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Max) show 3949 815 moveto 224 ns (Y) show 4128 815 moveto 224 /Symbol f1 (\316) show 4269 815 moveto 224 /Times-Bold f1 (V) show end pse gR gS 0 0 552 730 rC 142 282 :M f0_12 sf 1.81 .181(|{X | Local\()J 205 282 :M f3_12 sf (Y)S 215 282 :M f0_12 sf 1.83 .183(,X\)} in \246D. Since m )J cF f1_12 sf .183A sf 1.83 .183( \(MaxDegree\()J 405 282 :M f4_12 sf (G)S 414 282 :M f0_12 sf <2929>S 422 279 :M f0_10 sf .682(2)A f0_12 sf 0 3 rm 1.798 .18(, this step is)J 0 -3 rm 60 287 44 17 rC 104 304 :M psb currentpoint pse 60 287 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1408 div 544 3 -1 roll exch div scale currentpoint translate 64 38 translate -12 378 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (O) show 266 378 moveto 384 /Times-Roman f1 (\() show 411 378 moveto 384 /Times-Roman f1 (n) show 631 207 moveto 224 ns (r) show 734 108 moveto 160 /Times-Roman f1 (2) show 872 207 moveto 224 /Symbol f1 (+) show 1022 207 moveto 224 /Times-Roman f1 (4) show 1180 378 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 104 300 :M f0_12 sf (. Again this is a loose bound.)S 60 324 :M 1.019 .102(Step \246E performs no tests of d)J 214 324 :M .938 .094(-separation, while step \246F performs at most one test for)J 60 342 :M 1.652 .165(each quadruple satisfying the conditions. Hence this step is O\(n)J f0_10 sf 0 -3 rm .417(4)A 0 3 rm f0_12 sf 1.584 .158(\), \(though in many)J 60 360 :M (graphs there may be very few quadruples satisfying all four conditions\).)S 60 387 :M f2_14 sf (5)S 67 387 :M (.)S 70 387 :M ( )S 96 387 :M (d)S 104 387 :M (-separation Equivalence)S 60 420 :M f0_12 sf .504 .05(Since the CCD algorithm is d)J 206 420 :M .421 .042(-separation complete, the orientation rules in the algorithm)J 60 438 :M .433 .043(may be used to construct a d)J 201 438 :M .322 .032(-separation equivalence algorithm. We present an algorithm)J 60 456 :M .132 .013(that, given as input a Directed Cyclic or Acyclic graph )J 327 456 :M f4_12 sf .187 .019(G )J 339 456 :M f0_12 sf .128 .013(will produce as output the same)J 60 474 :M .751 .075(PAG that the CCD algorithm outputs given only a d)J 320 474 :M .664 .066(-separation oracle for )J 429 474 :M f4_12 sf (G)S 438 474 :M f0_12 sf .606 .061(. However,)J 60 492 :M 1.403 .14(this algorithm, unlike the CCD algorithm, runs in time polynomial in the number of)J 60 510 :M .231 .023(vertices, even if MaxDegree\()J 202 510 :M f4_12 sf (G)S 211 510 :M f0_12 sf .295 .029(\) is not kept fixed. Thus this algorithm can be used to test)J 60 528 :M .013 .001(for d-separation equivalence of two graphs in polynomial time. Let Children\()J f2_12 sf (X)S 440 528 :M f0_12 sf .017 .002(\) be the set)J 60 546 :M (of children of members of )S f2_12 sf (X )S 200 546 :M f0_12 sf (in )S f4_12 sf (G)S 221 546 :M f0_12 sf (.)S 60 574 :M f2_12 sf (Cyclic PAG-from-Graph Algorithm)S 60 598 :M (Input: )S 96 598 :M f0_12 sf (Directed Cyclic or Acyclic graph )S f4_12 sf (G)S 60 622 :M f2_12 sf (Output:)S f0_12 sf ( The CCD PAG )S f3_12 sf (Y)S 190 622 :M f0_12 sf ( for )S 210 622 :M f4_12 sf (G)S 219 622 :M f0_12 sf (.)S 60 642 :M f2_12 sf .204A f0_12 sf .616 .062( Form the complete undirected PAG )J f3_12 sf (Y)S 265 642 :M f0_12 sf .797 .08(, whch has an edge )J 365 642 :M f0_10 sf .176(o)A f0_12 sf .211A f0_10 sf .176(o)A f0_12 sf .619 .062( between every pair of)J 60 658 :M (vertices in the vertex set )S 180 658 :M f2_12 sf (V.)S endp %%Page: 24 24 %%BeginPageSetup initializepage (peter; page: 24 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (24)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf (For each ordered pair of vertices form the following sets:)S 60 76 :M f2_12 sf (S)S 67 78 :M f2_9 sf (A,B)S 82 76 :M f2_12 sf ( )S f0_12 sf (= Children\(A\) )S f1_12 sf S f0_12 sf ( An\({A,B}\))S 60 92 :M f2_12 sf (T)S f2_9 sf 0 2 rm (A,B)S 0 -2 rm 83 92 :M f2_12 sf ( = )S 96 92 :M f0_12 sf (\(Parents\()S f2_12 sf (S)S 146 94 :M f2_9 sf (A,B)S 161 92 :M f2_12 sf ( )S f1_12 sf S 182 92 :M f0_12 sf (A}\) )S f1_12 sf S f2_12 sf (S)S 222 94 :M f2_9 sf (A,B)S 237 92 :M f0_12 sf (\)\\\(Descendants\(Children\(A\))S 372 92 :M f1_12 sf S f0_12 sf (Children\(B\)\) )S 446 92 :M f1_12 sf S f0_12 sf ({A,B}\))S 60 110 :M (For each ordered pair :)S 60 126 :M 1.574 .157(If A and B are d)J 149 126 :M 1.137 .114(-separated given )J 234 126 :M f2_12 sf (T)S f2_9 sf 0 2 rm (A,B)S 0 -2 rm 257 126 :M f0_12 sf 1.037 .104( then record )J f2_12 sf .592(T)A f2_9 sf 0 2 rm .545(A,B)A 0 -2 rm 346 126 :M f2_12 sf .233 .023( )J f0_12 sf .809 .081(in )J f2_12 sf .557(Sepset)A 398 126 :M f0_12 sf 1.412 .141( and )J 459 126 :M f2_12 sf (Sepset)S 60 142 :M f0_12 sf ( and remove the edge A)S f0_10 sf (o)S f0_12 sf S f0_10 sf (o)S f0_12 sf (B from )S f3_12 sf (Y)S 270 142 :M f1_12 sf (.)S 60 158 :M f0_12 sf .383 .038(else if A and B are d)J 162 158 :M .283 .028(-separated given )J 244 158 :M f2_12 sf (T)S f2_9 sf 0 2 rm (B,A)S 0 -2 rm 267 158 :M f0_12 sf .364 .036( then record )J 329 158 :M f2_12 sf (T)S f2_9 sf 0 2 rm (B,A)S 0 -2 rm 352 158 :M f2_12 sf .181 .018( )J f0_12 sf .684 .068(in )J 368 158 :M f2_12 sf (Sepset)S 401 158 :M f0_12 sf .352 .035( and )J 459 158 :M f2_12 sf (Sepset)S 60 174 :M f0_12 sf ( and remove the edge A)S f0_10 sf (o)S f0_12 sf S f0_10 sf (o)S f0_12 sf (B from )S f3_12 sf (Y.)S 60 208 :M f2_12 sf .833 .083J f0_12 sf 1.083 .108(For each triple of vertices A,B,C such that the pair A,B and the pair B,C are each)J 60 225 :M (p-adjacent in )S f3_12 sf (Y)S 134 225 :M f0_12 sf ( but the pair A, C are not p-adjacent in )S f3_12 sf (Y)S 330 225 :M f0_12 sf (, orient A)S 376 225 :M f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (B)S f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (C as)S 60 243 :M <41D1>S 81 243 :M f1_12 sf (>)S 88 243 :M f0_12 sf (B)S f1_12 sf (<)S 103 243 :M f0_12 sf (\321C if and only if B is not in )S 244 243 :M f2_12 sf (Sepset)S 277 243 :M f0_12 sf (; orient A)S f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (B)S f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (C as)S 60 261 :M (A)S 69 261 :M f1_12 sf (*)S f0_12 sf S 87 0 6 730 rC 87 261 :M f1_12 sf 12 f7_1 :p 6 :m ( )S 90 261 :M 6 :m ( )S gR gS 0 0 552 730 rC 87 261 :M f1_12 sf 12 f7_1 :p 6 :m (*)S 87 0 6 730 rC 87 261 :M 6 :m ( )S 90 261 :M 6 :m ( )S gR gS 93 0 8 730 rC 93 261 :M f0_12 sf 12 f6_1 :p 6 :m ( )S 98 261 :M 6 :m ( )S gR gS 0 0 552 730 rC 93 261 :M f0_12 sf 12 f6_1 :p 8.004 :m (B)S 93 0 8 730 rC 93 261 :M 6 :m ( )S 98 261 :M 6 :m ( )S gR gS 101 0 6 730 rC 101 261 :M f1_12 sf 12 f7_1 :p 6 :m ( )S 104 261 :M 6 :m ( )S gR gS 0 0 552 730 rC 101 261 :M f1_12 sf 12 f7_1 :p 6 :m (*)S 101 0 6 730 rC 101 261 :M 6 :m ( )S 104 261 :M 6 :m ( )S gR gS 0 0 552 730 rC 107 261 :M f0_12 sf S f1_12 sf (*)S f0_12 sf (C if and only if B is in )S 236 261 :M f2_12 sf (Sepset)S 269 261 :M f0_12 sf (.)S 60 291 :M f2_12 sf S 75 291 :M f0_12 sf (For each triple of vertices in )S f3_12 sf (Y)S 272 291 :M f14_13 sf ( )S 275 291 :M f0_12 sf (such that)S 96 309 :M (\(a\) A is not p-adjacent to X or Y in )S 266 309 :M f3_12 sf (Y)S 96 327 :M f0_12 sf (\(b\) X and Y are p-adjacent in )S f3_12 sf (Y)S 96 345 :M f0_12 sf (\(c\) X )S 124 345 :M f1_12 sf S 133 345 :M f0_12 sf ( )S f2_12 sf (Sepset)S 169 345 :M f0_12 sf ()S 78 363 :M (Orient X )S f0_10 sf (o)S f0_12 sf S f1_12 sf (*)S f0_12 sf (Y as X<\321Y if A and X are d-connected given )S f2_12 sf (Sepset)S 398 363 :M f0_12 sf ()S 60 388 :M f2_12 sf 1.365 .136J f0_12 sf 2.154 .215(For each triple or such that A\321)J 352 388 :M f1_12 sf (>)S 359 388 :M f0_12 sf (B)S f1_12 sf (<)S 374 388 :M f0_12 sf 2.362 .236(\321C, A and C are not)J 60 406 :M (p-adjacent, form the following set:)S 60 430 :M f2_12 sf (Q)S f2_7 sf 0 3 rm (A,B,C)S 0 -3 rm f2_12 sf ( )S f0_12 sf (= Children\(A\) )S f1_12 sf S f0_12 sf ( An\({A,B,C}\))S 60 446 :M f2_12 sf (R)S 69 449 :M f2_7 sf .346(A,B,C)A f2_12 sf 0 -3 rm .529 .053( = )J 0 3 rm f0_12 sf 0 -3 rm .456(\(Parents\()A 0 3 rm f2_12 sf 0 -3 rm .885(Q)A 0 3 rm f2_7 sf .346(A,B,C)A f2_12 sf 0 -3 rm .284 .028( )J 0 3 rm 183 446 :M f1_12 sf 2.666 .267J f0_12 sf 2.949 .295(A}\) )J f1_12 sf 2.141 .214J 244 446 :M f2_12 sf (Q)S f2_7 sf 0 3 rm (A,B,C)S 0 -3 rm f0_12 sf (\)\\\(Descendants\(Children\(A\))S 406 446 :M f1_12 sf .366A f0_12 sf 1.123 .112(Children\(C\)\) )J f1_12 sf S 105 462 :M f0_12 sf ({A,C}\))S 60 478 :M .475 .048(If A and C are d-separated given )J 224 478 :M f2_12 sf (R)S 233 481 :M f2_7 sf .083(A,B,C)A f1_12 sf 0 -3 rm .253 .025<20C8>J 0 3 rm 264 478 :M f0_12 sf .433 .043(\312{B} then orient A\321)J 367 478 :M f1_12 sf (>)S 374 478 :M f0_12 sf (B)S f1_12 sf (<)S 389 478 :M f0_12 sf .513 .051(\321C as A\321)J 447 0 7 730 rC 447 478 :M f1_12 sf .693 .069( )J 450 478 :M .659 .066( )J gR gS 0 0 552 730 rC 447 478 :M f1_12 sf (>)S 447 0 7 730 rC 447 478 :M .693 .069( )J 450 478 :M .659 .066( )J gR gS 454 0 8 730 rC 454 478 :M f0_12 sf .659 .066( )J 458 478 :M .659 .066( )J gR gS 0 0 552 730 rC 454 478 :M f0_12 sf (B)S 454 0 8 730 rC 454 478 :M .659 .066( )J 458 478 :M .659 .066( )J gR gS 462 0 7 730 rC 462 478 :M f1_12 sf .693 .069( )J 465 478 :M .659 .066( )J gR gS 0 0 552 730 rC 462 478 :M f1_12 sf (<)S 462 0 7 730 rC 462 478 :M .693 .069( )J 465 478 :M .659 .066( )J gR gS 0 0 552 730 rC 469 478 :M f0_12 sf S 1 G 0 0 1 1 rF 492 478 :M psb /wp$x1 447 def /wp$x2 468 def /wp$y 481 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 494 :M 0 G (and record )S 114 494 :M f2_12 sf (R)S 123 497 :M f2_7 sf (A,B,C)S f0_12 sf 0 -3 rm S 0 3 rm f1_12 sf 0 -3 rm S 0 3 rm f0_12 sf 0 -3 rm (\312{B}\312in )S 0 3 rm 191 494 :M f2_12 sf (SupSepset)S 244 494 :M f0_12 sf (.)S 60 524 :M f2_12 sf S 75 524 :M f0_12 sf (If there is a quadruple of distinct vertices such that)S 73 541 :M (\(i\) A\321)S 108 0 7 730 rC 108 541 :M f1_12 sf ( )S 111 541 :M ( )S gR gS 0 0 552 730 rC 108 541 :M f1_12 sf (>)S 108 0 7 730 rC 108 541 :M ( )S 111 541 :M ( )S gR gS 115 0 8 730 rC 115 541 :M f0_12 sf ( )S 119 541 :M ( )S gR gS 0 0 552 730 rC 115 541 :M f0_12 sf (B)S 115 0 8 730 rC 115 541 :M ( )S 119 541 :M ( )S gR gS 123 0 7 730 rC 123 541 :M f1_12 sf ( )S 126 541 :M ( )S gR gS 0 0 552 730 rC 123 541 :M f1_12 sf (<)S 123 0 7 730 rC 123 541 :M ( )S 126 541 :M ( )S gR gS 0 0 552 730 rC 130 541 :M f0_12 sf (\321C in )S f3_12 sf (Y)S 175 541 :M f1_12 sf (,)S 1 G 0 0 1 1 rF 178 541 :M psb /wp$x1 108 def /wp$x2 129 def /wp$y 544 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 73 559 :M 0 G f0_12 sf (\(ii\) A\321)S f1_12 sf (>)S 118 559 :M f0_12 sf (D)S 127 559 :M f1_12 sf (<)S 134 559 :M f0_12 sf (\321C or A\321)S 191 0 7 730 rC 191 559 :M f1_12 sf ( )S 194 559 :M ( )S gR gS 0 0 552 730 rC 191 559 :M f1_12 sf (>)S 191 0 7 730 rC 191 559 :M ( )S 194 559 :M ( )S gR gS 198 0 9 730 rC 198 559 :M f0_12 sf ( )S 203 559 :M ( )S gR gS 0 0 552 730 rC 198 559 :M f0_12 sf (D)S 198 0 9 730 rC 198 559 :M ( )S 203 559 :M ( )S gR gS 207 0 7 730 rC 207 559 :M f1_12 sf ( )S 210 559 :M ( )S gR gS 0 0 552 730 rC 207 559 :M f1_12 sf (<)S 207 0 7 730 rC 207 559 :M ( )S 210 559 :M ( )S gR gS 0 0 552 730 rC 214 559 :M f0_12 sf (\321C in )S f3_12 sf (Y)S 259 559 :M f1_12 sf (,)S 1 G 0 0 1 1 rF 262 559 :M psb /wp$x1 191 def /wp$x2 213 def /wp$y 562 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 73 577 :M 0 G f0_12 sf (\(iii\) B and D are p)S 161 577 :M (-adjacent in )S f3_12 sf (Y)S 229 577 :M f1_12 sf (,)S 60 595 :M f0_12 sf (then orient B)S 123 595 :M f1_12 sf (*)S f0_12 sf S f0_10 sf (o)S f0_12 sf (D as B\321)S 185 595 :M f1_12 sf (>)S 192 595 :M f0_12 sf (D in )S 216 595 :M f3_12 sf (Y)S 226 595 :M f14_13 sf ( )S 229 595 :M f0_12 sf (if D is not in )S 293 595 :M f2_12 sf (SupSepset)S 346 595 :M f0_12 sf ()S 60 613 :M (else orient B)S 121 613 :M f1_12 sf (*)S f0_12 sf S f0_10 sf (o)S f0_12 sf (D as B)S 171 613 :M f1_12 sf (*)S f0_12 sf (\321D in )S 213 613 :M f3_12 sf (Y)S 223 613 :M f0_12 sf ( if D is in )S f2_12 sf (SupSepset)S 324 613 :M f0_12 sf (.)S endp %%Page: 25 25 %%BeginPageSetup initializepage (peter; page: 25 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (25)S gR gS 0 0 552 730 rC 60 50 :M f2_12 sf S f0_12 sf (For each quadruple of distinct vertices such that)S 73 67 :M (\(i\) A\321)S 108 0 7 730 rC 108 67 :M f1_12 sf ( )S 112 67 :M ( )S gR gS 0 0 552 730 rC 108 67 :M f1_12 sf (>)S 108 0 7 730 rC 108 67 :M ( )S 112 67 :M ( )S gR gS 115 0 8 730 rC 115 67 :M f0_12 sf ( )S 120 67 :M ( )S gR gS 0 0 552 730 rC 115 67 :M f0_12 sf (B)S 115 0 8 730 rC 115 67 :M ( )S 120 67 :M ( )S gR gS 123 0 7 730 rC 123 67 :M f1_12 sf ( )S 127 67 :M ( )S gR gS 0 0 552 730 rC 123 67 :M f1_12 sf (<)S 123 0 7 730 rC 123 67 :M ( )S 127 67 :M ( )S gR gS 0 0 552 730 rC 130 67 :M f0_12 sf (\321C in )S f3_12 sf (Y)S 1 G 0 0 1 1 rF 174 67 :M psb /wp$x1 108 def /wp$x2 129 def /wp$y 70 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 73 85 :M 0 G f0_12 sf (\(ii\) D is not p)S 138 85 :M (-adjacent to both A and C in )S 276 85 :M f3_12 sf (Y)S 286 85 :M f0_12 sf (, and)S 60 103 :M .667 .067(if A and D are d-connected given )J 230 103 :M f2_12 sf (SupSepset)S 283 103 :M f0_12 sf .116(\312)A f1_12 sf .181A f0_12 sf .376 .038(\312{D}, then orient B)J 434 103 :M f1_12 sf .209(*)A f0_12 sf .419A f0_10 sf .175(o)A f0_12 sf .456 .046(D as B)J 60 121 :M (\321>D in )S 103 121 :M f3_12 sf (Y)S 113 121 :M f1_12 sf (.)S 60 164 :M f0_12 sf 1.003 .1(We do not include the proof that this algorithm is correct, but it is very similar to the)J 60 182 :M 1.47 .147(proof that the CCD algorithm itself is correct. The main difference between the two)J 60 200 :M .587 .059(algorithms lies in the fact the CCD algorithm must search for the Sepset and SupSepset)J 60 218 :M .576 .058(sets, testing many different candidates, whereas the PAG from graph algorithm is faced)J 60 236 :M (with the much simpler task of constructing these sets, given the graph itself.)S 60 260 :M .274 .027(Since, by )J 109 260 :M .258 .026(Theorem 8, given two graphs )J 255 260 :M f4_12 sf (G)S 264 263 :M f0_7 sf (1)S 268 260 :M f0_12 sf .135 .014(, )J f4_12 sf (G)S 283 263 :M f0_7 sf (2)S 287 260 :M f0_12 sf .253 .025( as input, the CCD algorithm will produce)J 60 278 :M 2.171 .217(the same PAG as output if and only if )J 272 278 :M f4_12 sf (G)S 281 281 :M f0_7 sf (1)S 285 278 :M f0_12 sf 1.787 .179( and )J f4_12 sf (G)S 323 281 :M f0_7 sf (2)S 327 278 :M f0_12 sf 2.45 .245( are d)J 360 278 :M 1.332 .133(-separation equivalent, the)J 60 296 :M .087 .009(algorithm given above provides an algorithm for deciding the d)J 366 296 :M .064 .006(-separation equivalence of)J 60 314 :M 1.627 .163(two directed graphs. Moreover the algorithm is of complexity O\(n)J 403 309 :M f0_7 sf (7)S 407 314 :M f0_12 sf 2.012 .201(\) where n is the)J 60 332 :M .469 .047(number of vertices in the graph. In this respect this algorithm is significantly faster than)J 60 350 :M (the procedure presented in Richardson \(1994b\) which was O\(n)S f0_7 sf 0 -5 rm (9)S 0 5 rm 365 350 :M f0_12 sf (\).)S 60 374 :M 1.327 .133(In addition, if a directed cyclic graph )J 256 374 :M f4_12 sf (G)S 265 374 :M f0_12 sf 1.292 .129( is provided as input to the PAG-from-graph)J 60 392 :M .453 .045(algorithm, then it is also possible to tell from the execution of the algorithm, whether or)J 60 410 :M .548 .055(not there is an directed acyclic graph that is d)J 286 410 :M .417 .042(-separation equivalent to )J f4_12 sf (G)S 418 410 :M f0_12 sf .556 .056(: If steps \246c\320\246f)J 60 428 :M .645 .065(perform no orientations then there is a directed acyclic graph d)J 372 428 :M .478 .048(-separation equivalent to)J 60 446 :M f4_12 sf (G)S 69 446 :M f0_12 sf .261 .026(. This follows from the fact that the combination of d)J 328 446 :M .213 .021(-separation relations that the rules)J 60 464 :M .915 .092(in \246c\320\246f require are not entailed by any directed acyclic graph. \(See Richardson 1994,)J 60 482 :M (1994b\).)S 60 527 :M f2_14 sf (6)S 67 527 :M (.)S 70 527 :M ( )S 96 527 :M (Conclusion)S 60 554 :M f0_12 sf .47 .047(These results raise a number of interesting questions whose answers may be of practical)J 60 572 :M .496 .05(importance. Are there other parameterizations of directed cyclic graphs which entail the)J 60 590 :M 1.309 .131(global Markov condition? Is there a polynomial algorithm for determining when two)J 60 608 :M 1.175 .118(arbitrary directed graphs \(cyclic or acyclic\) linearly entail the same set of conditional)J 60 626 :M .352 .035(independence relations over a common subset of variables )J f2_12 sf .159(O)A f0_12 sf .264 .026(? As we have seen there are)J 60 644 :M 1.865 .187(correct, polynomial time algorithms for inferring features of sparse directed graphs)J 60 662 :M 1.708 .171(\(cyclic or acyclic\) from a probability distribution when there are no latent common)J 60 680 :M 1.324 .132(causes. There are similarly correct, but not polynomial time, algorithms for inferring)J endp %%Page: 26 26 %%BeginPageSetup initializepage (peter; page: 26 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (26)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf .702 .07(features of directed acyclic graphs from a probability distribution even when there may)J 60 74 :M .733 .073(be latent common causes \(see Spirtes, 1992, Spirtes, Glymour and Scheines, 1993, and)J 60 92 :M 1.482 .148(Spirtes, Meek, and Richardson 1995\). Are there comparable algorithms for inferring)J 60 110 :M .599 .06(features of directed graphs \(cyclic or acyclic\) from a probability distribution even when)J 60 128 :M (there may be latent common causes?)S endp %%Page: 27 27 %%BeginPageSetup initializepage (peter; page: 27 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (27)S gR gS 0 0 552 730 rC 60 65 :M f2_14 sf (7)S 67 65 :M (.)S 70 65 :M ( )S 96 65 :M (Proofs)S 60 92 :M f0_12 sf 1.142 .114(A )J 74 92 :M f2_12 sf .546 .055(directed graph)J 151 92 :M f0_12 sf .88 .088( is an ordered pair of a finite set of vertices )J 374 92 :M f2_12 sf (V)S 383 92 :M f0_12 sf .872 .087(, and a set of directed)J 60 110 :M .444 .044(edges )J 91 110 :M f2_12 sf .233(E)A f0_12 sf .436 .044(. A directed edge from A to B is an ordered pair of distinct vertices in )J f2_12 sf (V)S 60 128 :M f0_12 sf .148 .015(\(depicted as A )J f1_12 sf S 145 128 :M f0_12 sf .198 .02( B\) in which A is the )J 250 128 :M f2_12 sf (tail)S 267 128 :M f0_12 sf .161 .016( of the edge and B is the )J f2_12 sf .118(head)A 413 128 :M f0_12 sf .194 .019(; the edge is )J 475 128 :M f2_12 sf (out)S 60 146 :M (of)S 70 146 :M f0_12 sf .202 .02( A and )J f2_12 sf .142(into)A 126 146 :M f0_12 sf .315 .032( B, and A is )J 188 146 :M f2_12 sf (parent)S 222 146 :M f0_12 sf .318 .032( of B and B is a )J 303 146 :M f2_12 sf .087(child)A f0_12 sf .187 .019( of A; also A and B are )J f2_12 sf .107(adjacent)A 489 146 :M f0_12 sf (.)S 60 164 :M .333 .033(A sequence of edges <)J 171 164 :M f4_12 sf (E)S f0_9 sf 0 2 rm (1)S 0 -2 rm 183 164 :M f0_12 sf (,...,)S f4_12 sf (E)S f4_9 sf 0 2 rm (n)S 0 -2 rm 210 164 :M f0_12 sf .352 .035(> in a directed graph )J 315 164 :M f4_12 sf (G)S 324 164 :M f0_12 sf .169 .017( is an )J f2_12 sf .534 .053(undirected path )J 439 164 :M f0_12 sf .333 .033(if and only)J 60 182 :M .18 .018(if there exists a sequence of vertices <)J 245 182 :M f4_12 sf (V)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf (,...,)S f4_12 sf (V)S f4_10 sf 0 2 rm (n)S 0 -2 rm f0_10 sf 0 2 rm (+1)S 0 -2 rm 295 182 :M f0_12 sf .209 .021(> such that for 1 )J cF f1_12 sf .021A sf .209 .021( )J f4_12 sf .061(i)A f0_12 sf .1 .01( )J cF f1_12 sf .01A sf .1 .01( )J f4_12 sf .11(n)A f0_12 sf .262 .026( either <)J 451 182 :M f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 461 182 :M f0_12 sf (,)S f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 474 184 :M f0_10 sf (+1)S 485 182 :M f0_12 sf (>)S 60 200 :M .123 .012(= )J f4_12 sf .11(E)A f4_10 sf 0 2 rm (i)S 0 -2 rm 80 200 :M f0_12 sf .249 .025( or <)J 104 200 :M f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 114 202 :M f0_10 sf (+1)S 125 200 :M f0_12 sf (,)S f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 138 200 :M f0_12 sf .164 .016(> = )J f4_12 sf .148(E)A f4_10 sf 0 2 rm (i)S 0 -2 rm 168 200 :M f0_12 sf .249 .025( and )J 192 200 :M f4_12 sf (E)S f4_10 sf 0 2 rm (i)S 0 -2 rm 202 200 :M f0_12 sf .085 .009( )J f1_12 sf S 212 200 :M f0_12 sf .062 .006( )J f4_12 sf .167(E)A f4_10 sf 0 2 rm (i)S 0 -2 rm 225 202 :M f0_10 sf (+1)S 236 200 :M f0_12 sf .208 .021(. A sequence of edges <)J 353 200 :M f4_12 sf (E)S f0_9 sf 0 2 rm (1)S 0 -2 rm 365 200 :M f0_12 sf (,...,)S f4_12 sf (E)S f4_9 sf 0 2 rm (n)S 0 -2 rm 392 200 :M f0_12 sf .199 .02(> in a directed graph)J 60 218 :M f4_12 sf (G)S 69 218 :M f0_12 sf .148 .015( is a )J f2_12 sf .502 .05(directed path )J f0_12 sf .375 .038(if and only if there exists a sequence of vertices <)J f4_12 sf .176(V)A f0_10 sf 0 2 rm .12(1)A 0 -2 rm f0_12 sf .072(,...,)A f4_12 sf .176(V)A f4_10 sf 0 2 rm .12(n)A 0 -2 rm f0_10 sf 0 2 rm .255(+1)A 0 -2 rm 459 218 :M f0_12 sf .407 .041(> such)J 60 236 :M .204 .02(that for 1 )J cF f1_12 sf .02A sf .204 .02( )J 118 236 :M f4_12 sf .084(i)A f0_12 sf .137 .014( )J cF f1_12 sf .014A sf .137 .014( )J f4_12 sf .151(n)A f0_12 sf .223 .022( <)J 151 236 :M f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 161 236 :M f0_12 sf (,)S f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 174 238 :M f0_10 sf (+1)S 185 236 :M f0_12 sf .146 .015(> = )J f4_12 sf .131(E)A f4_10 sf 0 2 rm (i)S 0 -2 rm 215 236 :M f0_12 sf .171 .017(. A \(directed or undirected\) path )J 375 236 :M f4_12 sf (U)S 384 236 :M f0_12 sf .064 .006( is )J f2_12 sf .063(acyclic)A 433 236 :M f0_12 sf .19 .019( if no vertex)J 60 254 :M .845 .085(occurring on an edge in the path occurs more than once. If there is an acyclic directed)J 60 272 :M .585 .059(path from A to B or B = A then A is an )J f2_12 sf .28(ancestor)A f0_12 sf .475 .047( of B, and B is a )J f2_12 sf .297(descendant)A f0_12 sf .55 .055( of A. A)J 60 290 :M (directed graph is )S 143 290 :M f2_12 sf (acyclic)S 178 290 :M f0_12 sf ( if and only if it contains no directed cyclic paths.)S 416 287 :M f0_8 sf (1)S 420 287 :M (4)S 60 314 :M f2_12 sf (7)S 66 314 :M (.)S 69 314 :M (1)S 75 314 :M (.)S 78 314 :M ( )S 96 314 :M (Proof of )S 141 314 :M (Theorem 3)S 60 341 :M f0_12 sf 1.614 .161(Some of the proofs are simplified by using a graphical relation \(which we will call)J 60 359 :M 1.441 .144(\322Lauritzen d-separation\323\) shown in Lauritzen )J f4_12 sf .927 .093(et al.)J f0_12 sf 1.074 .107( \(1990\) to be equivalent to Pearl\325s)J 60 377 :M .668 .067(d-separation relation defined in Section 2. Several preliminary definitions are needed to)J 60 395 :M .486 .049(define Lauritzen d)J 150 395 :M .513 .051(-separation. An )J 228 395 :M f2_12 sf .359 .036(undirected graph)J 318 395 :M f0_12 sf .637 .064( is an ordered pair of a finite set of)J 60 413 :M .414 .041(vertices )J f2_12 sf (V)S 110 413 :M f0_12 sf .604 .06(, and a set of undirected edges )J f2_12 sf .333(E)A f0_12 sf .672 .067(. An undirected edge between A and B is an)J 60 431 :M 1.115 .112(unordered pair of distinct vertices {A,B} in )J 283 431 :M f2_12 sf (V)S 292 431 :M f0_12 sf 1.261 .126(. A sequence of edges <)J 416 431 :M f4_12 sf (E)S f0_9 sf 0 2 rm (1)S 0 -2 rm 428 431 :M f0_12 sf (,...,)S f4_12 sf (E)S f4_9 sf 0 2 rm (n)S 0 -2 rm 455 431 :M f0_12 sf 1.397 .14(> in an)J 60 449 :M 1.238 .124(undirected graph )J f4_12 sf (H)S 158 449 :M f0_12 sf 2.106 .211( is an )J 194 449 :M f2_12 sf 1.444 .144(undirected path )J 284 449 :M f0_12 sf 1.685 .169(if and only if there exists a sequence of)J 60 467 :M (vertices <)S f4_12 sf (V)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf (,...,)S f4_12 sf (V)S f4_10 sf 0 2 rm (n)S 0 -2 rm f0_10 sf 0 2 rm (+1)S 0 -2 rm 157 467 :M f0_12 sf (> such that for 1 )S cF f1_12 sf S sf ( )S f4_12 sf (i)S f0_12 sf ( )S cF f1_12 sf S sf ( )S 264 467 :M f4_12 sf (n)S f0_12 sf ( {)S 279 467 :M f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 289 467 :M f0_12 sf (,)S f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 302 469 :M f0_10 sf (+1)S 313 467 :M f0_12 sf (} = )S 332 467 :M f4_12 sf (E)S f4_10 sf 0 2 rm (i)S 0 -2 rm 342 467 :M f0_12 sf ( and )S f4_12 sf (E)S f4_10 sf 0 2 rm (i)S 0 -2 rm 375 467 :M f0_12 sf ( )S 378 467 :M f1_12 sf S 385 467 :M f0_12 sf ( )S 388 467 :M f4_12 sf (E)S f4_10 sf 0 2 rm (i)S 0 -2 rm 398 469 :M f0_10 sf (+1)S 409 467 :M f0_12 sf (. Let )S 434 467 :M f4_12 sf (G)S 443 467 :M f0_12 sf <28>S 447 467 :M f2_12 sf (X)S 456 467 :M f0_12 sf (\) be the)S 60 485 :M .308 .031(subgraph of directed graph )J 194 485 :M f4_12 sf (G)S 203 485 :M f0_12 sf .312 .031( that contains only vertices in )J f2_12 sf (X)S 359 485 :M f0_12 sf .374 .037(, with an edge from A to B)J 60 503 :M .481 .048(in )J 73 503 :M f4_12 sf (G)S 82 503 :M f0_12 sf <28>S 86 503 :M f2_12 sf (X)S 95 503 :M f0_12 sf .412 .041(\) if and only if there is an edge from A to B in )J f4_12 sf (G)S 336 503 :M f0_12 sf .263 .026(. Moral\()J f4_12 sf (G)S 384 503 :M f0_12 sf .525 .052(\) )J 392 503 :M f2_12 sf .064(moralizes)A f0_12 sf .194 .019( a directed)J 60 521 :M .044 .004(graph )J f4_12 sf (G)S 99 521 :M f0_12 sf .065 .006( if and only if Moral\()J 202 521 :M f4_12 sf (G)S 211 521 :M f0_12 sf .059 .006(\) is an undirected graph with the same vertices as )J f4_12 sf (G)S 460 521 :M f0_12 sf .07 .007(, and a)J 60 539 :M 1.357 .136(pair of vertices X and Y are adjacent in Moral\()J 304 539 :M f4_12 sf (G)S 313 539 :M f0_12 sf 1.489 .149(\) if and only if either X and Y are)J 60 557 :M .265 .027(adjacent in )J f4_12 sf (G)S 125 557 :M f0_12 sf .35 .035(, or they have a common child in )J 290 557 :M f4_12 sf (G)S 299 557 :M f0_12 sf .305 .03(. In an undirected graph )J f4_12 sf (H)S 427 557 :M f0_12 sf .257 .026(, if )J f2_12 sf (X)S 453 557 :M f0_12 sf .446 .045(, )J 460 557 :M f2_12 sf (Y)S 469 557 :M f0_12 sf .35 .035(, and)J 60 575 :M f2_12 sf .368(Z)A f0_12 sf .697 .07( are disjoint sets of vertices, then )J 237 575 :M f2_12 sf (X)S 246 575 :M f0_12 sf .685 .068( is separated from )J f2_12 sf (Y)S 348 575 :M f0_12 sf .853 .085( given )J 383 575 :M f2_12 sf .431(Z)A f0_12 sf .73 .073( if and only if every)J 60 593 :M .408 .041(undirected path between a member of )J 248 593 :M f2_12 sf (X)S 257 593 :M f0_12 sf .439 .044( and a member of )J f2_12 sf (Y)S 356 593 :M f0_12 sf .455 .046( contains a member of )J 469 593 :M f2_12 sf .235(Z)A f0_12 sf .316 .032(. If)J 60 611 :M f2_12 sf (X)S 69 611 :M f0_12 sf .107 .011(, )J f2_12 sf (Y)S 84 611 :M f0_12 sf .261 .026( and )J 108 611 :M f2_12 sf .094(Z)A f0_12 sf .181 .018( are disjoint sets of variables, )J f2_12 sf (X)S 269 611 :M f0_12 sf .261 .026( and )J 293 611 :M f2_12 sf (Y)S 302 611 :M f0_12 sf .261 .026( are )J 324 611 :M f2_12 sf (Lauritzen)S 375 611 :M f0_12 sf .088 .009( )J f2_12 sf (d)S 385 611 :M (-separated)S 438 611 :M f0_12 sf .241 .024( given )J 472 611 :M f2_12 sf .123(Z)A f0_12 sf .159 .016( in)J 60 642 :M ( )S 60 639.48 -.48 .48 204.48 639 .48 60 639 @a 60 651 :M f0_8 sf (1)S 64 651 :M (4)S 68 654 :M f0_10 sf .566 .057(An undirected path is often defined as a sequence of vertices rather than a sequence of edges. The two)J 60 665 :M .07 .007(definitions are essentially equivalent for acyclic directed graphs, because a pair of vertices can be identified)J 60 676 :M .337 .034(with a unique edge in the graph. However, a cyclic graph may contain more than one edge between a pair)J 60 687 :M (of vertices. In that case it is no longer possible to identify a pair of vertices with a unique edge.)S endp %%Page: 28 28 %%BeginPageSetup initializepage (peter; page: 28 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (28)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf 1.044 .104(a directed graph )J f4_12 sf (G)S 154 56 :M f0_12 sf 1.359 .136( just when )J 212 56 :M f2_12 sf (X)S 221 56 :M f0_12 sf 1.057 .106( and )J f2_12 sf (Y)S 256 56 :M f0_12 sf 1.208 .121( are separated given )J 363 56 :M f2_12 sf .444(Z)A f0_12 sf .9 .09( in Moral\()J f4_12 sf (G)S 432 56 :M f0_12 sf (\(An\()S 455 56 :M f2_12 sf (X)S 464 56 :M f1_12 sf 2.425 .243<20C8>J f0_12 sf .715 .071( )J 483 56 :M f2_12 sf (Y)S 60 74 :M f1_12 sf .493A f0_12 sf .16A f2_12 sf .428(Z)A f0_12 sf .855 .085(\)\)\). We will show that Pearl d)J 231 74 :M .776 .078(-separation and Lauritzen d)J 366 74 :M .644 .064(-separation are equivalent)J 60 92 :M (even in cyclic graphs.)S 60 116 :M .865 .087(Lemma 1 states conditions under which a set of \324short\325 d-connecting paths may be put)J 60 134 :M .525 .052(together to form a single d-connecting path. Since some of the vertices in the proofs are)J 60 152 :M .236 .024(defined as satisfying certain properties in the graph, if A and B are vertices, we write A)J f1_12 sf S 60 170 :M f0_12 sf .183 .018(B if and only if A and B are different names for the same vertex. If there is an undirected)J 60 188 :M .209 .021(path )J f4_12 sf (U)S 93 188 :M f0_12 sf .271 .027( containing vertices A and B in directed graph )J f4_12 sf (G)S 330 188 :M f0_12 sf .291 .029(, and there is only one subpath of)J 60 206 :M f4_12 sf (U)S 69 206 :M f0_12 sf ( between A and B, then )S 185 206 :M f4_12 sf (U)S 194 206 :M f0_12 sf (\(A,B\) is the subpath of )S 307 206 :M f4_12 sf (U)S 316 206 :M f0_12 sf ( between A and B.)S 60 234 :M f2_12 sf (Lemma 1:)S 112 234 :M f0_12 sf ( \(Richardson 1994b\))S 60 254 :M 1.792 .179(In a directed \(cyclic or acyclic\) graph )J 263 254 :M f4_12 sf (G)S 272 254 :M f0_12 sf 1.841 .184( over a set of vertices )J f2_12 sf (V)S 403 254 :M f0_12 sf 1.781 .178(, if the following)J 60 270 :M (conditions hold:)S 78 286 :M .46 .046(\(a\) )J 95 286 :M f4_12 sf .203(R)A f0_12 sf .374 .037( is a sequence of vertices in )J f2_12 sf (V)S 250 286 :M f0_12 sf .486 .049( from A to B, )J f4_12 sf .339(R)A f0_12 sf .139 .014( )J 331 286 :M f1_12 sf .276 .028J f0_12 sf .49 .049(< A)J f1_12 sf S 367 286 :M f0_12 sf (X)S 376 288 :M f0_9 sf (0)S 381 286 :M f0_12 sf <2CC958>S 405 288 :M f0_10 sf (n)S f0_9 sf (+1)S 420 286 :M f1_12 sf S 427 286 :M f0_12 sf .374 .037(B>, such that)J 78 302 :M f1_12 sf (")S 87 302 :M f0_12 sf 1.448 .145<692CCA30CA>J cF f1_12 sf .145A sf 1.448 .145( i )J cF f1_12 sf .145A sf 1.448 .145( n, X)J 160 304 :M f0_10 sf (i)S 163 302 :M f0_12 sf .556 .056( )J f1_12 sf S 174 302 :M f0_12 sf 1.778 .178( X)J 188 304 :M f0_10 sf (i)S 191 304 :M f0_9 sf (+1)S 201 302 :M f0_12 sf 1.467 .147( i.e. the X)J 254 304 :M f0_10 sf (i)S 257 302 :M f0_12 sf 1.504 .15( are only )J 308 302 :M f4_12 sf .752 .075(pairwise distinct)J 390 302 :M f0_12 sf 1.15 .115( \(i.e. not necessarily)J 78 318 :M (distinct\),)S 78 334 :M <286229CA>S 95 334 :M f2_12 sf (Z )S f1_12 sf S 115 334 :M f0_12 sf ( )S f2_12 sf (V)S 127 334 :M f0_12 sf (\\{A,B},)S 78 350 :M (\(c\) )S f14_13 sf (T)S 101 350 :M f0_12 sf ( is a set of undirected paths such that)S 96 366 :M 1.702 .17(\(i\)\312for each pair of consecutive vertices in )J f4_12 sf .706(R)A f0_12 sf 1.177 .118(, X)J 343 368 :M f0_10 sf (i)S 346 366 :M f0_12 sf 1.919 .192( and X)J f0_10 sf 0 2 rm (i)S 0 -2 rm 386 368 :M f0_9 sf (+1)S 396 366 :M f0_12 sf 1.87 .187(, there is a unique)J 96 382 :M (undirected path in )S 186 382 :M f14_13 sf (T)S 193 382 :M f0_12 sf ( that d-connects X)S 281 384 :M f0_10 sf (i)S 284 382 :M f0_12 sf ( and X)S 316 384 :M f0_10 sf (i)S 319 384 :M f0_9 sf (+1)S 329 382 :M f0_12 sf ( given )S 362 382 :M f2_12 sf (Z)S f0_12 sf (\\{X)S 388 384 :M f0_10 sf (i)S 391 382 :M f0_12 sf ( , X)S 409 384 :M f0_10 sf (i)S 412 384 :M f0_9 sf (+1)S 422 382 :M f0_12 sf (},)S 96 398 :M 1.958 .196(\(ii\)\312if some vertex X)J f0_10 sf 0 2 rm .51(k)A 0 -2 rm f0_12 sf .651 .065( in )J f4_12 sf .747(R)A f0_12 sf .787 .079(, is in )J f2_12 sf .816(Z)A f0_12 sf 1.281 .128(, then the paths in )J f14_13 sf (T)S 379 398 :M f0_12 sf 1.541 .154( that contain X)J f0_10 sf 0 2 rm .488(k)A 0 -2 rm f0_12 sf 1.067 .107( as an)J 96 414 :M (endpoint collide at X)S 197 416 :M f0_10 sf (k)S f0_12 sf 0 -2 rm (, \(i.e. both paths are directed into X)S 0 2 rm 372 416 :M f0_10 sf (k)S f0_12 sf 0 -2 rm <29>S 0 2 rm 96 430 :M (\(iii\)\312if for three vertices X)S f0_10 sf 0 2 rm (k)S 0 -2 rm f0_9 sf 0 2 rm S 0 -2 rm f0_12 sf (, X)S 249 432 :M f0_10 sf (k)S f0_12 sf 0 -2 rm (, X)S 0 2 rm 269 432 :M f0_10 sf (k)S f0_9 sf (+1)S 284 430 :M f0_12 sf ( occurring in )S f4_12 sf (R)S f0_12 sf (, the d-connecting paths in )S 485 430 :M f14_13 sf (T)S 96 446 :M f0_12 sf .103 .01(between X)J 148 448 :M f0_10 sf (k)S f0_9 sf .036A f0_12 sf 0 -2 rm .103 .01( and X)J 0 2 rm 195 448 :M f0_10 sf (k)S f0_12 sf 0 -2 rm .105 .01(, and X)J 0 2 rm f0_10 sf (k)S f0_12 sf 0 -2 rm .099 .01( and X)J 0 2 rm f0_10 sf (k)S f0_9 sf .077(+1)A 287 446 :M f0_12 sf .129 .013(, collide at X)J 350 448 :M f0_10 sf (k)S f0_12 sf 0 -2 rm .103 .01( then X)J 0 2 rm f0_10 sf (k)S f0_12 sf 0 -2 rm .128 .013( has a descendant in)J 0 2 rm 96 462 :M f2_12 sf (Z)S f0_12 sf (,)S 60 490 :M (then there is a path )S 154 490 :M f4_12 sf (U)S 163 490 :M f0_12 sf ( in )S f4_12 sf (G )S 190 490 :M f0_12 sf (that d-connects A)S 275 490 :M f1_12 sf S 282 490 :M f0_12 sf (X)S 291 492 :M f0_9 sf (0)S 296 490 :M f0_12 sf ( and B)S f1_12 sf S 334 490 :M f0_12 sf (X)S 343 492 :M f0_10 sf (n)S f0_9 sf (+1)S 358 490 :M f0_12 sf ( given )S 391 490 :M f2_12 sf (Z)S f0_12 sf (.)S 60 518 :M f2_12 sf (Proof:)S 93 518 :M f0_12 sf .094 .009(\312Let )J f4_12 sf .081<55AB>A 128 518 :M f0_12 sf .149 .015( be the concatenation of all of the paths in )J f14_13 sf (T)S 341 518 :M f0_12 sf .157 .016( in the order of the sequence )J 482 518 :M f4_12 sf (R)S f0_12 sf (.)S 60 534 :M f4_12 sf <55AB>S 73 534 :M f0_12 sf .893 .089( may not be acyclic because it may contain vertices more than once. In particular it)J 60 550 :M .395 .039(may contain the endpoints A and B more than once. Let )J f4_12 sf (U)S 346 547 :M f4_10 sf .138(*)A f0_12 sf 0 3 rm .364 .036( be a subsection of )J 0 -3 rm f4_12 sf 0 3 rm .35<55AB>A 0 -3 rm 459 550 :M f0_12 sf .405 .04( which)J 60 566 :M (begins with A, and ends with B, and has no occurences of A or B in between.)S 60 594 :M .36 .036(We now form the path )J f4_12 sf (U)S 182 594 :M f0_12 sf .348 .035( by removing all \(undirected\) cycles from )J 389 594 :M f4_12 sf (U)S 398 591 :M f4_10 sf .09(*)A f0_12 sf 0 3 rm .333 .033(. The process here)J 0 -3 rm 60 610 :M .079 .008(is simply as follows: If Y)J 183 612 :M f0_10 sf (i )S f0_12 sf 0 -2 rm .126 .013(= Y)J 0 2 rm f0_10 sf (j)S 209 610 :M f0_12 sf .073 .007(, then we remove the subpath associated with the sequence)J 60 626 :M .26 .026(of vertices S 157 628 :M f0_10 sf (j)S 160 626 :M f0_12 sf .271 .027(>. Carry out this process until there are no repetitions among the Y)J f0_10 sf 0 2 rm (i)S 0 -2 rm 489 626 :M f0_12 sf (.)S 60 642 :M (Let us then rename the sequence of vertices associated with )S f4_12 sf (U)S 358 642 :M f0_12 sf ( as S 415 644 :M f0_10 sf (p)S f0_12 sf 0 -2 rm (>.)S 0 2 rm endp %%Page: 29 29 %%BeginPageSetup initializepage (peter; page: 29 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (29)S gR gS 0 0 552 730 rC 60 54 :M f0_12 sf .342 .034(We will call an edge in )J 177 54 :M f4_12 sf (U)S 186 54 :M f0_12 sf .297 .03( containing a given vertex V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 327 54 :M f0_12 sf .125 .013(, an )J f2_12 sf .503 .05(endpoint edge)J 421 54 :M f0_12 sf .379 .038(, if V)J 447 56 :M f0_10 sf (i)S 450 54 :M f0_12 sf .364 .036( is in the)J 60 70 :M 1.639 .164(sequence )J f4_12 sf .571 .057(R )J f0_12 sf 1.285 .128(and the edge containing V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 258 70 :M f0_12 sf 1.486 .149( is either the first or last edge on a path in )J 485 70 :M f14_13 sf (T)S 60 86 :M f0_12 sf (between V)S 112 88 :M f0_10 sf (i)S 115 86 :M f0_12 sf ( and its predecessor or successor in )S 287 86 :M f4_12 sf (R)S f0_12 sf (; otherwise the edge is an )S 419 86 :M f2_12 sf (internal edge)S 486 86 :M f0_12 sf (.)S 60 114 :M (It now remains to show that )S 197 114 :M f4_12 sf (U)S 206 114 :M f0_12 sf ( is a d-connecting path.)S 60 142 :M (First we prove that if V)S f0_10 sf 0 2 rm (i)S 0 -2 rm 175 142 :M f0_12 sf ( )S f1_12 sf S 187 142 :M f0_12 sf S f2_12 sf (Z)S f0_12 sf ( then V)S f0_10 sf 0 2 rm (i)S 0 -2 rm 236 142 :M f0_12 sf ( is a collider on )S 313 142 :M f4_12 sf (U)S 322 142 :M f0_12 sf (:)S 73 162 :M .532 .053(Any edge in )J f4_12 sf (U)S 146 162 :M f0_12 sf .577 .058( is either an internal edge or an endpoint edge. If V)J 401 164 :M f0_10 sf (i)S 404 162 :M f0_12 sf .596 .06( occurrs on a path)J 73 178 :M 3.056 .306(between two vertices in )J 209 178 :M f4_12 sf 1.33(R)A f0_12 sf 3.022 .302(, then since the path is d-connecting given )J 460 178 :M f2_12 sf (Z)S f0_12 sf (\\{the)S 73 194 :M .08 .008(endpoints}, V)J 141 196 :M f0_10 sf (i )S f0_12 sf 0 -2 rm .123 .012(is a collider on this path. Hence any internal edge between V)J 0 2 rm 440 196 :M f0_10 sf (i)S 443 194 :M f0_12 sf .126 .013( and some)J 73 210 :M (other vertex is into V)S 175 212 :M f0_10 sf (i)S 178 210 :M f0_12 sf (.)S 73 238 :M 1.595 .159(If V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 98 238 :M f0_12 sf 1.911 .191( occurs in )J 156 238 :M f4_12 sf .783(R)A f0_12 sf 1.589 .159(, then by condition c\(ii\) any paths in )J f14_13 sf (T)S 367 238 :M f0_12 sf 1.699 .17( which contain V)J 457 240 :M f0_10 sf (i)S 460 238 :M f0_12 sf 2.039 .204( as an)J 73 254 :M (endpoint collide there. Hence any endpoint edges containing V)S f0_10 sf 0 2 rm (i)S 0 -2 rm 378 254 :M f0_12 sf ( are into V)S 429 256 :M f0_10 sf (i)S 432 254 :M f0_12 sf (.)S 73 282 :M .377 .038(Since the edges in )J f4_12 sf (U)S 174 282 :M f0_12 sf .417 .042( are a subset of the internal and endpoint edges in )J 421 282 :M f4_12 sf <55AB>S 434 282 :M f0_12 sf .436 .044(, and all the)J 73 298 :M (edges in )S 116 298 :M f4_12 sf <55AB>S 129 298 :M f0_12 sf ( that contained V)S f0_10 sf 0 2 rm (i)S 0 -2 rm 214 298 :M f0_12 sf ( are into V)S 265 300 :M f0_10 sf (i)S 268 298 :M f0_12 sf ( it follows that V)S f0_10 sf 0 2 rm (i)S 0 -2 rm 352 298 :M f0_12 sf ( is a collider on )S 429 298 :M f4_12 sf (U)S 438 298 :M f0_12 sf (.)S 60 326 :M (Next we prove that every collider V)S 233 328 :M f0_10 sf (j)S 236 326 :M f0_12 sf ( on )S f4_12 sf (U)S 263 326 :M f0_12 sf ( has some descendant in )S 382 326 :M f2_12 sf (Z)S f0_12 sf (:)S 73 354 :M .206 .021(Since the edges in )J f4_12 sf (U)S 173 354 :M f0_12 sf .246 .025( are a subset of those in )J 292 354 :M f4_12 sf <55AB>S 305 354 :M f0_12 sf .225 .022(, it follows that if V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 404 354 :M f0_12 sf .249 .025( is a collider on )J 483 354 :M f4_12 sf (U)S 73 370 :M f0_12 sf .428 .043(then there was some edge Y)J 212 372 :M f0_10 sf (h)S f0_9 sf S f1_12 sf 0 -2 rm S 0 2 rm 238 370 :M f0_12 sf (Y)S 247 372 :M f0_10 sf .124(h)A f0_12 sf 0 -2 rm .39 .039( where V)J 0 2 rm f0_10 sf (i)S 300 372 :M f0_9 sf .536 .054J 312 370 :M f1_12 sf .281 .028J f0_12 sf (Y)S 331 372 :M f0_10 sf .132(h)A f0_9 sf .119A f0_12 sf 0 -2 rm .324 .032( and V)J 0 2 rm f0_10 sf (i)S 381 370 :M f0_12 sf .643 .064( )J 385 370 :M f1_12 sf S 392 370 :M f0_12 sf .584 .058( Y)J 405 372 :M f0_10 sf .075(h)A f0_12 sf 0 -2 rm .327 .033(. Similarly, there)J 0 2 rm 73 386 :M (was some edge Y)S f0_10 sf 0 2 rm (k)S 0 -2 rm f1_12 sf S 174 386 :M f0_12 sf (Y)S 183 388 :M f0_10 sf (k)S f0_9 sf (+1 )S 200 386 :M f0_12 sf (where V)S 241 388 :M f0_10 sf (i)S 244 388 :M f0_9 sf ( )S f1_12 sf 0 -2 rm S 0 2 rm 256 386 :M f0_12 sf (Y)S 265 388 :M f0_10 sf (k)S f0_12 sf 0 -2 rm ( and V)S 0 2 rm 302 388 :M f0_10 sf (i)S 305 388 :M f0_9 sf (+1)S 315 386 :M f0_12 sf ( )S f1_12 sf S 325 386 :M f0_12 sf ( Y)S 337 388 :M f0_10 sf (k)S f0_9 sf (+1)S 352 386 :M f0_12 sf ( \(h )S cF f1_12 sf S sf ( k\).)S 73 414 :M .155 .016(Suppose that no descendant of V)J 232 416 :M f0_10 sf (i)S 235 414 :M f0_12 sf .249 .025( )J 239 414 :M f1_12 sf S 246 414 :M f0_12 sf .226 .023( Y)J 258 416 :M f0_10 sf .075(h)A f0_12 sf 0 -2 rm .107 .011( is in )J 0 2 rm f2_12 sf 0 -2 rm .12(Z)A 0 2 rm f0_12 sf 0 -2 rm .148 .015(. If Y)J 0 2 rm f0_10 sf .075(h)A f0_12 sf 0 -2 rm .186 .019( occurs on a path )J 0 2 rm f4_12 sf 0 -2 rm .11(P)A 0 2 rm f0_9 sf (1)S 426 414 :M f0_12 sf .216 .022( in )J 442 414 :M f14_13 sf (T)S 449 414 :M f0_12 sf .146 .015( between)J 73 430 :M 1.428 .143(two vertices X)J f1_10 sf 0 2 rm .438(a)A 0 -2 rm f0_12 sf .888 .089( and X)J 188 432 :M f1_10 sf (a)S f0_9 sf (+1)S 204 430 :M f0_12 sf 1 .1( in )J f2_12 sf (R)S 232 430 :M f0_12 sf 1.004 .1( which was d-connecting given )J f2_12 sf .422(Z)A f0_12 sf .468(\\{X)A 419 432 :M f1_10 sf .565(a)A f0_12 sf 0 -2 rm 1.094 .109(, X)J 0 2 rm 442 432 :M f1_10 sf (a)S f0_9 sf (+1)S 458 430 :M f0_12 sf 1.174 .117(}, then)J 73 446 :M 1.18 .118(either there is a subpath of )J f4_12 sf .593(P)A f0_9 sf 0 2 rm (1)S 0 -2 rm 226 446 :M f0_12 sf 1.337 .134( that is a directed path from Y)J 384 448 :M f0_10 sf .587(h)A f0_12 sf 0 -2 rm 1.127 .113( to X)J 0 2 rm f1_10 sf .741(a)A f0_12 sf 0 -2 rm 1.355 .135(, or there is a)J 0 2 rm 73 462 :M .888 .089(subpath of )J f4_12 sf .378(P)A f0_9 sf 0 2 rm (1)S 0 -2 rm 141 462 :M f0_12 sf 1.178 .118( that is a directed path from Y)J 297 464 :M f0_10 sf .375(h)A f0_12 sf 0 -2 rm .719 .072( to X)J 0 2 rm f1_10 sf .473(a)A f0_9 sf .718(+1)A 345 462 :M f0_12 sf 1.189 .119(, or it is an ancestor of some)J 73 478 :M .495 .049(element of )J f2_12 sf .227(Z)A f0_12 sf .252(\\{X)A 155 480 :M f1_10 sf .341(a)A f0_12 sf 0 -2 rm .66 .066(, X)J 0 2 rm 177 480 :M f1_10 sf (a)S f0_9 sf (+1)S 193 478 :M f0_12 sf (}.)S 202 475 :M f0_8 sf (1)S 206 475 :M (5)S 210 478 :M f0_12 sf .733 .073( Since we supposed that no descendant of V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 433 478 :M f0_12 sf .944 .094( is in )J 463 478 :M f2_12 sf .324(Z)A f0_12 sf .622 .062(, we)J 73 494 :M .11 .011(must conclude that either there is a subpath of )J 299 494 :M f4_12 sf (P)S f0_9 sf 0 2 rm (1)S 0 -2 rm 311 494 :M f0_12 sf .116 .012( that is a directed path from V)J 456 496 :M f0_10 sf (i)S 459 494 :M f0_12 sf .096 .01( to X)J f1_10 sf 0 2 rm .063(a)A 0 -2 rm f0_12 sf (,)S 73 510 :M .595 .06(or there is a subpath of )J 191 510 :M f4_12 sf (P)S f0_9 sf 0 2 rm (1)S 0 -2 rm 203 510 :M f0_12 sf .583 .058( that is a directed path from V)J 353 512 :M f0_10 sf (i)S 356 510 :M f0_12 sf .694 .069( to X)J 382 512 :M f1_10 sf (a)S f0_9 sf (+1)S 398 510 :M f0_12 sf .516 .052(. However since V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 73 526 :M f0_12 sf .787 .079(occurs on an edge of the form Y)J 236 528 :M f0_10 sf (h)S f0_9 sf S f1_12 sf 0 -2 rm S 0 2 rm 262 526 :M f0_12 sf (Y)S 271 528 :M f0_10 sf (h)S f1_12 sf 0 -2 rm S 0 2 rm 283 526 :M f0_12 sf (V)S 292 528 :M f0_10 sf (i)S 295 526 :M f0_12 sf .73 .073(, it follows that there is a subpath of )J f4_12 sf .397(P)A f0_9 sf 0 2 rm (1)S 0 -2 rm 73 542 :M f0_12 sf 2.275 .228(that is a directed path from V)J 234 544 :M f0_10 sf (i)S 237 542 :M f0_12 sf 2.812 .281( to X)J 268 544 :M f1_10 sf (a)S f0_9 sf (+1)S 284 542 :M f0_12 sf 2.259 .226(, and hence V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 363 542 :M f0_12 sf 2.446 .245( is an ancestor of X)J 473 544 :M f1_10 sf (a)S f0_9 sf (+1)S 489 542 :M f0_12 sf (.)S 73 558 :M -.005(\(Alternatively, if Y)A f0_10 sf 0 2 rm (h)S 0 -2 rm f0_12 sf ( is in )S f4_12 sf (R)S f0_12 sf (, then let )S 248 558 :M f1_12 sf (a)S 256 558 :M f0_12 sf -.008( be s.t. X)A 299 560 :M f1_10 sf (a)S f0_9 sf (+1)S 315 558 :M f0_12 sf S f1_12 sf S 325 558 :M f0_12 sf S 337 560 :M f0_10 sf (h)S f0_12 sf 0 -2 rm -.004(. The next step of the argument)A 0 2 rm 73 574 :M (then follows.\))S 73 602 :M .947 .095(But we can now carry out a similar argument with X)J 340 604 :M f1_10 sf (a)S f0_9 sf (+1)S 356 602 :M f0_12 sf 1.129 .113( and X)J 391 604 :M f1_10 sf (a)S f0_9 sf (+2)S 407 602 :M f0_12 sf 1.069 .107(. If the path in )J 485 602 :M f14_13 sf (T)S 73 618 :M f0_12 sf .278 .028(from X)J f1_10 sf 0 2 rm .08(a)A 0 -2 rm f0_9 sf 0 2 rm .122(+1)A 0 -2 rm 124 618 :M f0_12 sf .54 .054( to X)J 150 620 :M f1_10 sf (a)S f0_9 sf (+2)S 166 618 :M f0_12 sf .338 .034(, say )J f4_12 sf .258(P)A f0_9 sf 0 2 rm (2)S 0 -2 rm 204 618 :M f0_12 sf .355 .036(, is into X)J f1_10 sf 0 2 rm .194(a)A 0 -2 rm f0_9 sf 0 2 rm .295(+1)A 0 -2 rm 269 618 :M f0_12 sf .463 .046(, then the two paths )J 369 618 :M f4_12 sf (P)S f0_9 sf 0 2 rm (1)S 0 -2 rm 381 618 :M f0_12 sf .344 .034( and )J f4_12 sf .27(P)A f0_9 sf 0 2 rm (2)S 0 -2 rm 417 618 :M f0_12 sf .564 .056( in )J 434 618 :M f14_13 sf (T)S 441 618 :M f0_12 sf .418 .042(, collide at)J 60 650 :M ( )S 60 647.48 -.48 .48 204.48 647 .48 60 647 @a 60 660 :M f0_8 sf (1)S 64 660 :M (5)S 68 663 :M f0_10 sf .202 .02(It is easy to show if A and B are d-connected given )J 279 663 :M f2_10 sf (Z)S 286 663 :M f0_10 sf .183 .018( by some path )J f4_10 sf .104(P)A f0_10 sf .198 .02(, then every vertex V on )J f4_10 sf .104(P)A f0_10 sf .201 .02( is either)J 60 675 :M .077 .008(an ancestor of )J 120 675 :M f2_10 sf (Z)S 127 675 :M f0_10 sf .079 .008(, or on a directed subpath to A, or on a directed subpath to B. For a proof see Lemma 20 of)J 60 687 :M (Richardson\(1994\).)S endp %%Page: 30 30 %%BeginPageSetup initializepage (peter; page: 30 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (30)S gR gS 0 0 552 730 rC 73 54 :M f0_12 sf (X)S 82 56 :M f1_10 sf (a)S f0_9 sf (+1)S 98 54 :M f0_12 sf 1.676 .168(. But in that case, by condition c\(iii\) of the antecedent, some descendant of)J 73 70 :M (X)S 82 72 :M f1_10 sf (a)S f0_9 sf (+1)S 98 70 :M f0_12 sf S f1_12 sf S 110 70 :M f0_12 sf S f2_12 sf .058(Z)A f0_12 sf .129 .013(, and hence some descendant of V)J 286 72 :M f0_10 sf (i)S 289 70 :M f0_12 sf .175 .017( is in )J 316 70 :M f2_12 sf (Z)S f0_12 sf .119 .012(, contrary to hypothesis. Hence the)J 73 86 :M .311 .031(path in )J 110 86 :M f14_13 sf (T)S 117 86 :M f0_12 sf .289 .029( between X)J 173 88 :M f1_10 sf (a)S f0_9 sf (+1)S 189 86 :M f0_12 sf .336 .034( and X)J 222 88 :M f1_10 sf (a)S f0_9 sf (+2)S 238 86 :M f0_12 sf .249 .025( is out of X)J f1_10 sf 0 2 rm .152(a)A 0 -2 rm f0_9 sf 0 2 rm .23(+1)A 0 -2 rm 309 86 :M f0_12 sf .305 .031(. Let W be the first vertex on )J 455 86 :M f4_12 sf (P)S f0_9 sf 0 2 rm (2)S 0 -2 rm 467 86 :M f0_12 sf .28 .028( after)J 73 102 :M (X)S 82 104 :M f1_10 sf (a)S f0_9 sf (+1)S 98 102 :M f0_12 sf .349 .035(. Since the first edge on )J f4_12 sf .189(P)A f0_9 sf 0 2 rm (2)S 0 -2 rm 229 102 :M f0_12 sf .432 .043( is out of X)J 286 104 :M f1_10 sf (a)S f0_9 sf (+1)S 302 102 :M f0_12 sf .361 .036( it follows that either there is a collider)J 73 118 :M .891 .089(on the path that is a descendant of W, or )J 282 118 :M f4_12 sf (P)S f0_9 sf 0 2 rm (2)S 0 -2 rm 294 118 :M f0_12 sf .547 .055(\(W, X)J f1_10 sf 0 2 rm .173(a)A 0 -2 rm f0_9 sf 0 2 rm .262(+2)A 0 -2 rm 341 118 :M f0_12 sf .888 .089(\) is a directed path from W to)J 73 134 :M (X)S 82 136 :M f1_10 sf (a)S f0_9 sf (+2)S 98 134 :M f0_12 sf .639 .064(. But W is a descendant of V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 245 134 :M f0_12 sf .656 .066(, so it is not an ancestor of any member of )J 460 134 :M f2_12 sf .233(Z)A f0_12 sf .485 .048(, and)J 73 150 :M .564 .056(hence is not an ancestor of a collider on )J 274 150 :M f4_12 sf (P)S f0_9 sf 0 2 rm (2)S 0 -2 rm 286 150 :M f0_12 sf .576 .058(. It follows that )J 366 150 :M f4_12 sf (P)S f0_9 sf 0 2 rm (2)S 0 -2 rm 378 150 :M f0_12 sf .587 .059(\(W, X)J 409 152 :M f1_10 sf (a)S f0_9 sf (+2)S 425 150 :M f0_12 sf .56 .056(\) is a directed)J 73 166 :M 1.003 .1(path from W to X)J f1_10 sf 0 2 rm .415(a)A 0 -2 rm f0_9 sf 0 2 rm .63(+2)A 0 -2 rm 180 166 :M f0_12 sf 1.207 .121(. Hence the path in )J 282 166 :M f14_13 sf (T)S 289 166 :M f0_12 sf 1.148 .115( between X)J 347 168 :M f1_10 sf (a)S f0_9 sf (+1)S 363 166 :M f0_12 sf .856 .086( and X)J f1_10 sf 0 2 rm .439(a)A 0 -2 rm f0_9 sf 0 2 rm .666(+2)A 0 -2 rm 414 166 :M f0_12 sf 1.211 .121( is of the form:)J 73 182 :M (X)S 82 184 :M f1_10 sf (a)S f0_9 sf (+1)S 98 182 :M f1_12 sf S 110 182 :M f0_12 sf S f1_12 sf S 146 182 :M f0_12 sf (X)S 155 184 :M f1_10 sf (a)S f0_9 sf (+2)S 171 182 :M f0_12 sf .3 .03(. By repeating this argument we can show that for every r s.t. )J 473 182 :M f1_12 sf (a)S 481 182 :M f0_12 sf (+r)S 73 198 :M cF f1_12 sf .012A sf .116 .012( n, the path in )J f14_13 sf (T)S 156 198 :M f0_12 sf .115 .012( between X)J 211 200 :M f1_10 sf .062(a)A f0_9 sf (+)S f0_10 sf (r)S f0_9 sf ( )S 228 198 :M f0_12 sf .06 .006(and X)J f1_10 sf 0 2 rm (a)S 0 -2 rm f0_9 sf 0 2 rm (+)S 0 -2 rm f0_10 sf 0 2 rm (r)S 0 -2 rm f0_9 sf 0 2 rm (+1)S 0 -2 rm 281 198 :M f0_12 sf .125 .012( is of the form: X)J 365 200 :M f1_10 sf (a)S f0_9 sf (+)S f0_10 sf (r)S f1_12 sf 0 -2 rm S 0 2 rm 391 198 :M f0_12 sf S f1_12 sf S 415 198 :M f0_12 sf (X)S 424 200 :M f1_10 sf (a)S f0_9 sf (+)S f0_10 sf (r)S f0_9 sf (+1)S 448 198 :M f0_12 sf .115 .012(. But this)J 73 214 :M .437 .044(is a contradiction since the edge Y)J 243 216 :M f0_10 sf (k)S f1_12 sf 0 -2 rm S 0 2 rm 260 214 :M f0_12 sf (Y)S 269 216 :M f0_10 sf .126(k)A f0_9 sf .229 .023(+1 )J f0_12 sf 0 -2 rm .463 .046(occurs on one of these paths. Hence some)J 0 2 rm 73 230 :M (descendant of V)S 151 232 :M f0_10 sf (i)S 154 230 :M f0_12 sf ( is in )S f2_12 sf (Z)S f0_12 sf (.)S 60 258 :M (This completes the proof of )S 196 258 :M (Lemma 1)S 242 258 :M (. )S f1_12 sf <5C>S 60 282 :M f2_12 sf .062 .006(Lemma 2:)J 112 282 :M f0_12 sf .075 .008( In a directed graph )J 209 282 :M f4_12 sf (G)S 218 282 :M f0_12 sf .072 .007( with vertices )J 286 282 :M f2_12 sf (V)S 295 282 :M f0_12 sf .055 .005(, if )J f2_12 sf (X)S 320 282 :M f0_12 sf (, )S f2_12 sf (Y)S 335 282 :M f0_12 sf .052 .005(, and )J f2_12 sf (Z)S f0_12 sf .076 .008( are disjoint subsets of )J 480 282 :M f2_12 sf (V)S 489 282 :M f0_12 sf (,)S 60 300 :M (and )S f2_12 sf (X)S 89 300 :M f0_12 sf -.004( is d-connected to )A 177 300 :M f2_12 sf (Y)S 186 300 :M f0_12 sf ( given )S 219 300 :M f2_12 sf (Z)S f0_12 sf ( in )S f4_12 sf (G)S 251 300 :M f0_12 sf (, then )S 281 300 :M f2_12 sf (X)S 290 300 :M f0_12 sf -.004( is d-connected to )A 378 300 :M f2_12 sf (Y)S 387 300 :M f0_12 sf ( given )S 420 300 :M f2_12 sf (Z)S f0_12 sf ( in an acyclic)S 60 318 :M (directed subgraph of )S 162 318 :M f4_12 sf (G)S 171 318 :M f0_12 sf (.)S 60 342 :M f2_12 sf (Proof.)S 92 342 :M f0_12 sf .219 .022( Suppose that )J 161 342 :M f4_12 sf (U)S 170 342 :M f0_12 sf .213 .021( is an undirected path that d-connects X and Y given )J 429 342 :M f2_12 sf .163(Z)A f0_12 sf .214 .021(, and C is a)J 60 360 :M .31 .031(collider on )J 116 360 :M f4_12 sf (U)S 125 360 :M f0_12 sf .385 .038(. Let )J 151 360 :M f4_12 sf (length)S 181 360 :M f0_12 sf (\(C,)S 196 360 :M f2_12 sf .258(Z)A f0_12 sf .351 .035(\) be 0 if C is a member of )J 335 360 :M f2_12 sf .175(Z)A f0_12 sf .313 .031(; otherwise it is the length of a)J 60 378 :M .025 .003(shortest directed path from C to a member of )J 280 378 :M f2_12 sf (Z)S f0_12 sf (. Let )S f4_12 sf (size)S 331 378 :M f0_12 sf <28>S 335 378 :M f4_12 sf (U)S 344 378 :M f0_12 sf .023 .002(\) equal the number of colliders)J 60 396 :M (on )S f4_12 sf (U)S 84 396 :M f0_12 sf .027 .003( plus the sum over all colliders C on )J 261 396 :M f4_12 sf (U)S 270 396 :M f0_12 sf ( of )S f4_12 sf (length)S 316 396 :M f0_12 sf <28>S 320 396 :M f4_12 sf (C)S f0_12 sf (,)S f2_12 sf (Z)S f0_12 sf (\). )S f4_12 sf (U)S 358 396 :M f0_12 sf ( is a )S f2_12 sf .043 .004(minimal d-connecting)J 60 414 :M .237(path)A f0_12 sf .584 .058( between X and Y given )J f2_12 sf .325(Z)A f0_12 sf .266 .027(, if )J f4_12 sf (U)S 242 414 :M f0_12 sf .63 .063( d-connects X and Y given )J 379 414 :M f2_12 sf .328(Z)A f0_12 sf .582 .058( and there is no other)J 60 432 :M .593 .059(path )J 85 432 :M f4_12 sf <55D5>S 98 432 :M f0_12 sf .551 .055( that d-connects X and Y given )J f2_12 sf .292(Z)A f0_12 sf .436 .044( such that )J f4_12 sf .219(size)A 333 432 :M f0_12 sf <28>S 337 432 :M f4_12 sf <55D5>S 350 432 :M f0_12 sf .722 .072(\) < )J 369 432 :M f4_12 sf (size)S 387 432 :M f0_12 sf <28>S 391 432 :M f4_12 sf (U)S 400 432 :M f0_12 sf .593 .059(\). If there is a path)J 60 450 :M .203 .02(that d-connects X and Y given )J f2_12 sf .098(Z)A f0_12 sf .228 .023( there is at least one minimal d-connecting path between)J 60 468 :M (X and Y given )S f2_12 sf (Z.)S 60 492 :M f0_12 sf .715 .071(Suppose )J 105 492 :M f2_12 sf (X)S 114 492 :M f0_12 sf .838 .084( is d-connected to )J 207 492 :M f2_12 sf (Y)S 216 492 :M f0_12 sf .772 .077( given )J f2_12 sf .351(Z.)A f0_12 sf .798 .08( Then for some X in )J f2_12 sf (X)S 378 492 :M f0_12 sf .822 .082( and Y in )J f2_12 sf (Y)S 439 492 :M f0_12 sf .889 .089(, there is a)J 60 510 :M .326 .033(minimal d)J 110 510 :M .287 .029(-connecting path )J f4_12 sf (U)S 202 510 :M f0_12 sf .443 .044( between X and Y given )J f2_12 sf .247(Z)A f0_12 sf .168 .017(. )J 339 510 :M f4_12 sf (U)S 348 510 :M f0_12 sf .313 .031( in )J f4_12 sf (G)S 373 510 :M f0_12 sf .396 .04(. First we will show that)J 60 528 :M (no shortest acyclic directed path )S 218 528 :M f4_12 sf (D)S 227 530 :M f4_10 sf (i)S 230 528 :M f0_12 sf ( from a collider C)S f0_10 sf 0 2 rm (i)S 0 -2 rm 318 528 :M f0_12 sf ( on )S f4_12 sf (U)S 345 528 :M f0_12 sf ( to a member of )S 424 528 :M f2_12 sf (Z)S f0_12 sf ( intersects )S f4_12 sf (U)S 60 546 :M f0_12 sf .409 .041(except at C)J f0_10 sf 0 2 rm (i)S 0 -2 rm 118 546 :M f0_12 sf .444 .044( by showing that if such a point of intersection exists then )J 406 546 :M f4_12 sf (U)S 415 546 :M f0_12 sf .444 .044( is not minimal,)J 60 564 :M (contrary to our assumption. See Figure 18.)S endp %%Page: 31 31 %%BeginPageSetup initializepage (peter; page: 31 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (31)S gR gS 64 75 203 13 rC 64 84 :M f0_12 sf (X)S 73 84 :M ( )S 76 84 :M ( )S 79 84 :M ( )S 82 84 :M ( )S 85 84 :M ( )S 88 84 :M ( )S 91 84 :M ( )S 94 84 :M ( )S 97 84 :M ( )S 100 84 :M ( )S 103 84 :M (W)S 114 84 :M ( )S 117 84 :M ( )S 120 84 :M ( )S 123 84 :M ( )S 126 84 :M ( )S 129 84 :M ( )S 132 84 :M ( )S 135 84 :M ( )S 138 84 :M ( )S 141 84 :M ( )S 144 84 :M ( )S 147 84 :M ( )S 150 84 :M ( )S 153 84 :M ( )S 156 84 :M ( )S 159 84 :M (C)S 167 84 :M ( )S 170 84 :M ( )S 173 84 :M ( )S 176 84 :M ( )S 179 84 :M ( )S 182 84 :M ( )S 185 84 :M ( )S 188 84 :M ( )S 191 84 :M ( )S 194 84 :M ( )S 197 84 :M ( )S 200 84 :M ( )S 203 84 :M ( )S 206 84 :M (W)S 217 84 :M ( )S 220 84 :M ( )S 223 84 :M ( )S 226 84 :M ( )S 229 84 :M ( )S 232 84 :M ( )S 235 84 :M ( )S 238 84 :M ( )S 241 84 :M ( )S 244 84 :M ( )S 247 84 :M ( )S 250 84 :M (Y)S gR gS 60 41 420 151 rC np 100 82 :M 87 88 :L 87 88 :L 87 87 :L 86 87 :L 86 87 :L 86 87 :L 86 86 :L 86 86 :L 86 86 :L 86 86 :L 86 85 :L 86 85 :L 86 85 :L 86 85 :L 86 84 :L 86 84 :L 86 84 :L 86 84 :L 86 83 :L 86 83 :L 85 83 :L 85 82 :L 85 82 :L 85 82 :L 85 82 :L 85 81 :L 85 81 :L 85 81 :L 85 81 :L 86 80 :L 86 80 :L 86 80 :L 86 80 :L 86 79 :L 86 79 :L 86 79 :L 86 79 :L 86 78 :L 86 78 :L 86 78 :L 86 78 :L 86 77 :L 86 77 :L 86 77 :L 86 77 :L 86 76 :L 87 76 :L 87 76 :L 100 82 :L 100 82 :L eofill 76 83 -2 2 87 81 2 76 81 @a np 155 83 :M 142 89 :L 141 89 :L 141 88 :L 141 88 :L 141 88 :L 141 88 :L 141 87 :L 141 87 :L 141 87 :L 141 87 :L 141 86 :L 141 86 :L 141 86 :L 141 86 :L 141 85 :L 140 85 :L 140 85 :L 140 85 :L 140 84 :L 140 84 :L 140 84 :L 140 83 :L 140 83 :L 140 83 :L 140 83 :L 140 82 :L 140 82 :L 140 82 :L 140 82 :L 140 81 :L 140 81 :L 140 81 :L 140 81 :L 140 80 :L 141 80 :L 141 80 :L 141 80 :L 141 79 :L 141 79 :L 141 79 :L 141 79 :L 141 78 :L 141 78 :L 141 78 :L 141 78 :L 141 77 :L 141 77 :L 141 77 :L 155 83 :L 155 83 :L 2 lw eofill 128 84 -2 2 142 82 2 128 82 @a np 178 83 :M 192 77 :L 192 77 :L 192 77 :L 192 77 :L 192 78 :L 192 78 :L 192 78 :L 192 78 :L 193 79 :L 193 79 :L 193 79 :L 193 79 :L 193 80 :L 193 80 :L 193 80 :L 193 80 :L 193 81 :L 193 81 :L 193 81 :L 193 81 :L 193 82 :L 193 82 :L 193 82 :L 193 82 :L 193 83 :L 193 83 :L 193 83 :L 193 83 :L 193 84 :L 193 84 :L 193 84 :L 193 85 :L 193 85 :L 193 85 :L 193 85 :L 193 86 :L 193 86 :L 193 86 :L 193 86 :L 193 87 :L 193 87 :L 192 87 :L 192 87 :L 192 88 :L 192 88 :L 192 88 :L 192 88 :L 192 89 :L 178 83 :L 178 83 :L eofill 189 84 -2 2 205 82 2 189 82 @a np 249 82 :M 235 88 :L 235 88 :L 235 87 :L 235 87 :L 235 87 :L 235 87 :L 235 86 :L 235 86 :L 235 86 :L 235 86 :L 235 85 :L 234 85 :L 234 85 :L 234 85 :L 234 84 :L 234 84 :L 234 84 :L 234 84 :L 234 83 :L 234 83 :L 234 83 :L 234 82 :L 234 82 :L 234 82 :L 234 82 :L 234 81 :L 234 81 :L 234 81 :L 234 81 :L 234 80 :L 234 80 :L 234 80 :L 234 80 :L 234 79 :L 234 79 :L 234 79 :L 234 79 :L 234 78 :L 235 78 :L 235 78 :L 235 78 :L 235 77 :L 235 77 :L 235 77 :L 235 77 :L 235 76 :L 235 76 :L 235 76 :L 249 82 :L 249 82 :L eofill 227 83 -2 2 236 81 2 227 81 @a 1 lw 90 180 46 42 197.5 89.5 @n np 214 89 :M 218 98 :L 218 98 :L 218 98 :L 218 98 :L 218 98 :L 217 98 :L 217 98 :L 217 98 :L 217 98 :L 217 98 :L 217 98 :L 216 98 :L 216 98 :L 216 98 :L 216 98 :L 216 98 :L 215 99 :L 215 99 :L 215 99 :L 215 99 :L 215 99 :L 215 99 :L 214 99 :L 214 99 :L 214 99 :L 214 99 :L 214 99 :L 214 99 :L 213 99 :L 213 99 :L 213 99 :L 213 99 :L 213 99 :L 213 98 :L 212 98 :L 212 98 :L 212 98 :L 212 98 :L 212 98 :L 212 98 :L 211 98 :L 211 98 :L 211 98 :L 211 98 :L 211 98 :L 211 98 :L 210 98 :L 210 98 :L 214 89 :L 214 89 :L eofill -1 -1 215 102 1 1 214 97 @b 0 90 36 24 196.5 98.5 @n np 112 72 :M 108 63 :L 108 63 :L 109 63 :L 109 63 :L 109 62 :L 109 62 :L 109 62 :L 109 62 :L 110 62 :L 110 62 :L 110 62 :L 110 62 :L 110 62 :L 110 62 :L 111 62 :L 111 62 :L 111 62 :L 111 62 :L 111 62 :L 111 62 :L 112 62 :L 112 62 :L 112 62 :L 112 62 :L 112 62 :L 112 62 :L 113 62 :L 113 62 :L 113 62 :L 113 62 :L 113 62 :L 114 62 :L 114 62 :L 114 62 :L 114 62 :L 114 62 :L 114 62 :L 115 62 :L 115 62 :L 115 62 :L 115 62 :L 115 62 :L 115 62 :L 116 62 :L 116 62 :L 116 63 :L 116 63 :L 116 63 :L 112 72 :L 112 72 :L eofill -1 -1 113 65 1 1 112 58 @b 113 80 16 12 rC 113 90 :M f0_10 sf (X)S gR gS 214 80 16 12 rC 214 90 :M f0_10 sf (Y)S gR gS 167 79 12 12 rC 167 89 :M f0_10 sf (i)S gR gS 60 41 420 151 rC np 116 124 :M 112 115 :L 112 115 :L 113 115 :L 113 115 :L 113 115 :L 113 115 :L 113 115 :L 113 115 :L 114 115 :L 114 115 :L 114 115 :L 114 115 :L 114 115 :L 114 115 :L 115 115 :L 115 115 :L 115 115 :L 115 115 :L 115 115 :L 115 115 :L 116 115 :L 116 115 :L 116 115 :L 116 115 :L 116 114 :L 116 115 :L 117 115 :L 117 115 :L 117 115 :L 117 115 :L 117 115 :L 118 115 :L 118 115 :L 118 115 :L 118 115 :L 118 115 :L 118 115 :L 119 115 :L 119 115 :L 119 115 :L 119 115 :L 119 115 :L 119 115 :L 120 115 :L 120 115 :L 120 115 :L 120 115 :L 120 115 :L 116 124 :L 116 124 :L eofill -1 -1 117 118 1 1 116 94 @b 110 127 16 13 rC 110 136 :M f0_12 sf (Z)S gR gS 60 41 420 151 rC -90 0 110 58 163.5 73.5 @n -180 -90 114 28 169.5 58.5 @n 72 155 -2 2 112 153 2 72 153 @a 75 178 -1 1 114 177 1 75 177 @a 122 149 18 13 rC 122 158 :M f4_12 sf (U)S gR gS 127 172 20 13 rC 127 181 :M f4_12 sf (D)S gR gS 135 176 12 12 rC 135 186 :M f4_10 sf (i)S gR gS 139 169 8 13 rC 139 178 :M f0_12 sf (')S gR gS 282 76 198 13 rC 282 85 :M f0_12 sf (X)S 291 85 :M ( )S 294 85 :M ( )S 297 85 :M ( )S 300 85 :M ( )S 303 85 :M ( )S 306 85 :M ( )S 309 85 :M ( )S 312 85 :M ( )S 315 85 :M ( )S 318 85 :M ( )S 321 85 :M (W)S 332 85 :M ( )S 335 85 :M ( )S 338 85 :M ( )S 341 85 :M ( )S 344 85 :M ( )S 347 85 :M ( )S 350 85 :M ( )S 353 85 :M ( )S 356 85 :M ( )S 359 85 :M ( )S 362 85 :M ( )S 365 85 :M ( )S 368 85 :M ( )S 371 85 :M ( )S 374 85 :M ( )S 377 85 :M (C)S 385 85 :M ( )S 388 85 :M ( )S 391 85 :M ( )S 394 85 :M ( )S 397 85 :M ( )S 400 85 :M ( )S 403 85 :M ( )S 406 85 :M ( )S 409 85 :M ( )S 412 85 :M ( )S 415 85 :M ( )S 418 85 :M ( )S 421 85 :M ( )S 424 85 :M (W)S 435 85 :M ( )S 438 85 :M ( )S 441 85 :M ( )S 444 85 :M ( )S 447 85 :M ( )S 450 85 :M ( )S 453 85 :M ( )S 456 85 :M ( )S 459 85 :M ( )S 462 85 :M ( )S 465 85 :M ( )S 468 85 :M (Y)S gR gS 60 41 420 151 rC np 318 83 :M 304 89 :L 304 89 :L 304 88 :L 304 88 :L 304 88 :L 304 88 :L 304 87 :L 304 87 :L 304 87 :L 303 87 :L 303 86 :L 303 86 :L 303 86 :L 303 86 :L 303 85 :L 303 85 :L 303 85 :L 303 85 :L 303 84 :L 303 84 :L 303 84 :L 303 83 :L 303 83 :L 303 83 :L 303 83 :L 303 82 :L 303 82 :L 303 82 :L 303 82 :L 303 81 :L 303 81 :L 303 81 :L 303 81 :L 303 80 :L 303 80 :L 303 80 :L 303 80 :L 303 79 :L 303 79 :L 303 79 :L 304 79 :L 304 78 :L 304 78 :L 304 78 :L 304 78 :L 304 77 :L 304 77 :L 304 77 :L 318 83 :L 318 83 :L eofill 294 84 -2 2 305 82 2 294 82 @a np 372 84 :M 363 88 :L 363 88 :L 363 87 :L 362 87 :L 362 87 :L 362 87 :L 362 87 :L 362 87 :L 362 86 :L 362 86 :L 362 86 :L 362 86 :L 362 86 :L 362 86 :L 362 85 :L 362 85 :L 362 85 :L 362 85 :L 362 85 :L 362 85 :L 362 84 :L 362 84 :L 362 84 :L 362 84 :L 362 84 :L 362 84 :L 362 83 :L 362 83 :L 362 83 :L 362 83 :L 362 83 :L 362 82 :L 362 82 :L 362 82 :L 362 82 :L 362 82 :L 362 82 :L 362 81 :L 362 81 :L 362 81 :L 362 81 :L 362 81 :L 362 81 :L 362 80 :L 362 80 :L 362 80 :L 363 80 :L 363 80 :L 372 84 :L 372 84 :L 2 lw eofill 349 85 -1 1 364 84 1 349 84 @a np 396 84 :M 405 80 :L 405 80 :L 405 80 :L 405 80 :L 405 80 :L 405 80 :L 405 81 :L 405 81 :L 405 81 :L 405 81 :L 405 81 :L 405 81 :L 405 82 :L 405 82 :L 405 82 :L 405 82 :L 406 82 :L 406 82 :L 406 83 :L 406 83 :L 406 83 :L 406 83 :L 406 83 :L 406 84 :L 406 84 :L 406 84 :L 406 84 :L 406 84 :L 406 84 :L 406 85 :L 406 85 :L 406 85 :L 406 85 :L 405 85 :L 405 85 :L 405 86 :L 405 86 :L 405 86 :L 405 86 :L 405 86 :L 405 86 :L 405 87 :L 405 87 :L 405 87 :L 405 87 :L 405 87 :L 405 87 :L 405 88 :L 396 84 :L 396 84 :L 1 lw eofill 403 85 -1 1 420 84 1 403 84 @a np 463 83 :M 449 89 :L 449 89 :L 449 88 :L 449 88 :L 449 88 :L 448 88 :L 448 87 :L 448 87 :L 448 87 :L 448 87 :L 448 86 :L 448 86 :L 448 86 :L 448 86 :L 448 85 :L 448 85 :L 448 85 :L 448 85 :L 448 84 :L 448 84 :L 448 84 :L 448 83 :L 448 83 :L 448 83 :L 448 83 :L 448 82 :L 448 82 :L 448 82 :L 448 82 :L 448 81 :L 448 81 :L 448 81 :L 448 81 :L 448 80 :L 448 80 :L 448 80 :L 448 80 :L 448 79 :L 448 79 :L 448 79 :L 448 79 :L 448 78 :L 448 78 :L 448 78 :L 449 78 :L 449 77 :L 449 77 :L 449 77 :L 463 83 :L 463 83 :L eofill 445 84 -2 2 450 82 2 445 82 @a np 329 73 :M 323 59 :L 323 59 :L 323 59 :L 324 59 :L 324 59 :L 324 59 :L 324 59 :L 325 59 :L 325 58 :L 325 58 :L 325 58 :L 326 58 :L 326 58 :L 326 58 :L 326 58 :L 327 58 :L 327 58 :L 327 58 :L 327 58 :L 328 58 :L 328 58 :L 328 58 :L 328 58 :L 329 58 :L 329 58 :L 329 58 :L 329 58 :L 330 58 :L 330 58 :L 330 58 :L 330 58 :L 331 58 :L 331 58 :L 331 58 :L 331 58 :L 332 58 :L 332 58 :L 332 58 :L 332 58 :L 333 58 :L 333 58 :L 333 59 :L 333 59 :L 334 59 :L 334 59 :L 334 59 :L 334 59 :L 335 59 :L 329 73 :L 329 73 :L 2 lw eofill -2 -2 329 61 2 2 327 54 @b 331 82 16 12 rC 331 92 :M f0_10 sf (X)S gR gS 432 81 16 12 rC 432 91 :M f0_10 sf (Y)S gR gS 385 80 12 12 rC 385 90 :M f0_10 sf (i)S gR gS 60 41 420 151 rC np 334 125 :M 330 116 :L 330 116 :L 330 116 :L 330 116 :L 330 116 :L 331 116 :L 331 116 :L 331 116 :L 331 116 :L 331 116 :L 331 116 :L 332 116 :L 332 116 :L 332 116 :L 332 116 :L 332 116 :L 332 116 :L 333 116 :L 333 116 :L 333 116 :L 333 116 :L 333 116 :L 334 116 :L 334 116 :L 334 115 :L 334 116 :L 334 116 :L 334 116 :L 334 116 :L 335 116 :L 335 116 :L 335 116 :L 335 116 :L 335 116 :L 336 116 :L 336 116 :L 336 116 :L 336 116 :L 336 116 :L 336 116 :L 337 116 :L 337 116 :L 337 116 :L 337 116 :L 337 116 :L 337 116 :L 338 116 :L 338 116 :L 334 125 :L 334 125 :L 2 lw eofill -1 -1 334 119 1 1 333 95 @b 328 128 16 13 rC 328 137 :M f0_12 sf (Z)S gR 2 lw gS 60 41 420 151 rC -90 0 98 58 384 74 @n -180 -90 116 20 386 55 @n 290 156 -2 2 330 154 2 290 154 @a 340 150 18 13 rC 340 159 :M f4_12 sf (U)S gR gS 354 148 8 13 rC 354 157 :M f0_12 sf (')S gR 1 lw gS 60 41 420 151 rC 90 180 46 42 413.5 91.5 @n np 431 91 :M 435 100 :L 434 100 :L 434 100 :L 434 100 :L 434 100 :L 434 100 :L 434 100 :L 434 100 :L 433 100 :L 433 100 :L 433 100 :L 433 100 :L 433 100 :L 432 100 :L 432 100 :L 432 100 :L 432 100 :L 432 100 :L 432 100 :L 431 100 :L 431 100 :L 431 100 :L 431 100 :L 431 100 :L 431 101 :L 430 100 :L 430 100 :L 430 100 :L 430 100 :L 430 100 :L 430 100 :L 429 100 :L 429 100 :L 429 100 :L 429 100 :L 429 100 :L 429 100 :L 428 100 :L 428 100 :L 428 100 :L 428 100 :L 428 100 :L 428 100 :L 427 100 :L 427 100 :L 427 100 :L 427 100 :L 427 100 :L 431 91 :L 431 91 :L eofill -1 -1 431 104 1 1 430 99 @b 0 90 36 24 412.5 100.5 @n gR gS 0 0 552 730 rC 251 213 :M f2_12 sf (Figure )S 288 213 :M (18)S 60 243 :M f0_12 sf .02 .002(Form the path )J f4_12 sf <55D5>S 143 243 :M f0_12 sf .024 .002( in the following way. If )J 264 243 :M f4_12 sf (D)S 273 245 :M f4_10 sf (i)S 276 243 :M f0_12 sf .018 .002( intersects )J f4_12 sf (U)S 336 243 :M f0_12 sf .024 .002( at a vertex other than C)J f0_10 sf 0 2 rm (i )S 0 -2 rm f0_12 sf .037 .004(then let)J 60 261 :M .955(W)A f0_10 sf 0 2 rm .609(X)A 0 -2 rm f0_12 sf 1.093 .109( be the vertex closest to X on )J 235 261 :M f4_12 sf (U)S 244 261 :M f0_12 sf 1.135 .113( that is on both )J f4_12 sf (D)S 335 263 :M f4_10 sf (i)S 338 261 :M f0_12 sf 1.415 .142( and )J 365 261 :M f4_12 sf (U)S 374 261 :M f0_12 sf 1.078 .108(, and W)J f0_10 sf 0 2 rm .558(Y)A 0 -2 rm f0_12 sf 1.168 .117( be the vertex)J 60 279 :M .191 .019(closest to Y on )J 136 279 :M f4_12 sf (U)S 145 279 :M f0_12 sf .176 .018( that is on both )J f4_12 sf (D)S 229 281 :M f4_10 sf (i)S 232 279 :M f0_12 sf .219 .022( and )J 256 279 :M f4_12 sf (U)S 265 279 :M f0_12 sf .171 .017(. Suppose without loss of generality that W)J f0_10 sf 0 2 rm .067(X)A 0 -2 rm f0_12 sf .086 .009( is)J 60 297 :M .371 .037(after W)J f0_10 sf 0 2 rm .125(Y)A 0 -2 rm f0_12 sf .136 .014( on )J 122 297 :M f4_12 sf (D)S 131 299 :M f4_10 sf (i)S 134 297 :M f0_12 sf .285 .028(. Let )J 160 297 :M f4_12 sf <55D5>S 173 297 :M f0_12 sf .24 .024( be the concatenation of )J 293 297 :M f4_12 sf (U)S 302 297 :M f0_12 sf (\(X,W)S 329 299 :M f0_10 sf .099(X)A f0_12 sf 0 -2 rm .106 .011(\), )J 0 2 rm f4_12 sf 0 -2 rm (D)S 0 2 rm 355 299 :M f4_10 sf (i)S 358 297 :M f0_12 sf .066(\(W)A f0_10 sf 0 2 rm .062(Y)A 0 -2 rm f0_12 sf .061(,W)A f0_10 sf 0 2 rm .062(X)A 0 -2 rm f0_12 sf .096 .01(\), and )J f4_12 sf (U)S 441 297 :M f0_12 sf .053(\(W)A f0_10 sf 0 2 rm (Y)S 0 -2 rm f0_12 sf .125 .013(,Y\). It)J 60 315 :M .209 .021(is now easy to show that )J 183 315 :M f4_12 sf .243 .024<55D520>J 199 315 :M f0_12 sf .197 .02(d-connects X and Y given )J 329 315 :M f2_12 sf .089(Z)A f0_12 sf .113 .011(, and )J f4_12 sf .067(size)A 381 315 :M f0_12 sf <28>S 385 315 :M f4_12 sf <55D5>S 398 315 :M f0_12 sf .254 .025(\) < )J 416 315 :M f4_12 sf (size)S 434 315 :M f0_12 sf <28>S 438 315 :M f4_12 sf (U)S 447 315 :M f0_12 sf .162 .016(\) because)J 60 333 :M f4_12 sf <55D5>S 73 333 :M f0_12 sf .325 .032( contains no more colliders than )J 233 333 :M f4_12 sf (U)S 242 333 :M f0_12 sf .332 .033( and a shortest directed path from W)J 421 335 :M f0_10 sf .154(X)A f0_12 sf 0 -2 rm .327 .033( to a member)J 0 2 rm 60 351 :M (of )S 73 351 :M f2_12 sf (Z)S f0_12 sf ( is shorter than )S 155 351 :M f4_12 sf (D)S 164 353 :M f4_10 sf (i)S 167 351 :M f0_12 sf (. Hence )S 207 351 :M f4_12 sf (U)S 216 351 :M f0_12 sf ( is not minimal, contrary to the assumption.)S 60 375 :M .132 .013(Next, we will show that if )J 190 375 :M f4_12 sf (U)S 199 375 :M f0_12 sf .128 .013( is minimal, then it does not contain a pair of colliders C and)J 60 393 :M .717 .072(D such that a shortest directed path from C to a member of )J 359 393 :M f2_12 sf .258(Z)A f0_12 sf .589 .059( intersects a shortest path)J 60 411 :M (from D to a member of )S 174 411 :M f2_12 sf (Z)S f0_12 sf (. Suppose this is false. See )S 312 411 :M (Figure 19.)S 64 429 232 13 rC 64 438 :M (X)S 73 438 :M ( )S 76 438 :M ( )S 79 438 :M ( )S 82 438 :M ( )S 85 438 :M ( )S 88 438 :M ( )S 91 438 :M ( )S 94 438 :M ( )S 97 438 :M ( )S 100 438 :M ( )S 103 438 :M ( )S 106 438 :M ( )S 109 438 :M ( )S 112 438 :M ( )S 115 438 :M ( )S 118 438 :M (C)S 126 438 :M ( )S 129 438 :M ( )S 132 438 :M ( )S 135 438 :M ( )S 138 438 :M ( )S 141 438 :M ( )S 144 438 :M ( )S 147 438 :M ( )S 150 438 :M ( )S 153 438 :M ( )S 156 438 :M ( )S 159 438 :M ( )S 162 438 :M ( )S 165 438 :M (M)S 175 438 :M ( )S 178 438 :M ( )S 181 438 :M ( )S 184 438 :M ( )S 187 438 :M ( )S 190 438 :M ( )S 193 438 :M ( )S 196 438 :M ( )S 199 438 :M ( )S 202 438 :M ( )S 205 438 :M ( )S 208 438 :M ( )S 211 438 :M ( )S 214 438 :M ( )S 217 438 :M ( )S 220 438 :M ( )S 223 438 :M ( )S 226 438 :M (D)S 234 438 :M ( )S 237 438 :M ( )S 240 438 :M ( )S 243 438 :M ( )S 246 438 :M ( )S 249 438 :M ( )S 252 438 :M ( )S 255 438 :M ( )S 258 438 :M ( )S 261 438 :M ( )S 264 438 :M ( )S 267 438 :M ( )S 270 438 :M ( )S 273 438 :M ( )S 276 438 :M ( )S 279 438 :M (Y)S gR gS 60 426 470 175 rC np 115 433 :M 102 439 :L 102 439 :L 101 438 :L 101 438 :L 101 438 :L 101 438 :L 101 438 :L 101 437 :L 101 437 :L 101 437 :L 101 437 :L 101 436 :L 101 436 :L 101 436 :L 101 435 :L 101 435 :L 101 435 :L 101 435 :L 100 434 :L 100 434 :L 100 434 :L 100 434 :L 100 433 :L 100 433 :L 100 433 :L 100 433 :L 100 432 :L 100 432 :L 100 432 :L 100 432 :L 100 431 :L 101 431 :L 101 431 :L 101 431 :L 101 430 :L 101 430 :L 101 430 :L 101 430 :L 101 429 :L 101 429 :L 101 429 :L 101 429 :L 101 428 :L 101 428 :L 101 428 :L 101 428 :L 101 427 :L 102 427 :L 115 433 :L 115 433 :L eofill 80 434 -2 2 102 432 2 80 432 @a np 134 434 :M 148 428 :L 148 428 :L 148 428 :L 148 429 :L 148 429 :L 148 429 :L 149 429 :L 149 430 :L 149 430 :L 149 430 :L 149 430 :L 149 431 :L 149 431 :L 149 431 :L 149 431 :L 149 432 :L 149 432 :L 149 432 :L 149 432 :L 149 433 :L 149 433 :L 149 433 :L 149 433 :L 149 434 :L 149 434 :L 149 434 :L 149 434 :L 149 435 :L 149 435 :L 149 435 :L 149 435 :L 149 436 :L 149 436 :L 149 436 :L 149 436 :L 149 437 :L 149 437 :L 149 437 :L 149 438 :L 149 438 :L 149 438 :L 149 438 :L 149 439 :L 148 439 :L 148 439 :L 148 439 :L 148 439 :L 148 440 :L 134 434 :L 134 434 :L 2 lw eofill 145 435 -2 2 163 433 2 145 433 @a np 223 434 :M 209 440 :L 209 440 :L 209 439 :L 209 439 :L 209 439 :L 209 439 :L 209 439 :L 209 438 :L 209 438 :L 209 438 :L 209 438 :L 209 437 :L 209 437 :L 208 437 :L 208 436 :L 208 436 :L 208 436 :L 208 436 :L 208 435 :L 208 435 :L 208 435 :L 208 435 :L 208 434 :L 208 434 :L 208 434 :L 208 434 :L 208 433 :L 208 433 :L 208 433 :L 208 433 :L 208 432 :L 208 432 :L 208 432 :L 208 432 :L 208 431 :L 208 431 :L 209 431 :L 209 431 :L 209 430 :L 209 430 :L 209 430 :L 209 430 :L 209 429 :L 209 429 :L 209 429 :L 209 429 :L 209 428 :L 209 428 :L 223 434 :L 223 434 :L eofill 183 435 -2 2 210 433 2 183 433 @a np 243 434 :M 257 428 :L 257 428 :L 257 428 :L 257 429 :L 257 429 :L 257 429 :L 257 429 :L 257 430 :L 257 430 :L 257 430 :L 258 430 :L 258 431 :L 258 431 :L 258 431 :L 258 431 :L 258 432 :L 258 432 :L 258 432 :L 258 432 :L 258 433 :L 258 433 :L 258 433 :L 258 433 :L 258 434 :L 258 434 :L 258 434 :L 258 434 :L 258 435 :L 258 435 :L 258 435 :L 258 435 :L 258 436 :L 258 436 :L 258 436 :L 258 436 :L 258 437 :L 258 437 :L 258 437 :L 258 438 :L 257 438 :L 257 438 :L 257 438 :L 257 439 :L 257 439 :L 257 439 :L 257 439 :L 257 439 :L 257 440 :L 243 434 :L 243 434 :L eofill 254 435 -2 2 278 433 2 254 433 @a np 175 483 :M 166 480 :L 166 480 :L 166 479 :L 166 479 :L 166 479 :L 166 479 :L 166 479 :L 166 479 :L 166 478 :L 166 478 :L 166 478 :L 166 478 :L 167 478 :L 167 478 :L 167 478 :L 167 477 :L 167 477 :L 167 477 :L 167 477 :L 167 477 :L 167 477 :L 167 477 :L 168 476 :L 168 476 :L 168 476 :L 168 476 :L 168 476 :L 168 476 :L 168 476 :L 168 476 :L 168 475 :L 169 475 :L 169 475 :L 169 475 :L 169 475 :L 169 475 :L 169 475 :L 169 475 :L 170 475 :L 170 474 :L 170 474 :L 170 474 :L 170 474 :L 170 474 :L 170 474 :L 171 474 :L 171 474 :L 171 474 :L 175 483 :L 175 483 :L eofill 131 446 -1 1 170 478 1 131 445 @a np 182 483 :M 187 474 :L 187 474 :L 187 474 :L 187 474 :L 188 474 :L 188 474 :L 188 475 :L 188 475 :L 188 475 :L 188 475 :L 188 475 :L 189 475 :L 189 475 :L 189 475 :L 189 475 :L 189 476 :L 189 476 :L 189 476 :L 190 476 :L 190 476 :L 190 476 :L 190 476 :L 190 476 :L 190 477 :L 190 477 :L 190 477 :L 190 477 :L 190 477 :L 191 477 :L 191 477 :L 191 478 :L 191 478 :L 191 478 :L 191 478 :L 191 478 :L 191 478 :L 191 478 :L 191 479 :L 191 479 :L 191 479 :L 192 479 :L 192 479 :L 192 479 :L 192 480 :L 192 480 :L 192 480 :L 192 480 :L 192 480 :L 182 483 :L 182 483 :L 1 lw eofill -1 -1 189 479 1 1 233 444 @b np 180 550 :M 176 541 :L 176 541 :L 177 541 :L 177 541 :L 177 541 :L 177 541 :L 177 541 :L 177 541 :L 178 541 :L 178 541 :L 178 541 :L 178 541 :L 178 541 :L 178 541 :L 179 541 :L 179 541 :L 179 540 :L 179 540 :L 179 540 :L 179 540 :L 180 540 :L 180 540 :L 180 540 :L 180 540 :L 180 540 :L 180 540 :L 181 540 :L 181 540 :L 181 540 :L 181 540 :L 181 540 :L 181 540 :L 182 540 :L 182 541 :L 182 541 :L 182 541 :L 182 541 :L 182 541 :L 183 541 :L 183 541 :L 183 541 :L 183 541 :L 183 541 :L 183 541 :L 184 541 :L 184 541 :L 184 541 :L 184 541 :L 180 550 :L 180 550 :L eofill -1 -1 181 544 1 1 180 502 @b 174 488 17 13 rC 174 497 :M f0_12 sf (R)S gR gS 174 557 16 13 rC 174 566 :M f0_12 sf (Z)S gR gS 60 426 470 175 rC np 191 589 :M 178 595 :L 177 595 :L 177 595 :L 177 594 :L 177 594 :L 177 594 :L 177 594 :L 177 593 :L 177 593 :L 177 593 :L 177 593 :L 177 592 :L 177 592 :L 177 592 :L 176 592 :L 176 591 :L 176 591 :L 176 591 :L 176 591 :L 176 590 :L 176 590 :L 176 590 :L 176 590 :L 176 589 :L 176 589 :L 176 589 :L 176 589 :L 176 588 :L 176 588 :L 176 588 :L 176 588 :L 176 587 :L 176 587 :L 176 587 :L 176 587 :L 177 586 :L 177 586 :L 177 586 :L 177 585 :L 177 585 :L 177 585 :L 177 585 :L 177 584 :L 177 584 :L 177 584 :L 177 584 :L 177 584 :L 177 583 :L 191 589 :L 191 589 :L eofill 162 590 -2 2 178 588 2 162 588 @a 143 585 18 13 rC 143 594 :M f4_12 sf (U)S gR gS 300 429 230 13 rC 300 438 :M f0_12 sf (X)S 309 438 :M ( )S 312 438 :M ( )S 315 438 :M ( )S 318 438 :M ( )S 321 438 :M ( )S 324 438 :M ( )S 327 438 :M ( )S 330 438 :M ( )S 333 438 :M ( )S 336 438 :M ( )S 339 438 :M ( )S 342 438 :M ( )S 345 438 :M ( )S 348 438 :M ( )S 351 438 :M ( )S 354 438 :M (C)S 362 438 :M ( )S 365 438 :M ( )S 368 438 :M ( )S 371 438 :M ( )S 374 438 :M ( )S 377 438 :M ( )S 380 438 :M ( )S 383 438 :M ( )S 386 438 :M ( )S 389 438 :M ( )S 392 438 :M ( )S 395 438 :M ( )S 398 438 :M ( )S 401 438 :M (M)S 411 438 :M ( )S 414 438 :M ( )S 417 438 :M ( )S 420 438 :M ( )S 423 438 :M ( )S 426 438 :M ( )S 429 438 :M ( )S 432 438 :M ( )S 435 438 :M ( )S 438 438 :M ( )S 441 438 :M ( )S 444 438 :M ( )S 447 438 :M ( )S 450 438 :M ( )S 453 438 :M ( )S 456 438 :M ( )S 459 438 :M ( )S 462 438 :M (D)S 470 438 :M ( )S 473 438 :M ( )S 476 438 :M ( )S 479 438 :M ( )S 482 438 :M ( )S 485 438 :M ( )S 488 438 :M ( )S 491 438 :M ( )S 494 438 :M ( )S 497 438 :M ( )S 500 438 :M ( )S 503 438 :M ( )S 506 438 :M ( )S 509 438 :M ( )S 512 438 :M ( )S 515 438 :M (Y)S gR gS 60 426 470 175 rC np 351 433 :M 337 439 :L 337 439 :L 337 438 :L 337 438 :L 337 438 :L 337 438 :L 337 438 :L 337 437 :L 337 437 :L 336 437 :L 336 437 :L 336 436 :L 336 436 :L 336 436 :L 336 435 :L 336 435 :L 336 435 :L 336 435 :L 336 434 :L 336 434 :L 336 434 :L 336 434 :L 336 433 :L 336 433 :L 336 433 :L 336 433 :L 336 432 :L 336 432 :L 336 432 :L 336 432 :L 336 431 :L 336 431 :L 336 431 :L 336 431 :L 336 430 :L 336 430 :L 336 430 :L 336 430 :L 336 429 :L 336 429 :L 337 429 :L 337 429 :L 337 428 :L 337 428 :L 337 428 :L 337 428 :L 337 427 :L 337 427 :L 351 433 :L 351 433 :L 2 lw eofill 316 434 -2 2 338 432 2 316 432 @a np 370 434 :M 379 430 :L 379 430 :L 379 430 :L 379 430 :L 379 431 :L 379 431 :L 379 431 :L 379 431 :L 379 431 :L 379 431 :L 379 432 :L 379 432 :L 380 432 :L 380 432 :L 380 432 :L 380 432 :L 380 433 :L 380 433 :L 380 433 :L 380 433 :L 380 433 :L 380 433 :L 380 434 :L 380 434 :L 380 434 :L 380 434 :L 380 434 :L 380 434 :L 380 435 :L 380 435 :L 380 435 :L 380 435 :L 380 435 :L 380 435 :L 380 436 :L 380 436 :L 380 436 :L 379 436 :L 379 436 :L 379 436 :L 379 437 :L 379 437 :L 379 437 :L 379 437 :L 379 437 :L 379 437 :L 379 438 :L 379 438 :L 370 434 :L 370 434 :L eofill 377 435 -1 1 399 434 1 377 434 @a np 459 434 :M 450 438 :L 449 438 :L 449 438 :L 449 437 :L 449 437 :L 449 437 :L 449 437 :L 449 437 :L 449 437 :L 449 436 :L 449 436 :L 449 436 :L 449 436 :L 449 436 :L 449 436 :L 449 435 :L 449 435 :L 449 435 :L 449 435 :L 449 435 :L 449 435 :L 449 434 :L 449 434 :L 449 434 :L 449 434 :L 449 434 :L 449 434 :L 449 433 :L 449 433 :L 449 433 :L 449 433 :L 449 433 :L 449 433 :L 449 432 :L 449 432 :L 449 432 :L 449 432 :L 449 432 :L 449 432 :L 449 431 :L 449 431 :L 449 431 :L 449 431 :L 449 431 :L 449 431 :L 449 430 :L 449 430 :L 449 430 :L 459 434 :L 459 434 :L 1 lw eofill 420 435 -1 1 451 434 1 420 434 @a np 479 434 :M 492 428 :L 492 428 :L 492 428 :L 493 429 :L 493 429 :L 493 429 :L 493 429 :L 493 430 :L 493 430 :L 493 430 :L 493 430 :L 493 431 :L 493 431 :L 493 431 :L 493 431 :L 493 432 :L 493 432 :L 493 432 :L 493 432 :L 493 433 :L 494 433 :L 494 433 :L 494 433 :L 494 434 :L 494 434 :L 494 434 :L 494 434 :L 494 435 :L 494 435 :L 493 435 :L 493 435 :L 493 436 :L 493 436 :L 493 436 :L 493 436 :L 493 437 :L 493 437 :L 493 437 :L 493 438 :L 493 438 :L 493 438 :L 493 438 :L 493 439 :L 493 439 :L 493 439 :L 493 439 :L 492 439 :L 492 440 :L 479 434 :L 479 434 :L eofill 490 435 -2 2 514 433 2 490 433 @a np 411 483 :M 396 478 :L 397 478 :L 397 478 :L 397 478 :L 397 477 :L 397 477 :L 397 477 :L 397 477 :L 397 476 :L 397 476 :L 397 476 :L 398 476 :L 398 475 :L 398 475 :L 398 475 :L 398 475 :L 398 475 :L 398 474 :L 399 474 :L 399 474 :L 399 474 :L 399 473 :L 399 473 :L 399 473 :L 399 473 :L 400 473 :L 400 472 :L 400 472 :L 400 472 :L 400 472 :L 401 472 :L 401 472 :L 401 471 :L 401 471 :L 401 471 :L 402 471 :L 402 471 :L 402 471 :L 402 470 :L 402 470 :L 403 470 :L 403 470 :L 403 470 :L 403 470 :L 404 470 :L 404 469 :L 404 469 :L 404 469 :L 411 483 :L 411 483 :L 2 lw eofill 365 446 -2 2 401 473 2 365 444 @a np 418 483 :M 425 470 :L 425 470 :L 425 470 :L 426 470 :L 426 470 :L 426 470 :L 426 470 :L 426 471 :L 427 471 :L 427 471 :L 427 471 :L 427 471 :L 428 471 :L 428 472 :L 428 472 :L 428 472 :L 428 472 :L 428 472 :L 429 472 :L 429 473 :L 429 473 :L 429 473 :L 429 473 :L 429 473 :L 430 474 :L 430 474 :L 430 474 :L 430 474 :L 430 475 :L 430 475 :L 431 475 :L 431 475 :L 431 475 :L 431 476 :L 431 476 :L 431 476 :L 431 476 :L 431 477 :L 431 477 :L 432 477 :L 432 477 :L 432 478 :L 432 478 :L 432 478 :L 432 478 :L 432 479 :L 432 479 :L 432 479 :L 418 483 :L 418 483 :L eofill -2 -2 428 476 2 2 467 443 @b np 416 550 :M 412 541 :L 412 541 :L 412 541 :L 412 541 :L 412 541 :L 413 541 :L 413 541 :L 413 541 :L 413 541 :L 413 541 :L 413 541 :L 414 541 :L 414 541 :L 414 541 :L 414 541 :L 414 541 :L 414 540 :L 415 540 :L 415 540 :L 415 540 :L 415 540 :L 415 540 :L 415 540 :L 416 540 :L 416 540 :L 416 540 :L 416 540 :L 416 540 :L 416 540 :L 417 540 :L 417 540 :L 417 540 :L 417 540 :L 417 541 :L 417 541 :L 418 541 :L 418 541 :L 418 541 :L 418 541 :L 418 541 :L 418 541 :L 419 541 :L 419 541 :L 419 541 :L 419 541 :L 419 541 :L 419 541 :L 420 541 :L 416 550 :L 416 550 :L eofill -1 -1 416 544 1 1 415 502 @b 410 488 17 13 rC 410 497 :M f0_12 sf (R)S gR gS 410 557 16 13 rC 410 566 :M f0_12 sf (Z)S gR gS 60 426 470 175 rC np 427 589 :M 413 595 :L 413 595 :L 413 595 :L 413 594 :L 413 594 :L 413 594 :L 413 594 :L 412 593 :L 412 593 :L 412 593 :L 412 593 :L 412 592 :L 412 592 :L 412 592 :L 412 592 :L 412 591 :L 412 591 :L 412 591 :L 412 591 :L 412 590 :L 412 590 :L 412 590 :L 412 590 :L 412 589 :L 412 589 :L 412 589 :L 412 589 :L 412 588 :L 412 588 :L 412 588 :L 412 588 :L 412 587 :L 412 587 :L 412 587 :L 412 587 :L 412 586 :L 412 586 :L 412 586 :L 412 585 :L 412 585 :L 412 585 :L 412 585 :L 413 584 :L 413 584 :L 413 584 :L 413 584 :L 413 584 :L 413 583 :L 427 589 :L 427 589 :L eofill 398 590 -2 2 414 588 2 398 588 @a 379 585 18 13 rC 379 594 :M f4_12 sf (U)S gR gS 390 582 8 13 rC 390 591 :M f0_12 sf (')S gR gS 0 0 552 730 rC 251 634 :M f2_12 sf (Figure )S 288 634 :M (19)S 60 664 :M f0_12 sf .336 .034(Let )J f4_12 sf (D)S 88 666 :M f0_10 sf .155(1)A f0_12 sf 0 -2 rm .465 .047( be a shortest directed acyclic path from C to a member of )J 0 2 rm 384 664 :M f2_12 sf .185(Z)A f0_12 sf .355 .036( that intersects )J f4_12 sf (D)S 475 666 :M f0_10 sf .21(2)A f0_12 sf 0 -2 rm .396 .04(, a)J 0 2 rm 60 682 :M .109 .011(shortest directed acyclic path from D to a member of )J 319 682 :M f2_12 sf .067(Z)A f0_12 sf .111 .011(. Let the vertex on )J 419 682 :M f4_12 sf (D)S 428 684 :M f0_10 sf (1)S f0_12 sf 0 -2 rm .109 .011( closest to C)J 0 2 rm endp %%Page: 32 32 %%BeginPageSetup initializepage (peter; page: 32 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (32)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf .515 .051(that is also on )J f4_12 sf (D)S 141 58 :M f0_10 sf .293(2)A f0_12 sf 0 -2 rm .59 .059( be R. Let )J 0 2 rm 200 56 :M f4_12 sf <55D5>S 213 56 :M f0_12 sf .548 .055( be the concatenation of )J 335 56 :M f4_12 sf (U)S 344 56 :M f0_12 sf .379 .038(\(X,C\), )J f4_12 sf (D)S 387 58 :M f0_10 sf .107(1)A f0_12 sf 0 -2 rm .44 .044(\(C,R\), )J 0 2 rm 426 56 :M f4_12 sf (D)S 435 58 :M f0_10 sf .072(2)A f0_12 sf 0 -2 rm .389 .039(\(D,R\), and)J 0 2 rm 60 74 :M f4_12 sf (U)S 69 74 :M f0_12 sf 1.393 .139(\(Y,D\). It is now easy to show that)J f4_12 sf 1.165 .116<2055D5>J 263 74 :M f0_12 sf 1.343 .134( d-connects X and Y given )J f2_12 sf .704(Z)A f0_12 sf .821 .082( and )J f4_12 sf .528(size)A 459 74 :M f0_12 sf <28>S 463 74 :M f4_12 sf <55D5>S 476 74 :M f0_12 sf 1.655 .165(\) <)J 60 92 :M f4_12 sf (size)S 78 92 :M f0_12 sf <28>S 82 92 :M f4_12 sf (U)S 91 92 :M f0_12 sf .338 .034(\) because C and D are not colliders on )J f4_12 sf .296<55D5>A 294 92 :M f0_12 sf .359 .036(, the only collider on )J 400 92 :M f4_12 sf <55D5>S 413 92 :M f0_12 sf .376 .038( that may not be)J 60 110 :M .681 .068(on )J f4_12 sf (U)S 85 110 :M f0_12 sf 1.006 .101( is R, and the length of a shortest path from R to a member of )J 405 110 :M f2_12 sf .55(Z)A f0_12 sf .899 .09( is less than the)J 60 128 :M .441 .044(length of a shortest path from D to a member of )J f2_12 sf .24(Z)A f0_12 sf .441 .044(. Hence )J 349 128 :M f4_12 sf (U)S 358 128 :M f0_12 sf .42 .042( is not minimal, contrary to)J 60 146 :M (the assumption.)S 60 170 :M .977 .098(For each collider C on a minimal path )J f4_12 sf (U)S 266 170 :M f0_12 sf 1.056 .106( that d-connects X and Y given )J 429 170 :M f2_12 sf .405(Z)A f0_12 sf .843 .084(, a shortest)J 60 188 :M .836 .084(directed path from C to a member of )J 249 188 :M f2_12 sf .393(Z)A f0_12 sf .748 .075( does not intersect )J 352 188 :M f4_12 sf (U)S 361 188 :M f0_12 sf .848 .085( except at C, and does not)J 60 206 :M .528 .053(intersect a shortest directed path from any other collider D to a member of )J f2_12 sf .254(Z)A f0_12 sf .536 .054(. It follows)J 60 224 :M .461 .046(that the subgraph consisting of )J f4_12 sf (U)S 223 224 :M f0_12 sf .496 .05( and a shortest directed acyclic path from each collider)J 60 242 :M (on )S f4_12 sf (U)S 84 242 :M f0_12 sf ( to a member of )S 163 242 :M f2_12 sf (Z)S f0_12 sf ( is acyclic. )S 225 233 9 9 rC gS 1.286 1 scale 175.001 242 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 60 272 :M f2_12 sf .069 .007(Lemma 3:)J 112 272 :M f0_12 sf .078 .008( In a directed \(cyclic or acyclic\) graph )J 299 272 :M f4_12 sf (G)S 308 272 :M f0_12 sf .075 .008(, disjoint sets of variables )J 434 272 :M f2_12 sf (X)S 443 272 :M f0_12 sf (, )S f2_12 sf (Y)S 458 272 :M f0_12 sf .064 .006( and )J f2_12 sf .055(Z)A f0_12 sf (,)S 60 290 :M f2_12 sf (X)S 69 290 :M f0_12 sf .306 .031( and )J 92 290 :M f2_12 sf (Y)S 101 290 :M f0_12 sf .236 .024( are Pearl d-connected given )J 243 290 :M f2_12 sf .173(Z)A f0_12 sf .224 .022( if and only if )J f2_12 sf (X)S 330 290 :M f0_12 sf .214 .021( and )J f2_12 sf (Y)S 363 290 :M f0_12 sf .2 .02( are Lauritzen d-connected)J 60 308 :M (given )S 90 308 :M f2_12 sf (Z.)S 60 338 :M (Proof)S 89 338 :M f0_12 sf .916 .092(: First we will show that Pearl d-connection entails)J 345 338 :M f1_12 sf .124 .012( )J f0_12 sf 1.047 .105(Lauritzen d-connection. If )J 483 338 :M f2_12 sf (X)S 60 356 :M f0_12 sf .127 .013(and )J 81 356 :M f2_12 sf (Y)S 90 356 :M f0_12 sf .106 .011( are Pearl d-connected given )J f2_12 sf (Z)S f0_12 sf .103 .01( then there is some Pearl d-connecting path )J 450 356 :M f4_12 sf .058(P)A f0_12 sf .105 .01( from a)J 60 373 :M 2.004 .2(vertex X)J 106 373 :M f1_12 sf S 115 373 :M f2_12 sf (X)S 124 373 :M f0_12 sf 2.231 .223( to a vertex Y)J f1_12 sf S 211 373 :M f2_12 sf (Y)S 220 373 :M f0_12 sf 1.987 .199(. The adjacencies in the moralized undirected graph)J 60 391 :M (Moral\()S f4_12 sf (G)S 102 391 :M f0_12 sf ( \(An\({X})S 148 391 :M f1_12 sf S 163 391 :M f0_12 sf (Y})S f1_12 sf S f2_12 sf (Z)S f0_12 sf (\)\)\) are a superset of the adjacencies in )S 379 391 :M f4_12 sf (G)S 388 391 :M f0_12 sf (, hence there is a path)S 60 409 :M f4_12 sf 1.155(P)A f0_10 sf 0 -3 rm .788(*)A 0 3 rm f0_12 sf 2.643 .264( in Moral\()J 131 409 :M f4_12 sf (G)S 140 409 :M f0_12 sf 2.826 .283( \(An\({X})J 191 409 :M f1_12 sf S 206 409 :M f0_12 sf 1.13(Y})A f1_12 sf 1.444A f2_12 sf 1.255(Z)A f0_12 sf 3.041 .304(\)\)\) corresponding to )J f4_12 sf 1.149(P)A f0_12 sf 1.045 .104( in )J 382 409 :M f4_12 sf (G)S 391 409 :M f0_12 sf 2.998 .3(. Since )J f4_12 sf 1.725(P)A f0_12 sf 3.249 .325( is Pearl)J 60 428 :M 1.232 .123(d-connecting the only vertices B on )J f4_12 sf .499(P)A f0_10 sf 0 -3 rm .34(*)A 0 3 rm f0_12 sf .724 .072( that can be in )J f2_12 sf .544(Z)A f0_12 sf .831 .083( occur on )J 392 428 :M f4_12 sf .47(P)A f0_12 sf .949 .095( in colliders of the)J 60 445 :M .046 .005(form A)J f1_12 sf S 107 445 :M f0_12 sf (B)S f1_12 sf S 127 445 :M f0_12 sf .075 .008(C. A and C are not in )J 233 445 :M f2_12 sf (Z)S f0_12 sf .059 .006( because they are non-colliders on )J 409 445 :M f4_12 sf (P)S f0_12 sf .06 .006(. Since B)J 460 445 :M f1_12 sf S 469 445 :M f2_12 sf (Z)S f0_12 sf .053 .005(, A)J 60 463 :M .143 .014(and C are adjacent in Moral\()J 199 463 :M f4_12 sf (G*)S 214 463 :M f0_12 sf .137 .014(\). Thus, for any vertex B)J f1_12 sf S 342 463 :M f2_12 sf .136(Z)A f0_12 sf .133 .013( on )J 369 463 :M f4_12 sf .079(P)A f0_10 sf 0 -3 rm .054(*)A 0 3 rm f0_12 sf .142 .014( it is possible to form a)J 60 482 :M .088 .009(path )J 84 482 :M f4_12 sf (Q)S 93 479 :M f0_10 sf (*)S f0_12 sf 0 3 rm .052 .005( from )J 0 -3 rm f4_12 sf 0 3 rm (P)S 0 -3 rm f0_10 sf (*)S f0_12 sf 0 3 rm .085 .008( by replacing the A\321B\321C sub-path in )J 0 -3 rm 332 482 :M f4_12 sf (P)S f0_10 sf 0 -3 rm (*)S 0 3 rm f0_12 sf .078 .008( with the edge A\321C. Since no)J 60 499 :M .71 .071(vertex in )J f2_12 sf .377(Z)A f0_12 sf .6 .06( occurs on )J f4_12 sf (Q)S 179 496 :M f0_10 sf .233(*)A f0_12 sf 0 3 rm .721 .072(, X and Y are not separated in Moral\()J 0 -3 rm 373 499 :M f4_12 sf (G*)S 388 499 :M f0_12 sf (\(An\({X})S 431 499 :M f1_12 sf S 446 499 :M f0_12 sf (Y})S f1_12 sf S f2_12 sf (Z)S f0_12 sf (\)\)\),)S 60 518 :M (hence )S 91 518 :M f2_12 sf (X)S 100 518 :M f0_12 sf ( and )S f2_12 sf (Y)S 132 518 :M f0_12 sf ( are Lauritzen d)S 208 518 :M (-connected given )S f2_12 sf (Z)S f0_12 sf (.)S 60 548 :M .07 .007(Next we will show that Lauritzen d-connection entails Pearl d-connection. If )J 433 548 :M f2_12 sf (X)S 442 548 :M f0_12 sf .069 .007( and )J f2_12 sf (Y)S 474 548 :M f0_12 sf .09 .009( are)J 60 566 :M .915 .092(Lauritzen d-connected given )J 206 566 :M f2_12 sf .627(Z)A f0_12 sf 1.074 .107( then there is some path )J f4_12 sf (Q)S 350 566 :M f0_12 sf 1.038 .104( in the moralized undirected)J 60 583 :M .88 .088(graph Moral\()J 126 583 :M f4_12 sf (G)S 135 583 :M f0_12 sf (\(An\()S 158 583 :M f2_12 sf (X)S 167 583 :M f1_12 sf S f2_12 sf (Y)S 185 583 :M f1_12 sf S f2_12 sf (Z)S f0_12 sf <292929>S 214 583 :M f4_12 sf .186 .019( )J f0_12 sf 1.437 .144(connecting a vertex X)J 330 583 :M f1_12 sf S 339 583 :M f2_12 sf (X)S 348 583 :M f0_12 sf 1.21 .121( to a vertex Y)J f1_12 sf S 429 583 :M f2_12 sf (Y)S 438 583 :M f0_12 sf 1.274 .127(, on which)J 60 601 :M .247 .025(there is no vertex in )J 159 601 :M f2_12 sf .101(Z)A f0_12 sf .217 .022(. The edges in Moral\()J 273 601 :M f4_12 sf (G)S 282 601 :M f0_12 sf (\(An\()S 305 601 :M f2_12 sf (X)S 314 601 :M f1_12 sf S f2_12 sf (Y)S 332 601 :M f1_12 sf S f2_12 sf (Z)S f0_12 sf <292929>S 361 601 :M f4_12 sf ( )S f0_12 sf .27 .027(are a superset of the edges)J 60 619 :M .422(in)A f4_12 sf .958 .096( G)J 83 619 :M f0_12 sf (\(An\()S 106 619 :M f2_12 sf (X)S 115 619 :M f1_12 sf S f2_12 sf (Y)S 133 619 :M f1_12 sf .598A f2_12 sf .52(Z)A f0_12 sf 1.124 .112(\)\), but whenever there is an A\321C edge in Moral\()J f4_12 sf (G)S 413 619 :M f0_12 sf (\(An\()S 436 619 :M f2_12 sf (X)S 445 619 :M f1_12 sf S f2_12 sf (Y)S 463 619 :M f1_12 sf S f2_12 sf (Z)S f0_12 sf <292929>S 60 637 :M .179 .018(which is not present in )J f4_12 sf (G)S 182 637 :M f0_12 sf (\(An\()S 205 637 :M f2_12 sf (X)S 214 637 :M f1_12 sf S f2_12 sf (Y)S 232 637 :M f1_12 sf .109A f2_12 sf .095(Z)A f0_12 sf .181 .018(\)\) then there is some vertex B such that A)J 452 637 :M f1_12 sf S 464 637 :M f0_12 sf (B)S f1_12 sf S 484 637 :M f0_12 sf (C)S 60 655 :M 2.088 .209(in )J 75 655 :M f4_12 sf (G)S 84 655 :M f0_12 sf 1.702 .17(, and B is an ancestor in )J f4_12 sf (G)S 228 655 :M f0_12 sf 1.843 .184( of some vertex in )J 331 655 :M f2_12 sf (X)S 340 655 :M f1_12 sf S f2_12 sf (Y)S 358 655 :M f1_12 sf .746A f2_12 sf .648(Z)A f0_12 sf 1.392 .139( \(since every vertex in)J 60 673 :M f4_12 sf (G)S 69 673 :M f0_12 sf (\(An\()S 92 673 :M f2_12 sf (X)S 101 673 :M f1_12 sf S f2_12 sf (Y)S 119 673 :M f1_12 sf S f2_12 sf (Z)S f0_12 sf (\)\) is an ancestor of )S f2_12 sf (X)S 237 673 :M f1_12 sf S f2_12 sf (Y)S 255 673 :M f1_12 sf S f2_12 sf (Z)S f0_12 sf (.\))S endp %%Page: 33 33 %%BeginPageSetup initializepage (peter; page: 33 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (33)S gR gS 0 0 552 730 rC 60 50 :M f0_12 sf 1.164 .116(Form an undirected path )J 188 50 :M f4_12 sf 1.186(P)A f0_12 sf 1.078 .108( in )J 214 50 :M f4_12 sf (G)S 223 50 :M f0_12 sf 1.128 .113( as follows. Replace each undirected edge on )J f4_12 sf (Q)S 465 50 :M f0_12 sf 1.527 .153( by a)J 60 68 :M .498 .05(directed edge in )J 143 68 :M f4_12 sf (G)S 152 68 :M f0_12 sf .527 .053( if such exists \(if there is more than one then pick one\). If there is an)J 60 86 :M .63 .063(undirected edge A\321C on )J 190 86 :M f4_12 sf (Q)S 199 86 :M f0_12 sf .697 .07( which is not present in )J 320 86 :M f4_12 sf (G)S 329 86 :M f0_12 sf .658 .066( then replace A\321C by a subpath)J 60 103 :M (A)S 69 103 :M f1_12 sf S 81 103 :M f0_12 sf (B)S f1_12 sf S 101 103 :M f0_12 sf 1.686 .169(C where B)J 156 103 :M f1_12 sf S 165 103 :M f0_12 sf (Ancestors\()S 217 103 :M f2_12 sf (X)S 226 103 :M f1_12 sf S f2_12 sf (Y)S 244 103 :M f1_12 sf .658A f2_12 sf .571(Z)A f0_12 sf 1.342 .134(\); by the construction of the moralized graph)J 60 122 :M 3.85 .385(there is guaranteed to be such a B. Note that since such vertices B are in)J 60 139 :M (Moral\()S f4_12 sf (G)S 102 139 :M f0_12 sf (\(An\()S 125 139 :M f2_12 sf (X)S 134 139 :M f1_12 sf S f2_12 sf (Y)S 152 139 :M f1_12 sf 4.07A f2_12 sf 3.536(Z)A f0_12 sf 6.887 .689(\)\)\), it follows that every vertex on )J 415 139 :M f4_12 sf 3.128(P)A f0_12 sf 6.864 .686( occurs in)J 60 157 :M (Moral\()S f4_12 sf (G)S 102 157 :M f0_12 sf (\(An\()S 125 157 :M f2_12 sf (X)S 134 157 :M f1_12 sf S f2_12 sf (Y)S 152 157 :M f1_12 sf S f2_12 sf (Z)S f0_12 sf (\)\)\).)S 60 187 :M 1.315 .132(Index the vertices on the path )J f4_12 sf .607(P)A f0_12 sf 1.387 .139( as follows: S 334 189 :M f0_10 sf (n)S f0_9 sf (+1)S 349 187 :M f0_12 sf 1.367 .137(> \(so V)J f0_9 sf 0 2 rm (0)S 0 -2 rm 394 187 :M f1_12 sf S 401 187 :M f0_12 sf .912 .091(X, V)J f0_10 sf 0 2 rm .274(n)A 0 -2 rm f0_9 sf 0 2 rm .524(+1)A 0 -2 rm 441 187 :M f1_12 sf S 448 187 :M f0_12 sf 1.297 .13(Y\). Now)J 60 206 :M (construct the following sets:)S 60 235 :M f2_12 sf (T)S f2_9 sf 0 2 rm (X)S 0 -2 rm f0_12 sf ( = {V)S f0_10 sf 0 2 rm (i)S 0 -2 rm 104 235 :M f0_12 sf ( | V)S f0_10 sf 0 2 rm (i)S 0 -2 rm 124 235 :M f1_12 sf S 133 235 :M f0_12 sf (An\()S 152 235 :M f2_12 sf (X)S 161 235 :M f0_12 sf (\)\\An\()S 187 235 :M f2_12 sf (Z)S f0_12 sf (\)} and )S f2_12 sf (T)S f2_9 sf 0 2 rm (Y)S 0 -2 rm f0_12 sf ( = {V)S f0_10 sf 0 2 rm (i)S 0 -2 rm 272 235 :M f0_12 sf ( | V)S f0_10 sf 0 2 rm (i)S 0 -2 rm 292 235 :M f1_12 sf S 301 235 :M f0_12 sf (An\()S 320 235 :M f2_12 sf (Y)S 329 235 :M f0_12 sf (\)\\An\()S 355 235 :M f2_12 sf (Z)S f0_12 sf (\)}.)S 60 265 :M .254 .025(Let )J 80 265 :M f1_12 sf (a)S 88 265 :M f0_12 sf .239 .024( be the largest k s.t. V)J 195 267 :M f0_10 sf .1(k)A f0_12 sf 0 -2 rm .054 .005( )J 0 2 rm f1_12 sf 0 -2 rm S 0 2 rm 212 265 :M f2_12 sf .152(T)A f2_9 sf 0 2 rm .123(X)A 0 -2 rm f0_12 sf .154 .015(, \(if )J 247 265 :M f2_12 sf .143(T)A f2_9 sf 0 2 rm .116(X)A 0 -2 rm f0_12 sf .099 .01( = )J f1_12 sf S 284 265 :M f0_12 sf .265 .026(, let )J 306 265 :M f1_12 sf .301 .03(a )J 317 265 :M f0_12 sf .254 .025(= 0\). Let )J 363 265 :M f1_12 sf (b)S 370 265 :M f0_12 sf .22 .022( be the smallest k greater)J 60 283 :M (than )S 84 283 :M f1_12 sf (a)S 92 283 :M f0_12 sf ( s.t. V)S 121 285 :M f0_10 sf (k)S f1_12 sf 0 -2 rm S 0 2 rm 135 283 :M f2_12 sf (T)S f2_9 sf 0 2 rm (Y)S 0 -2 rm f0_12 sf (, \(if )S f2_12 sf (T)S f2_9 sf 0 2 rm (Y)S 0 -2 rm f0_12 sf ( = )S 196 283 :M f1_12 sf S 206 283 :M f0_12 sf (, let )S 227 283 :M f1_12 sf (b )S 237 283 :M f0_12 sf (= n+1\).)S 60 313 :M 1.639 .164(We will now show that every vertex V)J 263 315 :M f0_9 sf (k)S 268 313 :M f0_12 sf 2.137 .214( on )J 291 313 :M f4_12 sf (P)S f0_12 sf (\(V)S 311 315 :M f1_10 sf (a)S f0_12 sf 0 -2 rm (,V)S 0 2 rm 329 315 :M f1_10 sf .593(b)A f0_12 sf 0 -2 rm 1.68 .168(\), except for V)J 0 2 rm f1_10 sf .682(a)A f0_12 sf 0 -2 rm 1.382 .138( and V)J 0 2 rm 454 315 :M f1_10 sf .798(b)A f0_12 sf 0 -2 rm 1.584 .158(, is an)J 0 2 rm 60 331 :M .07 .007(ancestor of )J f2_12 sf (Z)S f0_12 sf .077 .008(. Suppose some vertex V)J 245 333 :M f0_10 sf (k)S f0_12 sf 0 -2 rm ( \()S 0 2 rm f1_12 sf 0 -2 rm (a)S 0 2 rm 265 331 :M f0_12 sf (S f2_12 sf (Y)S 275 349 :M f1_12 sf .072A f2_12 sf .063(Z)A f0_12 sf .13 .013(\)\)\), V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 322 349 :M f1_12 sf S 331 349 :M f0_12 sf (An\()S 350 349 :M f2_12 sf (X)S 359 349 :M f1_12 sf S f2_12 sf (Y)S 377 349 :M f1_12 sf .144A f2_12 sf .125(Z)A f0_12 sf .178 .018(\). If V)J 425 351 :M f0_10 sf (i)S 428 349 :M f1_12 sf S 437 349 :M f0_12 sf (An\()S 456 349 :M f2_12 sf .082(Z)A f0_12 sf .189 .019(\) then)J 60 367 :M (V)S 69 369 :M f0_10 sf (i)S 72 367 :M f1_12 sf S 81 367 :M f0_12 sf (An\()S 100 367 :M f2_12 sf (X)S 109 367 :M f1_12 sf S f2_12 sf (Y)S 127 367 :M f0_12 sf .409 .041(\). However, if V)J 208 369 :M f0_10 sf (i)S 211 367 :M f0_12 sf S f1_12 sf S 223 367 :M f0_12 sf (An\()S 242 367 :M f2_12 sf (X)S 251 367 :M f0_12 sf (\)\\An\()S 277 367 :M f2_12 sf .196(Z)A f0_12 sf .344 .034(\) then V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 328 367 :M f1_12 sf S 337 367 :M f2_12 sf .287(T)A f2_9 sf 0 2 rm .233(X)A 0 -2 rm f0_12 sf .294 .029(, so )J 372 367 :M f1_12 sf (a)S 380 367 :M f0_12 sf .433 .043( is not the largest k s.t.)J 60 385 :M (V)S 69 387 :M f0_10 sf .719(k)A f0_12 sf 0 -2 rm .392 .039( )J 0 2 rm f1_12 sf 0 -2 rm S 0 2 rm 88 385 :M f2_12 sf .656(T)A f2_9 sf 0 2 rm .532(X)A 0 -2 rm f0_12 sf 1.323 .132(. Similarly if V)J 182 387 :M f0_10 sf (i)S 185 385 :M f1_12 sf S 194 385 :M f0_12 sf (An\()S 213 385 :M f2_12 sf (Y)S 222 385 :M f0_12 sf (\)\\An\()S 248 385 :M f2_12 sf .691(Z)A f0_12 sf 1.213 .121(\) then V)J f0_10 sf 0 2 rm (m)S 0 -2 rm 308 385 :M f1_12 sf S 317 385 :M f2_12 sf 1.115(T)A f2_9 sf 0 2 rm .905(Y)A 0 -2 rm f0_12 sf 1.141 .114(, so )J 356 385 :M f1_12 sf (b)S 363 385 :M f0_12 sf 1.66 .166( is not the smallest k s.t.)J 60 403 :M (V)S 69 405 :M f0_10 sf (k)S f1_12 sf 0 -2 rm S 0 2 rm 83 403 :M f2_12 sf (T)S f2_9 sf 0 2 rm (Y)S 0 -2 rm f0_12 sf (. Hence V)S f0_10 sf 0 2 rm (i)S 0 -2 rm 148 403 :M f0_12 sf ( )S f1_12 sf S 160 403 :M f0_12 sf ( An\()S 182 403 :M f2_12 sf (Z)S f0_12 sf (\). )S 60 433 :M (Since V)S f1_10 sf 0 2 rm (a)S 0 -2 rm f1_12 sf S 113 433 :M f2_12 sf (T)S f2_9 sf 0 2 rm (X)S 0 -2 rm f0_12 sf (, V)S 142 435 :M f1_10 sf (a)S f1_12 sf 0 -2 rm S 0 2 rm 157 433 :M f0_12 sf (An\()S 176 433 :M f2_12 sf (X)S 185 433 :M f0_12 sf -.005(\) hence there is a directed path )A 335 433 :M f4_12 sf (D)S 344 435 :M f5_10 sf (a)S f0_12 sf 0 -2 rm ( in )S 0 2 rm f4_12 sf 0 -2 rm (G)S 0 2 rm 374 433 :M f0_12 sf ( from V)S 412 435 :M f1_10 sf (a)S f0_12 sf 0 -2 rm -.004( to some vertex)A 0 2 rm 60 451 :M (X)S 69 448 :M f0_10 sf .152A f0_12 sf 0 3 rm .292 .029( in )J 0 -3 rm f2_12 sf 0 3 rm (X)S 0 -3 rm 97 451 :M f0_12 sf .479 .048(. \(If )J f1_12 sf .571 .057(a )J 131 451 :M f0_12 sf .424 .042(= 0 then this is trivially true since V)J 309 453 :M f1_10 sf (a)S f1_12 sf 0 -2 rm S 0 2 rm 322 451 :M f0_12 sf (V)S 331 453 :M f0_9 sf (0)S 336 451 :M f1_12 sf S 343 451 :M f0_12 sf (X)S 352 451 :M f1_12 sf S 361 451 :M f2_12 sf (X)S 370 451 :M f0_12 sf .39 .039(.\) Further, since V)J 460 453 :M f1_10 sf (a)S f1_12 sf 0 -2 rm S 0 2 rm 475 451 :M f2_12 sf (T)S f2_9 sf 0 2 rm (X)S 0 -2 rm f0_12 sf (,)S 60 469 :M (V)S 69 471 :M f1_10 sf (a)S f1_12 sf 0 -2 rm S 0 2 rm 84 469 :M f0_12 sf .66 .066(An \()J 107 469 :M f2_12 sf .373(Z)A f0_12 sf .576 .058(\), so no vertex in )J 204 469 :M f2_12 sf .338(Z)A f0_12 sf .573 .057( occurs on the path )J 310 469 :M f4_12 sf (D)S 319 471 :M f1_10 sf .193(a)A f0_12 sf 0 -2 rm .527 .053(. Likewise there is a directed path)J 0 2 rm 60 487 :M f4_12 sf (D)S 69 489 :M f1_10 sf (b)S f0_12 sf 0 -2 rm ( in)S 0 2 rm f4_12 sf 0 -2 rm (G)S 0 2 rm 95 487 :M f0_12 sf ( from V)S 133 489 :M f1_10 sf (b)S f0_12 sf 0 -2 rm ( to some vertex Y)S 0 2 rm f0_10 sf 0 -5 rm S 0 5 rm f0_12 sf 0 -2 rm ( in )S 0 2 rm f2_12 sf 0 -2 rm (Y)S 0 2 rm 250 487 :M f0_12 sf (, and no vertex in )S 337 487 :M f2_12 sf (Z)S f0_12 sf ( occurs on the path )S 439 487 :M f4_12 sf (D)S 448 489 :M f1_10 sf (b)S f0_12 sf 0 -2 rm (.)S 0 2 rm 60 517 :M .69 .069(We now construct a set )J f14_13 sf (T)S 187 517 :M f0_12 sf .733 .073( of paths as follows: For each edge V)J 375 519 :M f0_10 sf (i)S 378 517 :M f1_12 sf S 390 517 :M f0_12 sf (V)S 399 519 :M f0_10 sf .25(i+1)A f0_12 sf 0 -2 rm .573 .057( or V)J 0 2 rm 439 519 :M f0_10 sf (i)S 442 517 :M f1_12 sf S 454 517 :M f0_12 sf (V)S 463 519 :M f0_10 sf .817 .082(i+1 )J 480 517 :M f0_12 sf (on)S 60 535 :M f4_12 sf (Q)S 69 535 :M f0_12 sf (\(V)S 82 537 :M f1_10 sf (a)S f0_12 sf 0 -2 rm (,V)S 0 2 rm 100 537 :M f1_10 sf .091(b)A f0_12 sf 0 -2 rm .301 .03(\) put the \(one-edge\) path between V)J 0 2 rm f0_10 sf (i)S 284 535 :M f0_12 sf .4 .04( and V)J 317 537 :M f0_10 sf .117(i+1)A f0_12 sf 0 -2 rm .259 .026( into )J 0 2 rm 356 535 :M f14_13 sf (T)S 363 535 :M f0_12 sf .315 .032(. If the directed path )J f4_12 sf (D)S 474 537 :M f1_10 sf .182(a)A f0_12 sf 0 -2 rm .265 .026( is)J 0 2 rm 60 553 :M (of length greater than 0 then put and )S f4_12 sf (D)S 246 555 :M f1_10 sf (a)S f0_12 sf 0 -2 rm ( into )S 0 2 rm 277 553 :M f14_13 sf (T)S 284 553 :M f0_12 sf (, and similarly for )S 373 553 :M f4_12 sf (D)S 382 555 :M f1_10 sf (b)S f0_12 sf 0 -2 rm (.)S 0 2 rm 60 583 :M 2.061 .206(We will now show that )J 189 583 :M f14_13 sf (T)S 196 583 :M f0_12 sf 1.829 .183( satisfies the conditions for Lemma 1)J 391 583 :M 1.828 .183(, with the sequence)J 60 601 :M f4_12 sf (R)S f1_12 sf S 74 601 :M f0_12 sf (S 135 603 :M f1_10 sf .313(b)A f0_12 sf 0 -2 rm .903 .09(,Y\253> \(in the cases in which V)J 0 2 rm f1_10 sf .36(a)A f1_12 sf 0 -2 rm S 0 2 rm 303 601 :M f0_12 sf (X)S 312 601 :M f1_12 sf S 319 601 :M f0_12 sf 1.01 .101(X\253 omit X\253, similarly if V)J 451 603 :M f1_10 sf (b)S f1_12 sf 0 -2 rm S 0 2 rm 463 601 :M f0_12 sf (Y)S 472 601 :M f1_12 sf S 479 601 :M f0_12 sf <59AB>S 60 620 :M (omit Y\253\).)S 60 650 :M .415 .042(Condition \(a\): \(Pairwise distinctness.\) This follows directly from the fact that there is an)J 60 667 :M (edge between V)S f0_10 sf 0 2 rm (k)S 0 -2 rm f0_12 sf ( and V)S 174 669 :M f0_10 sf (k+1)S 190 667 :M f0_12 sf (.)S endp %%Page: 34 34 %%BeginPageSetup initializepage (peter; page: 34 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (34)S gR gS 0 0 552 730 rC 60 54 :M f0_12 sf (Condition \(b\): )S f2_12 sf (Z)S f1_12 sf S 148 54 :M f2_12 sf (V)S 157 54 :M f0_12 sf (\\{X\253,Y\253\). This follows since )S f2_12 sf (X)S 304 54 :M f0_12 sf (, )S f2_12 sf (Y)S 319 54 :M f0_12 sf ( and )S f2_12 sf (Z)S f0_12 sf ( are pairwise disjoint.)S 60 84 :M .462 .046(Condition \(c\(i\)\): A directed path )J f4_12 sf .179(P)A f0_12 sf .163 .016( in )J 246 84 :M f14_13 sf (T)S 253 84 :M f15_12 sf .053 .005( )J f0_12 sf .413 .041(between each pair of consecutive vertices which)J 60 102 :M .853 .085(d-connects given )J 148 102 :M f2_12 sf .308(Z)A f0_12 sf .812 .081(\\{endpoints of )J 231 102 :M f4_12 sf .405(P)A f0_12 sf .932 .093(}. This holds trivially for V)J 378 104 :M f0_10 sf (i)S 381 102 :M f0_12 sf (,V)S 393 104 :M f0_10 sf .294(i+1)A f0_12 sf 0 -2 rm .923 .092( since there is an)J 0 2 rm 60 120 :M 1.222 .122(edge between V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 144 120 :M f0_12 sf 1.369 .137( and V)J f0_10 sf 0 2 rm .498(i+1)A 0 -2 rm f0_12 sf 1.428 .143(. The path )J 251 120 :M f4_12 sf (D)S 260 122 :M f1_10 sf .504(a)A f0_12 sf 0 -2 rm 1.367 .137( d-connects X\253 and V)J 0 2 rm f1_10 sf .504(a)A f0_12 sf 0 -2 rm 1.358 .136( since, as we showed)J 0 2 rm 60 138 :M .565 .056(above, it is a directed path no vertex of which is in )J 316 138 :M f2_12 sf .311(Z)A f0_12 sf .544 .054(. \(In the case in which V)J 447 140 :M f1_10 sf (a)S f1_12 sf 0 -2 rm S 0 2 rm 460 138 :M f0_12 sf (X)S 469 138 :M f1_12 sf S 476 138 :M f0_12 sf <58AB2C>S 60 156 :M .025 .003(this case does not arise.\) Similarly the path )J 270 156 :M f4_12 sf (D)S 279 158 :M f1_10 sf (b)S f0_12 sf 0 -2 rm .026 .003( is d-connecting given )J 0 2 rm f2_12 sf 0 -2 rm (Z)S 0 2 rm f0_12 sf 0 -2 rm .023 .002( since no vertex on)J 0 2 rm 60 175 :M (the path is in )S 125 175 :M f2_12 sf (Z)S f0_12 sf (.)S 60 204 :M ( )S 63 204 :M .569 .057(Condition\(c\(ii\)\): If some vertex W in )J f4_12 sf .225(R)A f0_12 sf .225 .022( is in )J 284 204 :M f2_12 sf .296(Z)A f0_12 sf .456 .046( then the paths in )J f14_13 sf (T)S 387 204 :M f0_12 sf .525 .052( that contain W as an)J 60 222 :M 1.065 .107(endpoint collide at W. The vertices on V)J 267 224 :M f1_10 sf (a)S f0_12 sf 0 -2 rm <2CC956>S 0 2 rm 297 224 :M f1_10 sf .399(b)A f0_12 sf 0 -2 rm .545 .054( on )J 0 2 rm f4_12 sf 0 -2 rm .532(P)A 0 2 rm f0_12 sf 0 -2 rm 1.186 .119( are either \(a\) vertices that were)J 0 2 rm 60 240 :M -.007(present on the path )A 154 240 :M f4_12 sf (Q)S 163 240 :M f0_12 sf -.007( in Moral\()A 212 240 :M f4_12 sf (G)S 221 240 :M f0_12 sf (\(An\()S 244 240 :M f2_12 sf (X)S 253 240 :M f1_12 sf S f2_12 sf (Y)S 271 240 :M f1_12 sf S f2_12 sf (Z)S f0_12 sf -.006(\)\)\), or \(b\) they are vertices that were added)A 60 258 :M .494 .049(when edges \(A\321C\) not present in )J 232 258 :M f4_12 sf (G)S 241 258 :M f0_12 sf .42 .042( were replaced by unshielded colliders \(A)J f1_12 sf S 457 258 :M f0_12 sf (B)S f1_12 sf S 477 258 :M f0_12 sf (C\),)S 60 277 :M .587 .059(\(or both, since the path )J f4_12 sf .285(P)A f0_12 sf .629 .063( is not guaranteed to be acyclic\). Any vertex in category \(a\) is)J 60 295 :M 1.25 .125(not in )J 95 295 :M f2_12 sf .58(Z)A f0_12 sf 1.016 .102(, since if it were then the original path )J f4_12 sf (Q)S 311 295 :M f0_12 sf 1.048 .105( would not have been connecting in)J 60 312 :M (Moral\()S f4_12 sf (G)S 102 312 :M f0_12 sf (\(An\()S 125 312 :M f2_12 sf (X)S 134 312 :M f1_12 sf S f2_12 sf (Y)S 152 312 :M f1_12 sf .119A f2_12 sf .104(Z)A f0_12 sf .199 .02(\)\)\) given )J 215 312 :M f2_12 sf .113(Z)A f0_12 sf .241 .024(. The vertices V)J 301 314 :M f1_10 sf .132(a)A f0_12 sf 0 -2 rm .267 .027( and V)J 0 2 rm 340 314 :M f1_10 sf .121(b)A f0_12 sf 0 -2 rm .217 .022( are not in )J 0 2 rm f2_12 sf 0 -2 rm .176(Z)A 0 2 rm f0_12 sf 0 -2 rm .321 .032(, since V)J 0 2 rm f1_10 sf .139(a)A f0_12 sf 0 -2 rm .282 .028( and V)J 0 2 rm 487 314 :M f1_10 sf (b)S 60 329 :M f0_12 sf 1.23 .123(are either not ancestors of )J f2_12 sf .605(Z)A f0_12 sf .992 .099(, or in the case in which V)J 341 331 :M f1_10 sf (a)S f1_12 sf 0 -2 rm S 0 2 rm 354 329 :M f0_12 sf 1.101 .11(X or V)J f1_9 sf 0 2 rm (b)S 0 -2 rm 395 329 :M f1_12 sf S 402 329 :M f0_12 sf 1.072 .107(Y are elements of)J 60 348 :M f2_12 sf (X)S 69 348 :M f1_12 sf S f2_12 sf (Y)S 87 348 :M f0_12 sf .583 .058(; in both cases V)J f1_10 sf 0 2 rm .262(a)A 0 -2 rm f0_12 sf .508 .051(, V)J 192 350 :M f1_10 sf .543(b)A f0_12 sf 0 -2 rm .297 .03( )J 0 2 rm 201 348 :M f1_12 sf S 210 348 :M f2_12 sf .29(Z)A f0_12 sf .506 .051(. Hence if V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 282 348 :M f0_12 sf .84 .084( )J 286 348 :M f1_12 sf S 295 348 :M f2_12 sf .274(Z)A f0_12 sf .468 .047(, then V)J f0_10 sf 0 2 rm (i)S 0 -2 rm 346 348 :M f0_12 sf .525 .052( occurs as a collider \(possibly)J 60 366 :M .24 .024(more than once\) on the path )J f2_12 sf .113(P)A f0_12 sf .203 .02(, hence all paths in )J 301 366 :M f14_13 sf (T)S 308 366 :M f0_12 sf .226 .023( that contain V)J 380 368 :M f0_10 sf (i)S 383 366 :M f0_12 sf .214 .021( as an endpoint collide)J 60 384 :M (at V)S f0_10 sf 0 2 rm (i)S 0 -2 rm 83 384 :M f0_12 sf (.)S 60 415 :M ( )S 63 415 :M 1.811 .181(Condition\(c\(iii\)\): If for three consecutive vert)J 300 415 :M 2.279 .228(ices in the sequence )J f4_12 sf .905(R)A f0_12 sf 1.677 .168( the)J 60 432 :M .184 .018(d-connecting paths between A and B, and between B and C in )J f14_13 sf (T)S 370 432 :M f15_12 sf .046 .005( )J f0_12 sf .226 .023(collide at B then B has a)J 60 450 :M .662 .066(descendant in )J 131 450 :M f2_12 sf .312(Z)A f0_12 sf .691 .069(. We have already shown that every vertex V)J 364 452 :M f0_9 sf (k)S 369 450 :M f0_12 sf .431 .043( on )J f4_12 sf .422(P)A f0_12 sf .728(\(V)A 409 452 :M f1_10 sf (a)S f0_12 sf 0 -2 rm (,V)S 0 2 rm 427 452 :M f1_10 sf .188(b)A f0_12 sf 0 -2 rm .624 .062(\), except for)J 0 2 rm 60 468 :M (V)S 69 470 :M f1_10 sf .454(a)A f0_12 sf 0 -2 rm .885 .089( and V)J 0 2 rm f1_10 sf .395(b)A f0_12 sf 0 -2 rm .892 .089( is an ancestor of )J 0 2 rm 205 468 :M f2_12 sf .49(Z)A f0_12 sf .85 .085(. Thus it is sufficient to show that the paths in )J f14_13 sf (T)S 456 468 :M f0_12 sf 1.013 .101( do not)J 60 486 :M 2.271 .227(collide at V)J f1_10 sf 0 2 rm .819(a)A 0 -2 rm f0_12 sf 1.334 .133( or V)J 158 488 :M f1_10 sf .501(b)A f0_12 sf 0 -2 rm 1.662 .166(. However, this follows immediately from the fact that )J 0 2 rm f4_12 sf 0 -2 rm (D)S 0 2 rm 461 488 :M f1_10 sf 1.389(a)A f0_12 sf 0 -2 rm 1.773 .177( is a)J 0 2 rm 60 504 :M 1.014 .101(directed path from V)J f1_10 sf 0 2 rm .319(a)A 0 -2 rm f0_12 sf .528 .053( to X\253, and )J f4_12 sf (D)S 237 506 :M f1_10 sf .315(b)A f0_12 sf 0 -2 rm .828 .083( is a directed path from V)J 0 2 rm 373 506 :M f1_10 sf .412(a)A f0_12 sf 0 -2 rm .836 .084( to Y\253. \(In the cases in)J 0 2 rm 60 522 :M .169 .017(which V)J 102 524 :M f1_10 sf (a)S f1_12 sf 0 -2 rm S 0 2 rm 115 522 :M f0_12 sf .193 .019(X, V)J f1_10 sf 0 2 rm .073(a)A 0 -2 rm f0_12 sf .162 .016( is the first vertex in the sequence )J f4_12 sf .085(R)A f0_12 sf .187 .019( so the case does not arise, similarly)J 60 540 :M (if V)S 79 542 :M f1_10 sf (b)S f1_12 sf 0 -2 rm S 0 2 rm 91 540 :M f0_12 sf (Y, then V)S 138 542 :M f1_10 sf (b)S f0_12 sf 0 -2 rm ( is the last vertex in the sequence.\))S 0 2 rm 60 571 :M 1.089 .109(We can now apply )J 159 571 :M .953 .095(Lemma 1)J 207 571 :M .984 .098( to construct an acyclic d-connecting path from X\253 to Y\253)J 60 588 :M (given )S 90 588 :M f2_12 sf (Z)S f0_12 sf (. )S f1_12 sf <5C>S 60 617 :M f0_12 sf 1.049 .105(Since Lauritzen d)J 149 617 :M 1.089 .109(-separation and Pearl d)J 264 617 :M .994 .099(-separation are equivalent, henceforth we will)J 60 633 :M 1.088 .109(simply refer to \322d-separation\323 when the context makes clear which definition is being)J 60 649 :M (used.)S endp %%Page: 35 35 %%BeginPageSetup initializepage (peter; page: 35 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (35)S gR gS 0 0 552 730 rC 60 56 :M f2_12 sf .846 .085(Theorem 3)J 118 56 :M f0_12 sf .812 .081(: If )J f2_12 sf (V)S 147 56 :M f0_12 sf 1.024 .102( is a set of random variables with a probability measure P that has a)J 60 74 :M .934 .093(positive density function )J 188 74 :M f4_12 sf (f)S f0_12 sf <28>S 195 74 :M f2_12 sf (V)S 204 74 :M f0_12 sf 1.048 .105(\), and P satisfies the global directed Markov property for)J 60 92 :M (directed \(cyclic or acyclic\) graph )S 222 92 :M f4_12 sf (G)S 231 92 :M f0_12 sf (, then )S 261 92 :M f4_12 sf (f)S f0_12 sf <28>S 268 92 :M f2_12 sf (V)S 277 92 :M f0_12 sf (\) factors according to )S f4_12 sf (G)S 391 92 :M f0_12 sf (.)S 60 116 :M f2_12 sf (Proof.)S 92 116 :M f0_12 sf 1.514 .151( Assume that probability measure over )J 296 116 :M f2_12 sf (V)S 305 116 :M f0_12 sf 1.423 .142( satisfies the global directed Markov)J 60 134 :M 1.26 .126(property for directed \(cyclic or acyclic\) graph )J 297 134 :M f4_12 sf (G)S 306 134 :M f0_12 sf 1.203 .12(. Let )J f2_12 sf (X)S 344 134 :M f0_12 sf .718 .072(, )J f2_12 sf (Y)S 361 134 :M f0_12 sf 1.685 .169(, and )J 392 134 :M f2_12 sf .54(Z)A f0_12 sf 1.204 .12( be three arbitrary)J 60 152 :M 1.156 .116(disjoint sets of vertices in )J 195 152 :M f4_12 sf (G)S 204 152 :M f0_12 sf 1.265 .126(. Since by Lemma 3)J 308 152 :M 1.048 .105(, Pearl d-separation and Lauritzen d-)J 60 170 :M .53 .053(separation are equivalent, we will now show that for any disjoint sets of variables )J f2_12 sf (R)S 475 170 :M f0_12 sf .788 .079(, )J 482 170 :M f2_12 sf (S)S 489 170 :M f0_12 sf (,)S 60 188 :M .056 .006(and )J f2_12 sf (T)S f0_12 sf .066 .007( included in An\()J f2_12 sf (X)S 175 188 :M f1_12 sf .101 .01<20C8>J 188 188 :M f0_12 sf ( )S f2_12 sf (Y)S 200 188 :M f0_12 sf ( )S f1_12 sf .059A f0_12 sf S f2_12 sf .051(Z)A f0_12 sf .05 .005(\), if )J f2_12 sf (R)S 252 188 :M f0_12 sf .069 .007( and )J f2_12 sf (S)S 282 188 :M f0_12 sf .074 .007( are separated given )J 382 188 :M f2_12 sf (T)S f0_12 sf .066 .007( in Moral\()J 439 188 :M f4_12 sf (G)S 448 188 :M f0_12 sf (\(An\()S 471 188 :M f2_12 sf (X)S 480 188 :M f1_12 sf .101 .01<20C8>J 60 206 :M f2_12 sf (Y)S 69 206 :M f0_12 sf .06 .006( )J f1_12 sf .202A f0_12 sf .066A f2_12 sf .175(Z)A f0_12 sf .314 .031(\)\)\), then )J 135 206 :M f2_12 sf (R)S 144 206 :M f0_12 sf .295 .029( and )J f2_12 sf (S)S 175 206 :M f0_12 sf .305 .031( are independent given )J 289 206 :M f2_12 sf .303(T)A f0_12 sf .268 .027(. If )J 315 206 :M f2_12 sf (R)S 324 206 :M f0_12 sf .431 .043(, )J 331 206 :M f2_12 sf (S)S 338 206 :M f0_12 sf .221 .022(, and )J f2_12 sf .175(T)A f0_12 sf .355 .035( are included in An\()J 471 206 :M f2_12 sf (X)S 480 206 :M f1_12 sf .431 .043<20C8>J 60 224 :M f2_12 sf (Y)S 69 224 :M f0_12 sf .347 .035( )J 73 224 :M f1_12 sf .088A f0_12 sf S f2_12 sf .077(Z)A f0_12 sf .167 .017(\), then An\()J 145 224 :M f2_12 sf (R)S 154 224 :M f1_12 sf .316 .032<20C8>J 167 224 :M f0_12 sf .098 .01( )J f2_12 sf (S)S 177 224 :M f0_12 sf ( )S f1_12 sf .137A f0_12 sf S f2_12 sf .119(T)A f0_12 sf .236 .024(\) is included in An\()J 295 224 :M f2_12 sf (X)S 304 224 :M f1_12 sf .465 .046<20C8>J f0_12 sf .137 .014( )J 320 224 :M f2_12 sf (Y)S 329 224 :M f0_12 sf ( )S f1_12 sf .139A f0_12 sf S f2_12 sf .121(Z)A f0_12 sf .236 .024(\). Any pair of vertices A and)J 60 242 :M .393 .039(B adjacent in Moral\()J 162 242 :M f4_12 sf (G)S 171 242 :M f0_12 sf (\(An\()S 194 242 :M f2_12 sf (R)S 203 242 :M f1_12 sf .583 .058<20C8>J 216 242 :M f0_12 sf .181 .018( )J f2_12 sf (S)S 226 242 :M f0_12 sf .641 .064( )J 230 242 :M f1_12 sf .178A f0_12 sf .058A f2_12 sf .155(T)A f0_12 sf .331 .033(\)\)\) is also adjacent in Moral\()J f4_12 sf (G)S 399 242 :M f0_12 sf (\(An\()S 422 242 :M f2_12 sf (X)S 431 242 :M f1_12 sf .583 .058<20C8>J 444 242 :M f0_12 sf .15 .015( )J f2_12 sf (Y)S 456 242 :M f0_12 sf .641 .064( )J 460 242 :M f1_12 sf S f0_12 sf S f2_12 sf (Z)S f0_12 sf <292929>S 60 260 :M .164 .016(because )J f4_12 sf (G)S 110 260 :M f0_12 sf (\(An\()S 133 260 :M f2_12 sf (R)S 142 260 :M f1_12 sf .325 .032<20C8>J 155 260 :M f0_12 sf .101 .01( )J f2_12 sf (S)S 165 260 :M f0_12 sf .048 .005( )J f1_12 sf .162A f0_12 sf .053A f2_12 sf .14(T)A f0_12 sf .226 .023(\)\) is a subgraph of )J f4_12 sf (G)S 289 260 :M f0_12 sf (\(An\()S 312 260 :M f2_12 sf (X)S 321 260 :M f1_12 sf .325 .032<20C8>J 334 260 :M f0_12 sf .083 .008( )J f2_12 sf (Y)S 346 260 :M f0_12 sf ( )S f1_12 sf .095A f0_12 sf S f2_12 sf .082(Z)A f0_12 sf .238 .024(\)\). Hence Moral\()J 451 260 :M f4_12 sf (G)S 460 260 :M f0_12 sf (\(An\()S 483 260 :M f2_12 sf (R)S 60 278 :M f1_12 sf .74A f0_12 sf .241 .024( )J 73 278 :M f2_12 sf (S)S 80 278 :M f0_12 sf .98 .098( )J 84 278 :M f1_12 sf .306A f0_12 sf .099A f2_12 sf .265(T)A f0_12 sf .557 .056(\)\)\) is a subgraph of Moral\()J 237 278 :M f4_12 sf (G)S 246 278 :M f0_12 sf (\(An\()S 269 278 :M f2_12 sf (X)S 278 278 :M f1_12 sf 1.312 .131<20C8>J f0_12 sf .387 .039( )J 295 278 :M f2_12 sf (Y)S 304 278 :M f0_12 sf .98 .098( )J 308 278 :M f1_12 sf .38A f0_12 sf .124A f2_12 sf .33(Z)A f0_12 sf .557 .056(\)\)\). It follows that if )J 432 278 :M f2_12 sf (R)S 441 278 :M f0_12 sf .61 .061( and )J f2_12 sf (S)S 473 278 :M f0_12 sf .754 .075( are)J 60 296 :M 3.928 .393(separated given )J f2_12 sf 1.452(T)A f0_12 sf 3.043 .304( in Moral\()J 219 296 :M f4_12 sf (G)S 228 296 :M f0_12 sf (\(An\()S 251 296 :M f2_12 sf (X)S 260 296 :M f1_12 sf 5.204 .52<20C8>J f0_12 sf 1.394 .139( )J f2_12 sf (Y)S 296 296 :M f0_12 sf .626 .063( )J f1_12 sf 2.116A f0_12 sf .689A f2_12 sf 1.838(Z)A f0_12 sf 3.961 .396(\)\)\) they are also separated in)J 60 314 :M (Moral\()S f4_12 sf (G)S 102 314 :M f0_12 sf (\(An\()S 125 314 :M f2_12 sf (R)S 134 314 :M f1_12 sf .706 .071<20C8>J 147 314 :M f0_12 sf .776 .078( )J 151 314 :M f2_12 sf (S)S 158 314 :M f0_12 sf .776 .078( )J 162 314 :M f1_12 sf .244A f0_12 sf .079A f2_12 sf .212(T)A f0_12 sf .457 .046(\)\)\). But by the global directed Markov property, if )J 433 314 :M f2_12 sf (R)S 442 314 :M f0_12 sf .483 .048( and )J f2_12 sf (S)S 473 314 :M f0_12 sf .597 .06( are)J 60 332 :M (separated given )S 138 332 :M f2_12 sf (T)S f0_12 sf ( in Moral\()S 195 332 :M f4_12 sf (G)S 204 332 :M f0_12 sf (\(An\()S 227 332 :M f2_12 sf (R)S 236 332 :M f1_12 sf <20C8>S f0_12 sf ( )S f2_12 sf (S)S 258 332 :M f0_12 sf ( )S f1_12 sf S f0_12 sf S f2_12 sf (T)S f0_12 sf (\)\)\) then )S 320 332 :M f2_12 sf (R)S 329 332 :M f0_12 sf ( and )S f2_12 sf (S)S 359 332 :M f0_12 sf .013 .001( are independent given )J f2_12 sf (T)S f0_12 sf (. It)S 60 350 :M 1.772 .177(follows from the Hammersly-Clifford Theorem \(see Lauritzen )J 385 350 :M f4_12 sf 2.076 .208(et al.)J f0_12 sf 2.191 .219( 1990\) that the)J 60 368 :M (density function )S 141 368 :M f4_12 sf (f)S f0_12 sf (\(An\()S 167 368 :M f2_12 sf (X)S 176 368 :M f1_12 sf <20C8>S f0_12 sf ( )S f2_12 sf (Y)S 200 368 :M f0_12 sf ( )S f1_12 sf S f0_12 sf S f2_12 sf (Z)S f0_12 sf (\)\) can be factored as)S 162 383 227 26 rC 389 409 :M psb currentpoint pse 162 383 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7264 div 832 3 -1 roll exch div scale currentpoint translate 64 40 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (f) 56 344 sh (g) 4473 344 sh 384 /Times-Roman f1 (\() 226 344 sh (\() 842 344 sh (\)\)) 2664 344 sh (\(V,) 4859 344 sh (\(V\)) 6522 344 sh (\)) 7053 344 sh 224 ns (V) 4659 440 sh (V) 3354 714 sh (\() 3964 714 sh (\)) 4955 714 sh 384 /Times-Roman f1 (An) 366 344 sh (Parents) 5400 344 sh 224 ns (An) 3675 714 sh 384 /Times-Bold f1 (X) 979 344 sh (Y) 1696 344 sh (Z) 2407 344 sh 224 ns (X) 4056 714 sh (Y) 4426 714 sh (Z) 4793 714 sh 384 /Symbol f1 (\310) 1326 344 sh (\310) 2044 344 sh (=) 3024 344 sh 224 ns (\316) 3533 714 sh (\310) 4235 714 sh (\310) 4606 714 sh 576 ns (\325) 3950 432 sh end MTsave restore pse gR gS 0 0 552 730 rC 60 436 :M f0_12 sf (where each )S f4_12 sf (g)S f4_10 sf 0 2 rm (V)S 0 -2 rm f0_12 sf ( is a positive function, i.e., the density function factors according to )S 456 436 :M f4_12 sf (G)S 465 436 :M f0_12 sf (. )S 471 427 9 9 rC gS 1.286 1 scale 366.336 436 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 60 460 :M f2_12 sf (7)S 66 460 :M (.)S 69 460 :M (2)S 75 460 :M (.)S 78 460 :M ( )S 96 460 :M (Proof of )S 141 460 :M (Theorem 4)S 60 487 :M .263 .026(Theorem 4:)J 120 487 :M f0_12 sf .32 .032( The probability measure P over the substantive variables of a linear SEM )J 485 487 :M f4_12 sf (L)S 60 505 :M f0_12 sf .695 .069(\(recursive or non-recursive\) with jointly independent error varaibles satisfies the global)J 60 523 :M .616 .062(directed Markov property for the directed \(cyclic or acyclic\) graph )J f4_12 sf (G)S 402 523 :M f0_12 sf .625 .062( of )J f4_12 sf (L)S 427 523 :M f0_12 sf .635 .063(, i.e. if )J f2_12 sf (X)S 473 523 :M f0_12 sf .362 .036(, )J f2_12 sf (Y)S 489 523 :M f0_12 sf (,)S 60 541 :M (and )S f2_12 sf (Z)S f0_12 sf ( are disjoint sets of variables in )S f4_12 sf (G)S 249 541 :M f0_12 sf ( and )S f2_12 sf (X)S 281 541 :M f0_12 sf ( is d-separated from )S 379 541 :M f2_12 sf (Y)S 388 541 :M f0_12 sf ( given )S 421 541 :M f2_12 sf (Z)S f0_12 sf ( in )S f4_12 sf (G)S 453 541 :M f0_12 sf (, then )S 483 541 :M f2_12 sf (X)S 60 559 :M f0_12 sf (and )S f2_12 sf (Y)S 89 559 :M f0_12 sf ( are independent given )S 201 559 :M f2_12 sf (Z)S f0_12 sf ( in P.)S 60 583 :M f2_12 sf (Proof.)S 92 583 :M f0_12 sf 1.698 .17( Let )J 119 583 :M f2_12 sf (Err)S 138 583 :M f0_12 sf <28>S 142 583 :M f2_12 sf (X)S 151 583 :M f0_12 sf 1.3 .13(\) be the set of error variables corresponding to a set of substantive)J 60 601 :M .35 .035(variables )J f2_12 sf (X)S 116 601 :M f0_12 sf .534 .053(. In order to distinguish the density function for )J 355 601 :M f2_12 sf (V)S 364 601 :M f0_12 sf .511 .051( from the density function)J 60 619 :M .027 .003(for the error variables we will use )J 226 619 :M f4_12 sf (f)S f2_10 sf 0 2 rm (V)S 0 -2 rm f0_12 sf .022 .002( to represent the density function \(including marginal)J 60 637 :M .062 .006(densities\) for the latter and )J 193 637 :M f4_12 sf (f)S f2_10 sf 0 2 rm (Err)S 0 -2 rm 212 637 :M f0_12 sf .064 .006( to represent the density function of the former. If )J 454 637 :M f2_12 sf (V)S 463 637 :M f0_12 sf .075 .008( is the)J 60 655 :M (set of variables in )S 148 655 :M f4_12 sf (G)S 157 655 :M f0_12 sf (, then by hypothesis,)S endp %%Page: 36 36 %%BeginPageSetup initializepage (peter; page: 36 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (36)S gR gS 209 41 133 26 rC 342 67 :M psb currentpoint pse 209 41 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4256 div 832 3 -1 roll exch div scale currentpoint translate 64 40 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (f) 56 344 sh (f) 3217 344 sh 224 /Times-Bold f1 (Err) 161 440 sh (Err) 3322 440 sh (Err) 2657 714 sh (V) 3110 714 sh 384 ns (Err) 682 344 sh (V) 1416 344 sh 384 /Times-Roman f1 (\() 545 344 sh (\() 1279 344 sh (\)\)) 1701 344 sh (\() 3706 344 sh (\)) 4038 344 sh 224 ns (\() 3017 714 sh (\)) 3288 714 sh 384 /Symbol f1 (=) 2061 344 sh 224 ns (\316) 2516 714 sh 576 ns (\325) 2635 432 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 3835 344 sh 224 ns (e) 2386 714 sh end MTsave restore pse gR gS 0 0 552 730 rC 60 94 :M f0_12 sf (It is possible to integrate out the error variables not in )S f2_12 sf (Err)S 339 94 :M f0_12 sf (\(An\()S 362 94 :M f2_12 sf (X)S 371 94 :M f0_12 sf (\)\) and obtain)S 194 115 164 26 rC 358 141 :M psb currentpoint pse 194 115 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5248 div 832 3 -1 roll exch div scale currentpoint translate 64 40 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (f) 56 344 sh (f) 4191 344 sh 224 /Times-Bold f1 (Err) 161 440 sh (Err) 4296 440 sh (Err) 3399 714 sh (n) 4014 714 sh (X) 4242 714 sh 384 ns (Err) 682 344 sh (X) 2032 344 sh 384 /Times-Roman f1 (\() 545 344 sh (\() 1279 344 sh (\() 1895 344 sh (\)\)\)) 2316 344 sh (\() 4680 344 sh (\)) 5012 344 sh 224 ns (\() 3759 714 sh (\() 4150 714 sh (\)\)) 4420 714 sh 384 /Times-Roman f1 (An) 1419 344 sh 224 ns (A) 3853 714 sh 384 /Symbol f1 (=) 2803 344 sh 224 ns (\316) 3258 714 sh 576 ns (\325) 3609 432 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 4809 344 sh 224 ns (e) 3128 714 sh end MTsave restore pse gR gS 0 0 552 730 rC 60 168 :M f0_12 sf .28 .028(Because for each variable X in )J 214 168 :M f2_12 sf (V)S 223 168 :M f0_12 sf .296 .03(, X is a linear function of its parents in )J 415 168 :M f4_12 sf (G)S 424 168 :M f0_12 sf .293 .029( plus a unique)J 60 186 :M .112 .011(error variable )J f1_12 sf (e)S f0_10 sf 0 2 rm (X)S 0 -2 rm f0_12 sf .076 .008(, it follows that )J 217 186 :M f1_12 sf (e)S f0_10 sf 0 2 rm .05(X)A 0 -2 rm f0_12 sf .093 .009( is a linear function )J 326 186 :M f4_12 sf .05(g)A f0_10 sf 0 2 rm .06(X)A 0 -2 rm f0_12 sf .097 .01( of X and the parents of X in )J f4_12 sf (G)S 489 186 :M f0_12 sf (.)S 60 204 :M .562 .056(Hence )J 95 204 :M f2_12 sf (Err)S 114 204 :M f0_12 sf (\(An\()S 137 204 :M f2_12 sf (X)S 146 204 :M f0_12 sf .533 .053(\)\) is a function of An\()J f2_12 sf (X)S 264 204 :M f0_12 sf .513 .051(\). Following Haavelmo \(1943\) it is possible to)J 60 222 :M .079 .008(derive the density function for the set of variables An\()J f2_12 sf (X)S 330 222 :M f0_12 sf .09 .009(\) by replacing each )J 426 222 :M f1_12 sf (e)S f0_10 sf 0 2 rm (X)S 0 -2 rm f0_12 sf .036 .004( in )J f4_12 sf (f)S f2_10 sf 0 2 rm .044(Err)A 0 -2 rm 472 222 :M f0_12 sf <28>S 476 222 :M f1_12 sf (e)S f0_10 sf 0 2 rm (X)S 0 -2 rm f0_12 sf <29>S 60 240 :M (by )S f4_12 sf (g)S f0_10 sf 0 2 rm (X)S 0 -2 rm f0_12 sf (\(X,Parents\(X\)\) and multiplying by the absolute value of the Jacobian:)S 164 255 224 26 rC 388 281 :M psb currentpoint pse 164 255 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7168 div 832 3 -1 roll exch div scale currentpoint translate 64 40 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 6752 51 moveto 0 388 rlineto stroke 7038 51 moveto 0 388 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (f) 56 344 sh (f) 2909 344 sh (g) 3540 344 sh (J) 6818 344 sh 224 ns (X) 3740 440 sh 224 /Times-Bold f1 (V) 161 440 sh (Err) 3014 440 sh (n) 2571 714 sh (X) 2799 714 sh 384 ns (X) 1112 344 sh 384 /Times-Roman f1 (\() 359 344 sh (\() 975 344 sh (\)\)) 1396 344 sh (\() 3398 344 sh (\(X,) 3924 344 sh (\(X\)\)\)) 5587 344 sh 224 ns (X) 2088 714 sh (\() 2707 714 sh (\)) 2977 714 sh 384 /Times-Roman f1 (An) 499 344 sh (Parents) 4465 344 sh 224 ns (A) 2410 714 sh 384 /Symbol f1 (=) 1756 344 sh (\264) 6444 344 sh 224 ns (\316) 2268 714 sh 576 ns (\325) 2327 432 sh end MTsave restore pse gR gS 0 0 552 730 rC 60 302 :M f0_12 sf .885 .088(where )J f4_12 sf .233(J)A f0_12 sf .826 .083( is the Jacobian of the transformation. Because the transformation is linear, the)J 60 320 :M .949 .095(Jacobian is a constant. All of the terms in the multiplication are non-negative because)J 60 338 :M .172 .017(they are either a density function or a positive constant. It follows from Theorem 2 that if)J 60 356 :M f2_12 sf (X )S 72 356 :M f0_12 sf (and )S f2_12 sf (Y )S 104 356 :M f0_12 sf (are d)S 128 356 :M (-separated given )S 209 356 :M f2_12 sf (Z)S f0_12 sf ( then )S 244 356 :M f2_12 sf (X)S 253 356 :M f0_12 sf ( and )S f2_12 sf (Y)S 285 356 :M f0_12 sf ( are independent given )S 397 356 :M f2_12 sf (Z)S f0_12 sf (. )S 411 347 9 9 rC gS 1.286 1 scale 319.669 356 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 60 380 :M f2_12 sf (7)S 66 380 :M (.)S 69 380 :M (3)S 75 380 :M (.)S 78 380 :M ( )S 96 380 :M (Proof of )S 141 380 :M (Theorem 5)S 60 407 :M 1.192 .119(Theorem 5)J 118 407 :M (:)S 122 407 :M f0_12 sf 1.481 .148( In a linear SEM )J f4_12 sf (L)S 222 407 :M f0_12 sf 1.221 .122( with jointly independent error variables and directed)J 60 425 :M .61 .061(\(cyclic or acyclic\) graph )J f4_12 sf (G)S 193 425 :M f0_12 sf .623 .062( containing disjoint sets of variables )J f2_12 sf (X)S 384 425 :M f0_12 sf .98 .098(, )J 392 425 :M f2_12 sf (Y)S 401 425 :M f0_12 sf .715 .071( and )J f2_12 sf .613(Z)A f0_12 sf .5 .05(, if )J f2_12 sf (X)S 461 425 :M f0_12 sf .83 .083( is not)J 60 443 :M .249 .025(d-separated from )J 145 443 :M f2_12 sf (Y)S 154 443 :M f0_12 sf .346 .035( given )J 188 443 :M f2_12 sf .225(Z)A f0_12 sf .179 .018( in )J f4_12 sf (G)S 221 443 :M f0_12 sf .359 .036( then )J 249 443 :M f4_12 sf (L)S 256 443 :M f0_12 sf .3 .03( does not linearly entail that )J 396 443 :M f2_12 sf (X)S 405 443 :M f0_12 sf .287 .029( is independent of)J 60 461 :M f2_12 sf (Y)S 69 461 :M f0_12 sf ( given )S 102 461 :M f2_12 sf (Z)S f0_12 sf (.)S 60 485 :M f2_12 sf (Proof.)S 92 485 :M f0_12 sf 1.409 .141( Suppose that )J 166 485 :M f2_12 sf (X)S 175 485 :M f0_12 sf 1.365 .136( is not d-separated from )J 301 485 :M f2_12 sf (Y)S 310 485 :M f0_12 sf 1.553 .155( given )J 347 485 :M f2_12 sf .633(Z)A f0_12 sf 1.323 .132(. By Lemma 2)J 430 485 :M 1.056 .106(, if )J f2_12 sf (X)S 459 485 :M f0_12 sf 1.553 .155( is not)J 60 503 :M .347 .035(d-separated from )J 145 503 :M f2_12 sf (Y)S 154 503 :M f0_12 sf .481 .048( given )J 188 503 :M f2_12 sf .267(Z)A f0_12 sf .42 .042( in a cyclic graph )J 286 503 :M f4_12 sf (G,)S 298 503 :M f0_12 sf .381 .038( then there is some acyclic subgraph )J f4_12 sf .292<47D5>A 60 521 :M f0_12 sf .225 .022(of )J 74 521 :M f4_12 sf (G)S 83 521 :M f0_12 sf .171 .017( in which )J f2_12 sf (X)S 140 521 :M f0_12 sf .182 .018( is not d-separated from )J 258 521 :M f2_12 sf (Y)S 267 521 :M f0_12 sf .117 .012( given )J f2_12 sf .078(Z)A f0_12 sf .186 .019(. Geiger and Pearl \(1988\) have shown)J 60 539 :M .105 .011(that if )J 92 539 :M f2_12 sf (X)S 101 539 :M f0_12 sf .096 .01( is not d-separated from )J 217 539 :M f2_12 sf (Y)S 226 539 :M f0_12 sf .109 .011( given )J 260 539 :M f2_12 sf (Z)S f0_12 sf .09 .009( in a directed acyclic graph, then there is some)J 60 557 :M .168 .017(distribution represented by the directed acyclic graph in which )J 366 557 :M f2_12 sf (X)S 375 557 :M f0_12 sf .181 .018( is not independent of )J f2_12 sf (Y)S 60 575 :M f0_12 sf .928 .093(given )J 92 575 :M f2_12 sf .41(Z)A f0_12 sf .88 .088(, and it has been shown \(Spirtes, Glymour and Scheines, 1993\) that there is in)J 60 593 :M .168 .017(particular a linear normal distribution P in which )J 300 593 :M f2_12 sf (X)S 309 593 :M f0_12 sf .173 .017( is not independent of )J f2_12 sf (Y)S 426 593 :M f0_12 sf .206 .021( given )J f2_12 sf .136(Z)A f0_12 sf .161 .016(. If P)J 60 611 :M -.004(satisfies the global directed Markov property for )A 296 611 :M f4_12 sf <47D5>S 309 611 :M f0_12 sf -.005( it also satisfies it for )A f4_12 sf (G)S 421 611 :M f0_12 sf -.004( because every)A 60 629 :M .57 .057(d-connecting path in )J 165 629 :M f4_12 sf <47D5>S 178 629 :M f0_12 sf .612 .061( is a d-connecting path in )J f4_12 sf (G)S 316 629 :M f0_12 sf .606 .061(. Hence there is some linear normal)J 60 647 :M (distribution represented by )S 192 647 :M f4_12 sf (G)S 201 647 :M f0_12 sf ( in which )S 249 647 :M f2_12 sf (X)S 258 647 :M f0_12 sf ( is not independent of )S 365 647 :M f2_12 sf (Y)S 374 647 :M f0_12 sf ( given )S 407 647 :M f2_12 sf (Z)S f0_12 sf (. )S 421 638 9 9 rC gS 1.286 1 scale 327.447 647 :M f1_10 sf <5C>S gR endp %%Page: 37 37 %%BeginPageSetup initializepage (peter; page: 37 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (37)S gR gS 0 0 552 730 rC 60 50 :M f2_12 sf (7)S 66 50 :M (.)S 69 50 :M (4)S 75 50 :M (.)S 78 50 :M ( )S 96 50 :M (Proof of )S 141 50 :M (Theorem 6)S 60 77 :M .829 .083(Theorem 6)J 118 77 :M (:)S 122 77 :M f0_12 sf 1.03 .103( In a linear SEM )J f4_12 sf (L)S 218 77 :M f0_12 sf .884 .088( with jointly independent error variables and \(cyclic or)J 60 95 :M .221 .022(acyclic\) directed graph )J f4_12 sf (G)S 183 95 :M f0_12 sf .267 .027( containing substantive variables X, Y and )J 393 95 :M f2_12 sf .177(Z)A f0_12 sf .284 .028(, where X )J 453 95 :M f1_12 sf S 460 95 :M f0_12 sf .285 .029(\312Y and)J 60 113 :M f2_12 sf .099(Z)A f0_12 sf .156 .016( does not contain X or Y, X is d)J 223 113 :M .135 .014(-separated from Y gi)J 323 113 :M .221 .022(ven )J f2_12 sf .122(Z)A f0_12 sf .101 .01( in )J 367 113 :M f4_12 sf (G)S 376 113 :M f0_12 sf .153 .015( if and only if )J f4_12 sf (L)S 452 113 :M f0_12 sf .123 .012( linearly)J 60 131 :M (entails that )S f5_12 sf (r)S 122 133 :M f0_10 sf (XY.)S 139 133 :M f2_10 sf (Z)S 146 131 :M f0_12 sf ( = 0.)S 60 155 :M f2_12 sf (Proof.)S 92 155 :M f0_12 sf .211 .021( \(This proof for cyclic or acyclic graphs is based on the proof for acyclic graphs in)J 60 173 :M .241 .024(Verma and Pearl, 1990.\) Let )J 202 173 :M f4_12 sf <4CD5>S 213 173 :M f0_12 sf .245 .025( be a linear SEM with the same directed graph )J f4_12 sf (G)S 450 173 :M f0_12 sf .265 .027( and that)J 60 191 :M .241 .024(is the same as )J 131 191 :M f4_12 sf (L)S 138 191 :M f0_12 sf .188 .019( except that the exogenous variables are jointly normally distributed with)J 60 209 :M .121 .012(the same variances as the corresponding variables in )J 317 209 :M f4_12 sf (L)S 324 209 :M f0_12 sf .147 .015(. By Theorem 4 and )J 423 209 :M .109 .011(Theorem 5, )J f4_12 sf .062<4CD5>A 60 227 :M f0_12 sf .319 .032(linearly entails that X is independent of Y given )J f2_12 sf .153(Z)A f0_12 sf .204 .02( if and only if X is d)J 405 227 :M .225 .022(-separated from Y)J 60 245 :M 2.2 .22(given )J 94 245 :M f2_12 sf 1.651(Z)A f0_12 sf 1.318 .132( in )J f4_12 sf (G)S 132 245 :M f0_12 sf 2.095 .21(. Hence for all values of the linear coefficients and all joint normal)J 60 263 :M .022 .002(distributions over the exogenous variables, )J 270 263 :M f5_12 sf (r)S 277 265 :M f0_10 sf (XY.)S 294 265 :M f2_10 sf (Z)S 301 263 :M f0_12 sf .029 .003( = 0 if and only if X is d-separated from)J 60 281 :M .583 .058(Y given )J f2_12 sf .316(Z)A f0_12 sf .263 .026( in )J 127 281 :M f4_12 sf (G)S 136 281 :M f0_12 sf .411 .041(. Because the value of a partial correlation in a linear SEM depends only)J 60 299 :M .619 .062(on the values of the linear coefficients and the variances of the exogenous variables, )J 481 299 :M f4_12 sf <4CD5>S 60 317 :M f0_12 sf .272 .027(linearly entails )J 135 317 :M f5_12 sf (r)S 142 319 :M f0_10 sf (XY.)S 159 319 :M f2_10 sf (Z)S 166 317 :M f0_12 sf .356 .036( = 0 if and only if X is d-separated from Y given )J 407 317 :M f2_12 sf .245(Z)A f0_12 sf .195 .02( in )J f4_12 sf (G)S 440 317 :M f0_12 sf .338 .034( and hence)J 60 335 :M f4_12 sf (L)S 67 335 :M f0_12 sf .122 .012( also linearly entails that )J f5_12 sf (r)S 195 337 :M f0_10 sf (XY.)S 212 337 :M f2_10 sf (Z)S 219 335 :M f0_12 sf .156 .016( = 0 if and only if X is d)J 337 335 :M .127 .013(-separated from Y given )J 457 335 :M f2_12 sf .1(Z)A f0_12 sf .08 .008( in )J f4_12 sf (G)S 489 335 :M f0_12 sf (.)S 60 344 9 9 rC gS 1.286 1 scale 46.667 353 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 60 401 :M f2_12 sf (7)S 66 401 :M (.)S 69 401 :M (5)S 75 401 :M (.)S 78 401 :M ( )S 96 401 :M (Proof of )S 141 401 :M (Theorem 7)S 60 428 :M .069 .007(Theorem 7)J 117 428 :M (:)S 121 428 :M f0_12 sf .075 .008( \(Soundness\) Given as input an oracle for testing d-separation relations in the)J 60 446 :M (directed \(cyclic or acyclic\) graph )S 222 446 :M f4_12 sf (G)S 231 446 :M f0_12 sf (, then the output is a PAG )S f3_12 sf (Y)S 368 446 :M f0_12 sf ( for )S 388 446 :M f4_12 sf (G)S 397 446 :M f0_12 sf (.)S 60 470 :M f2_12 sf (Proof.)S 92 470 :M f0_12 sf .958 .096( The proof proceeds by showing that each section of the CCD algorithm makes)J 60 488 :M 1.048 .105(correct inferences from the d)J 206 488 :M 1.028 .103(-separation oracle for )J 317 488 :M f4_12 sf (G)S 326 488 :M f0_12 sf 1.211 .121(, to the structure of any graph in)J 60 506 :M f2_12 sf (Equiv)S 91 506 :M f0_12 sf <28>S 95 506 :M f4_12 sf (G)S 104 506 :M f0_12 sf (\).)S 60 0 7 730 rC 60 530 :M f2_12 sf 12 f8_1 :p 6 :m ( )S 64 530 :M 6 :m ( )S gR gS 0 0 552 730 rC 60 530 :M f2_12 sf 12 f8_1 :p 55.462 :m (Section \246A)S 107 0 8 730 rC 107 530 :M 6 :m ( )S 112 530 :M 6 :m ( )S gR gS 0 0 552 730 rC 60 546 :M f2_12 sf .092 .009(Lemma )J 103 546 :M (4:)S 113 546 :M f0_12 sf .091 .009( Given a PAG )J f3_12 sf (Y)S 194 546 :M f0_12 sf .086 .009( for graph )J f4_12 sf (G)S 253 546 :M f0_12 sf .098 .01( with vertex set )J 331 546 :M f2_12 sf (V)S 340 546 :M f0_12 sf .093 .009(, if at least one of the following)J 60 562 :M (holds:)S 73 578 :M (\(i\) X is a parent of Y in )S f4_12 sf (G)S 197 578 :M f0_12 sf (, or)S 73 594 :M (\(ii\) Y is a parent of X in )S 192 594 :M f4_12 sf (G)S 201 594 :M f0_12 sf (, or)S 73 610 :M .134 .013(\(iii\) there is some vertex Z which is a child of both X and Y, such that Z is an ancestor)J 82 626 :M (of either X or Y \(or both\))S 60 642 :M 2.126 .213(then X and Y are p)J 166 642 :M 1.78 .178(-adjacent in )J 229 642 :M f3_12 sf (Y)S 239 642 :M f0_12 sf 1.885 .188(, i.e. X and Y are d-connected given any subset)J 60 658 :M f2_12 sf (S)S 67 658 :M f1_12 sf S 76 658 :M f2_12 sf (V)S 85 658 :M f0_12 sf (\\{X,Y} of the other vertices in )S f4_12 sf (G)S 243 658 :M f0_12 sf (.)S endp %%Page: 38 38 %%BeginPageSetup initializepage (peter; page: 38 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (38)S gR gS 0 0 552 730 rC 60 56 :M f2_12 sf .049 .005(Proof. )J 95 56 :M f0_12 sf .049 .005(If \(i\) holds then the path X)J f1_12 sf S 235 56 :M f0_12 sf .054 .005(Y d-connects X and Y given any subset )J 429 56 :M f2_12 sf (S)S 436 56 :M f1_12 sf S 445 56 :M f2_12 sf (V)S 454 56 :M f0_12 sf (\\{X,Y},)S 60 74 :M .28 .028(hence X and Y are p-adjacent in any PAG )J 268 74 :M f3_12 sf (Y)S 278 74 :M f0_12 sf .25 .025( for graph )J f4_12 sf (G)S 338 74 :M f0_12 sf .275 .027(. The case in which \(ii\) holds is)J 60 92 :M (equally trivial: X)S 142 92 :M f1_12 sf S 154 92 :M f0_12 sf (Y is a d-connecting path given any set )S 341 92 :M f2_12 sf (S)S 348 92 :M f1_12 sf S 357 92 :M f2_12 sf (V)S 366 92 :M f0_12 sf (\\{X,Y}.)S 60 116 :M .933 .093(If \(iii\) holds there is a common child \(Z\) of X and Y which is an ancestor of X or Y;)J 60 134 :M 1.694 .169(therefore either there is a directed path X)J 276 134 :M f1_12 sf S 288 134 :M f0_12 sf (Z)S f1_12 sf S 307 134 :M f0_12 sf (A)S 316 136 :M f0_9 sf (1)S 321 134 :M f1_12 sf S 333 134 :M f0_12 sf S 354 136 :M f0_10 sf (n)S f1_12 sf 0 -2 rm S 0 2 rm 371 134 :M f0_12 sf 1.857 .186(Y \(n\312)J cF f1_12 sf .186A sf 1.857 .186(\3120\), or there is a)J 60 152 :M .585 .059(directed path Y)J 136 152 :M f1_12 sf S 148 152 :M f0_12 sf (Z)S f1_12 sf S 167 152 :M f0_12 sf (A)S 176 154 :M f0_9 sf (1)S 181 152 :M f1_12 sf S 193 152 :M f0_12 sf S 214 154 :M f0_10 sf (n)S f1_12 sf 0 -2 rm S 0 2 rm 231 152 :M f0_12 sf .631 .063(X. Suppose without much loss of generality that it is)J 60 170 :M .752 .075(the former. Let )J 139 170 :M f2_12 sf (S)S 146 170 :M f0_12 sf .741 .074( be an arbitrary subset of the other variables \()J 374 170 :M f2_12 sf (S)S 381 170 :M f1_12 sf S 390 170 :M f2_12 sf (V)S 399 170 :M f0_12 sf .61 .061(\\{X,Y}\). There are)J 60 188 :M (two cases to consider:)S 60 212 :M f2_12 sf (Case 1)S 94 212 :M f0_12 sf (: )S f2_12 sf (S)S 107 212 :M f0_12 sf ( )S f1_12 sf S f2_12 sf ( )S f0_12 sf ({Z, A)S 150 214 :M f0_9 sf (1)S 155 212 :M f0_12 sf S 176 214 :M f0_10 sf (n)S f0_12 sf 0 -2 rm (})S 0 2 rm 187 212 :M f1_12 sf S f0_12 sf (; in this case X)S f1_12 sf S 289 212 :M f0_12 sf (Z )S f1_12 sf S 311 212 :M f0_12 sf (Y is a d-connecting path.)S 60 236 :M f2_12 sf (Case 2: S)S f0_12 sf ( )S f1_12 sf S f2_12 sf S f0_12 sf ({Z, A)S 150 238 :M f0_9 sf (1)S 155 236 :M f0_12 sf S 176 238 :M f0_10 sf (n)S f0_12 sf 0 -2 rm (})S 0 2 rm 187 236 :M f1_12 sf <3D20C6>S f0_12 sf (; then X)S 245 236 :M f1_12 sf S 257 236 :M f0_12 sf (Z)S f1_12 sf S 276 236 :M f0_12 sf (A)S 285 238 :M f0_9 sf (1)S 290 236 :M f1_12 sf S 302 236 :M f0_12 sf S 323 238 :M f0_10 sf (n)S f1_12 sf 0 -2 rm S 0 2 rm 340 236 :M f0_12 sf (Y is a d)S 377 236 :M (-connecting path. )S f1_12 sf <5C>S 60 268 :M f2_12 sf (Lemma 5:)S 112 268 :M f0_12 sf ( In a graph )S 167 268 :M f4_12 sf (G)S 176 268 :M f0_12 sf (, with vertices )S 247 268 :M f2_12 sf (V)S 256 268 :M f0_12 sf (, if the following hold:)S 364 265 :M f0_9 sf (1)S 368 265 :M (6)S 69 284 :M f0_12 sf (\(i\) X is not a parent of Y in )S 203 284 :M f4_12 sf (G)S 69 300 :M f0_12 sf (\(ii\) Y is not a parent of X in )S 206 300 :M f4_12 sf (G)S 215 300 :M f0_12 sf (, and)S 69 316 :M (\(iii\) there is no vertex Z s.t. Z is a common child of X and Y, and an ancestor of X or Y,)S 60 338 :M (then for any set )S f2_12 sf (Q)S f0_12 sf (, X and Y are d-separated given )S 300 338 :M f2_12 sf (T)S f0_12 sf (, defined as follows:)S 60 356 :M f2_12 sf (S )S 70 356 :M f0_12 sf (= Children\(X\) )S f1_12 sf S f0_12 sf ( An\({X,Y})S 204 356 :M f1_12 sf S f0_12 sf ( )S f2_12 sf (Q)S f0_12 sf <29>S 60 374 :M f2_12 sf (T = )S 81 374 :M f0_12 sf (\(Parents\()S f2_12 sf (S )S 134 374 :M f1_12 sf S 152 374 :M f0_12 sf (X}\) )S f1_12 sf S f2_12 sf (S)S 192 374 :M f0_12 sf (\)\\\(Descendants\(Children\(X\))S 327 374 :M f1_12 sf S f0_12 sf (Children\(Y\)\) )S 402 374 :M f1_12 sf S f0_12 sf ({X,Y}\).)S 60 398 :M f2_12 sf (Proof.)S 92 398 :M f0_12 sf 1.023 .102( Every vertex in )J 178 398 :M f2_12 sf (S)S 185 398 :M f0_12 sf 1.092 .109( is an ancestor of X or Y or )J 332 398 :M f2_12 sf .579(Q)A f0_12 sf .903 .09(. Every vertex in )J 430 398 :M f2_12 sf .596(T)A f0_12 sf .902 .09( is either a)J 60 416 :M 1.027 .103(parent of X, a vertex in )J f2_12 sf (S)S 190 416 :M f0_12 sf 1.073 .107(, or a parent of a vertex in )J f2_12 sf (S)S 336 416 :M f0_12 sf 1.054 .105(, hence every vertex in )J 456 416 :M f2_12 sf .739(T)A f0_12 sf .936 .094( is an)J 60 434 :M .066 .007(ancestor of X or Y or )J f2_12 sf (Q)S f0_12 sf .071 .007(. We will now show that if \(i\), \(ii\), and \(iii\) hold then X and Y are)J 60 452 :M (d-separated given )S 147 452 :M f2_12 sf (T)S f0_12 sf (.)S 60 480 :M .166 .017(Suppose, on the contrary that there is a path d-connecting X and Y given )J f2_12 sf .08(T)A f0_12 sf .129 .013(. Let W be the)J 60 496 :M .71 .071(first vertex on the path from X to Y. \(It follows from \(i\) and \(ii\) that W)J 418 496 :M f1_12 sf S 425 496 :M f0_12 sf (Y)S 434 496 :M f1_12 sf .286 .029(.\) )J f0_12 sf .943 .094(There are)J 60 512 :M (two cases to consider:)S 60 537 :M f2_12 sf (Case 1)S 94 537 :M f0_12 sf ( The path contains X)S 194 537 :M f1_12 sf S 206 537 :M f0_12 sf (W\311Y.)S 74 562 :M f2_12 sf 1.031 .103(Subcase A:)J 133 562 :M f0_12 sf 1.439 .144( W is not a descendant of a common child of X and Y. If W is not a)J 74 580 :M .034 .003(descendant of a common child, then W)J 262 580 :M f1_12 sf S 271 580 :M f2_12 sf (T)S f0_12 sf .037 .004( \(Since W is a parent of X\). Thus since W is)J 74 598 :M (a non-collider on the path, the path is not d-connecting given )S 368 598 :M f2_12 sf (T.)S 60 673 :M f0_12 sf ( )S 60 670.48 -.48 .48 204.48 670 .48 60 670 @a 60 684 :M f0_9 sf (1)S 64 684 :M (6)S 68 687 :M f0_10 sf (i.e. None of the conditions in the antecedent of )S 258 687 :M (Lemma 4 hold.)S endp %%Page: 39 39 %%BeginPageSetup initializepage (peter; page: 39 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (39)S gR gS 0 0 552 730 rC 73 50 :M f2_12 sf .118 .012(Subcase B: )J 133 50 :M f0_12 sf .126 .013(W is a descendant of a common child of X and Y. In this case since X is a)J 73 66 :M .142 .014(child of W, it follows that X is a descendant of some common child Z of X and Y. But)J 73 82 :M (this is contrary to the assumption that \(iii\) holds.)S 60 110 :M f2_12 sf (Case 2 )S 97 110 :M f0_12 sf (The path contains X)S 194 110 :M f1_12 sf S 206 110 :M f0_12 sf (W\311 Y.)S 73 139 :M f2_12 sf .435 .043(Subcase A)J 127 139 :M f0_12 sf .564 .056(: W is not a descendant of a common child of X and Y. Let V be the next)J 73 155 :M (vertex on the path.)S 87 183 :M f2_12 sf .222 .022(Sub-subcase a:)J 164 183 :M f0_12 sf .304 .03( The path contains X)J f1_12 sf S 278 183 :M f0_12 sf (W)S f1_12 sf S 301 183 :M f0_12 sf .315 .032(V\311Y. If this path is d-connecting then)J 87 200 :M .471 .047(some descendant of W is in )J f2_12 sf .242(T)A f0_12 sf .473 .047(, but then some descendant of W is an ancestor of X)J 87 216 :M .285 .028(or Y or )J f2_12 sf .268(Q)A f0_12 sf .372 .037(. Hence W is an ancestor of X, Y or )J 315 216 :M f2_12 sf .211(Q)A f0_12 sf .328 .033(. So if some descendant of W is in)J 87 232 :M f2_12 sf .973(T)A f0_12 sf 1.259 .126(, then W is in )J f2_12 sf (S)S 179 232 :M f0_12 sf 1.297 .13(. Moreover, since W is \(by hypothesis\) not a descendant of a)J 87 248 :M (common child, V )S 174 248 :M f1_12 sf S 184 248 :M f0_12 sf (Y.)S 87 277 :M .501 .05(Now V is a parent of W, and W)J f1_12 sf S 254 277 :M f2_12 sf (S)S 261 277 :M f0_12 sf .488 .049(. Moreover V is not a descendant of a common)J 87 294 :M 1.544 .154(child since in that instance W would also be a descendant of a common child,)J 87 310 :M .106 .011(contrary to hypothesis. X )J 212 310 :M f1_12 sf S 219 310 :M f0_12 sf .136 .014( V )J f1_12 sf .186 .019J 244 310 :M f0_12 sf .133 .013(Y, so V)J 282 310 :M f1_12 sf S 291 310 :M f2_12 sf .055(T)A f0_12 sf .111 .011(. Thus V occurs as a non-collider, but V)J 87 327 :M f1_12 sf S 96 327 :M f0_12 sf ( )S f2_12 sf (T)S f0_12 sf (, hence the path fails to d-connect given )S f2_12 sf (T)S f0_12 sf (.)S 87 356 :M f2_12 sf .548 .055(Sub-subcase b: )J 169 356 :M f0_12 sf .508 .051(The path contains X)J f1_12 sf S 280 356 :M f0_12 sf (W)S f1_12 sf S 303 356 :M f0_12 sf .643 .064(V\311Y. If some path X)J 414 356 :M f1_12 sf S 426 356 :M f0_12 sf (W)S f1_12 sf S 449 356 :M f0_12 sf .621 .062(V\311Y d-)J 87 373 :M .186 .019(connects given )J f2_12 sf .07(T)A f0_12 sf .12 .012( then W is either an ancestor of Y or some vertex in )J 424 373 :M f2_12 sf (T)S f0_12 sf .11 .011(. However if)J 87 389 :M .837 .084(W is an ancestor of some vertex in )J 266 389 :M f2_12 sf .553(T)A f0_12 sf .848 .085(, then W is an ancestor of X, Y or )J 451 389 :M f2_12 sf .278(Q)A f0_12 sf .571 .057(, since)J 87 405 :M .196 .02(every vertex in )J f2_12 sf .096(T)A f0_12 sf .141 .014( is an ancestor of X, Y or )J 296 405 :M f2_12 sf .07(Q)A f0_12 sf .137 .014(. Hence W)J 357 405 :M f1_12 sf S 366 405 :M f2_12 sf (S)S 373 405 :M f0_12 sf .162 .016(, and thus since W is \(by)J 87 422 :M .909 .091(hypothesis\) not a descendant of a common child of X and Y, and X\312)J 431 422 :M f1_12 sf S 438 422 :M f0_12 sf S f1_12 sf S 462 422 :M f0_12 sf .969 .097(\312Y, W)J 87 439 :M f1_12 sf S 96 439 :M f2_12 sf 1.102(T)A f0_12 sf 2.207 .221(. Since W occurs as a non-collider on this path, it follows that any path)J 87 456 :M (X)S 96 456 :M f1_12 sf S 108 456 :M f0_12 sf (W)S f1_12 sf S 131 456 :M f0_12 sf .43 .043(V\311Y fails to d-connect given )J 283 456 :M f2_12 sf .211(T)A f0_12 sf .423 .042(. \(This allows for the possibility that V =)J 87 473 :M (Y\).)S 73 501 :M f2_12 sf .812 .081(Subcase B: )J f0_12 sf .8 .08(W is a descendant of a common child. Thus Descendants\(W\) )J f1_12 sf .391A f0_12 sf .127 .013( )J 456 501 :M f2_12 sf .571(T)A f0_12 sf .396 .04( = )J f1_12 sf .919A 73 518 :M f0_12 sf .604 .06(since descendants of W are also descendants of common children of X and Y and so)J 73 534 :M (cannot occur in )S 150 534 :M f2_12 sf (T)S f0_12 sf (.)S 73 562 :M 1.033 .103(Since no descendant of W has been conditioned on, if W occurs on a d-connecting)J 73 578 :M 1.886 .189(path then W is a non-collider. We can show that any other vertex on such a d-)J 73 594 :M .842 .084(connecting path must be a non-collider. Suppose that there is a collider on the path,)J 73 610 :M .651 .065(then take the first collider on the path after W, let us say , so that the path)J 73 626 :M .038 .004(now takes the form: X)J 181 626 :M f1_12 sf S 193 626 :M f0_12 sf (W)S f1_12 sf S 216 626 :M f0_12 sf (V)S 225 626 :M f1_12 sf S 237 626 :M f0_12 sf S f1_12 sf S 261 626 :M f0_12 sf S f1_12 sf S 285 626 :M f0_12 sf (A)S 294 626 :M f1_12 sf S 306 626 :M f0_12 sf (B)S f1_12 sf S 326 626 :M f0_12 sf .035 .003(C\311Y. Since is the first)J 73 643 :M .253 .025(collider after V, it follows that B is a descendant of W. But if the path is d-connecting)J 73 659 :M .23 .023(then some descendant of B, say D, has been conditioned on, i.e. D)J 396 659 :M f1_12 sf S 405 659 :M f0_12 sf ( )S f2_12 sf .132(T)A f0_12 sf .274 .027(. But then since)J endp %%Page: 40 40 %%BeginPageSetup initializepage (peter; page: 40 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (40)S gR gS 0 0 552 730 rC 73 54 :M f0_12 sf .786 .079(D is a descendant of B, and B is a descendant of W, D)J 348 54 :M f1_12 sf S 357 54 :M f0_12 sf .611 .061(Descendants\(W\) which is a)J 73 71 :M (contradiction since Descendants\(W\) )S 250 71 :M f1_12 sf S f0_12 sf ( )S f2_12 sf (T)S f0_12 sf ( = )S 283 71 :M f1_12 sf S 73 100 :M f0_12 sf .248 .025(As there are no colliders on the path it follows that W is an ancestor of Y. But then W)J 73 116 :M 1.905 .19(is a descendant of a common child of X and Y, )J 332 116 :M f4_12 sf .673(and)A f0_12 sf 1.638 .164( an ancestor of Y. But this)J 73 132 :M (contradicts \(iii\).)S 60 160 :M (This completes the proof of )S 196 160 :M (Lemma 5)S 242 160 :M (.)S f1_12 sf <5C>S 60 193 :M f2_12 sf .543 .054(Corollary 1:)J 124 193 :M f0_12 sf .844 .084( Given a graph )J 203 193 :M f4_12 sf (G)S 212 193 :M f0_12 sf .725 .072(, and PAG )J f3_12 sf (Y)S 278 193 :M f0_12 sf .95 .095( for )J 301 193 :M f4_12 sf (G)S 310 193 :M f0_12 sf .804 .08(, X and Y are p-adjacent i)J 439 193 :M .461 .046(n )J f3_12 sf (Y)S 459 193 :M f0_12 sf .877 .088( if and)J 60 209 :M (only if at least one of the following holds in )S f4_12 sf (G)S 282 209 :M f0_12 sf (:)S 73 225 :M (\(i\) X is a parent of Y, or)S 73 241 :M (\(ii\) Y is a parent of X, or)S 73 257 :M .134 .013(\(iii\) there is some vertex Z which is a child of both X and Y, such that Z is an ancestor)J 82 273 :M (of either X or Y \(or both\).)S 60 289 :M f2_12 sf (Proof.)S 92 289 :M f0_12 sf 1.73 .173( 'If' is proved by )J 185 289 :M 1.551 .155(Lemma 4. 'Only if' follows from )J 358 289 :M 2.1 .21(Lemma 5 with )J f2_12 sf 1.272 .127(Q )J f0_12 sf 1.099 .11(= )J 465 289 :M f1_12 sf S 475 289 :M f0_12 sf 1.961 .196( by)J 60 305 :M (contraposition.)S 132 305 :M f1_12 sf ( \\)S 60 329 :M f0_12 sf .327 .033(We have obtained necessary and sufficient conditions on a graph )J f4_12 sf (G)S 389 329 :M f0_12 sf .383 .038( for a pair of vertices)J 60 347 :M .441 .044(to be p)J 94 347 :M (-adjacen)S 134 347 :M .434 .043(t in any PAG for )J 220 347 :M f4_12 sf (G)S 229 347 :M f0_12 sf .395 .04(. Thus it makes sense to speak of a pair of vertices X,)J 60 365 :M .895 .09(Y being )J 104 365 :M f2_12 sf (p)S 111 365 :M .679 .068(-adjacent in graph )J f4_12 sf (G)S 220 365 :M f0_12 sf .815 .081(, where this means that at least one of \(i\), \(ii\) and \(iii\))J 60 383 :M (holds.)S 60 407 :M .773 .077(The previous Corollary tells us that a pair of vertices are p)J 353 407 :M .632 .063(-adjacent in )J f4_12 sf (G)S 422 407 :M f0_12 sf .885 .089( if and only if)J 60 425 :M .711 .071(they are p)J 110 425 :M .629 .063(-adjacent in every PAG for )J f4_12 sf (G)S 256 425 :M f0_12 sf .713 .071(. For this reason we will often refer to a pair of)J 60 443 :M .871 .087(variables as p-adjacent without specifying whether we are referring to the graph or the)J 60 461 :M (PAG.)S 60 485 :M f2_12 sf .341 .034(Corollary )J 114 485 :M (2:)S 124 485 :M f0_12 sf .43 .043( In a graph )J f4_12 sf (G)S 190 485 :M f0_12 sf .505 .05(, if X and Y are d)J 278 485 :M .418 .042(-separated by some set )J 392 485 :M f2_12 sf (R)S 401 485 :M f0_12 sf .484 .048(, then X and Y are)J 60 503 :M .125 .012(d-separated by a set )J 158 503 :M f2_12 sf .059(T)A f0_12 sf .12 .012( in which every vertex is an ancestor of X or Y. Furthermore, either)J 60 521 :M f2_12 sf (T)S f0_12 sf ( is a subset of the vertices p-adjacent to X or X is an ancestor of Y in )S 400 521 :M f4_12 sf (G)S 409 521 :M f0_12 sf (.)S 60 545 :M f2_12 sf (Proof)S 89 545 :M f0_12 sf .75 .075(. Since X and Y are d)J 198 545 :M .645 .065(-separated by some set )J 313 545 :M f2_12 sf (R)S 322 545 :M f0_12 sf .769 .077(, X and Y are not p)J 420 545 :M .529 .053(-adjacent in )J f4_12 sf (G)S 489 545 :M f0_12 sf (.)S 60 563 :M (Apply )S 93 563 :M (Lemma 5)S 139 563 :M (, with )S f2_12 sf (Q )S f0_12 sf (= )S 191 563 :M f1_12 sf S 201 563 :M f0_12 sf (. In that case)S 60 592 :M f2_12 sf (S )S 70 592 :M f0_12 sf (= Children\(X\) )S f1_12 sf S f0_12 sf ( An\({X,Y}\))S 60 617 :M f2_12 sf (T = )S 81 617 :M f0_12 sf (\(Parents\()S f2_12 sf (S )S 134 617 :M f1_12 sf S 152 617 :M f0_12 sf (X}\) )S f1_12 sf S f2_12 sf (S)S 192 617 :M f0_12 sf (\)\\\(Descendants\(Children\(X\))S 327 617 :M f1_12 sf S f0_12 sf (Children\(Y\)\) )S 402 617 :M f1_12 sf S f0_12 sf ({X,Y}\))S 60 641 :M .549 .055(It follows from Lemma 5)J 185 641 :M .592 .059( that X and Y are d-separated given )J 364 641 :M f2_12 sf .279(T)A f0_12 sf .499 .05(. Every vertex in )J f2_12 sf (S)S 465 641 :M f0_12 sf .681 .068( is an)J 60 659 :M .285 .028(ancestor of X or Y. Every vertex in )J f2_12 sf .154(T)A f0_12 sf .232 .023( is either a parent of X, a vertex in )J f2_12 sf (S)S 420 659 :M f0_12 sf .274 .027(, or a parent of)J 60 677 :M (a vertex in )S 114 677 :M f2_12 sf (S)S 121 677 :M f0_12 sf (, and hence every vertex in )S f2_12 sf (T)S f0_12 sf ( is an ancestor of X or Y.)S endp %%Page: 41 41 %%BeginPageSetup initializepage (peter; page: 41 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (41)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf 1.075 .107(We will now show that either )J 215 56 :M f2_12 sf .635(T)A f0_12 sf 1.079 .108( is a subset of the vertices p-adjacent to X or X is an)J 60 74 :M (ancestor of Y in )S 140 74 :M f4_12 sf (G)S 149 74 :M f0_12 sf (. Every vertex in )S 232 74 :M f2_12 sf (T)S f0_12 sf .006 .001( is either a parent of X, a child of X, or a parent V of)J 60 92 :M .186 .019(some vertex C in )J 146 92 :M f2_12 sf (S)S 153 92 :M f0_12 sf .179 .018(, where C is also a child of X. Any vertex in the first two categories is)J 60 110 :M .199 .02(clearly p)J 102 110 :M .254 .025(-adjacent to X. Since C is in )J 242 110 :M f2_12 sf (S)S 249 110 :M f0_12 sf .263 .026(, C is an ancestor of X or Y. If C is an ancestor of)J 60 128 :M (X, then V is p)S 127 128 :M (-adjacent to X. If C is an ancestor of Y, then X is an ancestor of Y.)S f1_12 sf <5C>S 60 152 :M f2_12 sf 4.054 .405(Lemma 6)J f0_12 sf 1.59 .159(: In a graph )J 182 152 :M f4_12 sf (G)S 191 152 :M f0_12 sf 2.131 .213(, if A and B are not p-adjacent, then either A and B are)J 60 170 :M 2.671 .267(d-separated given a set )J f2_12 sf 1.215(T)A f2_9 sf 0 2 rm .986(A)A 0 -2 rm f0_12 sf 2.016 .202( of vertices p-adjacent to A or by a set )J 418 170 :M f2_12 sf .807(T)A f2_9 sf 0 2 rm .605(B)A 0 -2 rm f0_12 sf 1.734 .173( of vertices)J 60 188 :M (p-adjacent to B.)S 60 212 :M f2_12 sf (Proof.)S 92 212 :M f0_12 sf .446 .045( By Corollary 2, if A and B are not p)J 274 212 :M .387 .039(-adjacent then A and B are d-separated gi)J 475 212 :M (ven)S 60 230 :M f2_12 sf (T)S f2_9 sf 0 2 rm (A)S 0 -2 rm f0_12 sf ( where:)S 60 254 :M f2_12 sf (S)S 67 256 :M f2_9 sf (A)S f2_12 sf 0 -2 rm ( )S 0 2 rm f0_12 sf 0 -2 rm (= Children\(A\) )S 0 2 rm f1_12 sf 0 -2 rm S 0 2 rm f0_12 sf 0 -2 rm ( An\({A,B}\))S 0 2 rm 60 278 :M f2_12 sf (T)S f2_9 sf 0 2 rm (A)S 0 -2 rm f2_12 sf ( = )S 87 278 :M f0_12 sf (\(Parents\()S f2_12 sf (S )S 140 278 :M f1_12 sf S 158 278 :M f0_12 sf (A}\) )S f1_12 sf S f2_12 sf (S)S 198 278 :M f0_12 sf (\)\\\(Descendants\(Children\(A\))S 333 278 :M f1_12 sf S f0_12 sf (Children\(B\)\) )S 407 278 :M f1_12 sf S f0_12 sf ({A,B}\),)S 60 302 :M f2_12 sf .273 .027(Case 1:)J 99 302 :M f0_12 sf .31 .031( A is not an ancestor of B. From Corollary 2, since A is not an ancestor of B, )J 478 302 :M f2_12 sf (T)S f2_9 sf 0 2 rm (A)S 0 -2 rm 60 320 :M f1_12 sf S 69 320 :M f0_12 sf ( {X\312| X\312p-adjacent to A}.)S 60 344 :M f2_12 sf 1.576 .158(Case 2)J 97 344 :M f0_12 sf 1.654 .165(: B is not an ancestor of A. It follows again by symmetry that A and B are)J 60 362 :M (d-separated given )S 147 362 :M f2_12 sf (T)S f2_9 sf 0 2 rm (B)S 0 -2 rm f0_12 sf (, where )S f2_12 sf (T)S f2_9 sf 0 2 rm (B)S 0 -2 rm f0_12 sf ( is defined symmetrically to )S 350 362 :M f2_12 sf (T)S f2_9 sf 0 2 rm (A)S 0 -2 rm f0_12 sf ( in Case 1.)S 60 386 :M f2_12 sf .037 .004(Case 3:)J 98 386 :M f0_12 sf .041 .004( B is an ancestor of A and A is an ancestor of B. Now any vertex V in )J f2_12 sf (T)S f2_9 sf 0 2 rm (A)S 0 -2 rm f0_12 sf .045 .005( is either)J 60 404 :M .071 .007(a child of A, a parent of A or a parent of some vertex C in )J f2_12 sf (S)S 349 406 :M f2_9 sf (A)S f0_12 sf 0 -2 rm .067 .007(, which is itself a child of A.)J 0 2 rm 60 422 :M .632 .063(Clearly vertices in the first two categories are p-adjacent to A; as before, vertices in the)J 60 440 :M .24 .024(last category are p)J 149 440 :M .263 .026(-adjacent to A if C is an ancestor of A. Any vertex in )J f2_12 sf (S)S 417 442 :M f2_9 sf .113 .011(A )J f0_12 sf 0 -2 rm .308 .031(is an ancestor)J 0 2 rm 60 458 :M .668 .067(of A or B. Since A is an ancestor of B, and B is an ancestor of A, it follows that every)J 60 476 :M .121 .012(vertex in )J f2_12 sf (S)S 112 478 :M f2_9 sf .064(A)A f0_12 sf 0 -2 rm .141 .014( is an ancestor of A, hence every vertex in )J 0 2 rm 326 476 :M f2_12 sf .07(T)A f2_9 sf 0 2 rm .057(A)A 0 -2 rm f0_12 sf .129 .013( is p-adjacent to A. \(Note that it)J 60 494 :M (is also the case that every vertex in )S 231 494 :M f2_12 sf (T)S f2_9 sf 0 2 rm (B)S 0 -2 rm f0_12 sf ( is p-adjacent to B.\) )S f1_12 sf <5C>S 60 518 :M f0_12 sf .173 .017(Suppose that the input to the algorithm is a d-separation oracle for a directed graph )J 464 518 :M f4_12 sf (G)S 473 518 :M f0_12 sf .208 .021(. To)J 60 536 :M 1.254 .125(find a set which d-separates some pair of variables A and B in )J 385 536 :M f4_12 sf (G)S 394 536 :M f14_13 sf 1.856 .186( )J 399 536 :M f0_12 sf .975 .098(the algorithm tests)J 60 554 :M .449 .045(subsets of the vertices which are p)J 229 554 :M .435 .044(-adjacent to A in )J f3_12 sf (Y)S 324 554 :M f0_12 sf .449 .045(, and subsets of vertices which are)J 60 572 :M 1.319 .132(p-adjacent to B in )J 156 572 :M f3_12 sf (Y)S 166 572 :M f0_12 sf 1.288 .129( to see if they d-separate A and B. Since the vertices which are)J 60 590 :M .368 .037(p-adjacent to A and B in )J 183 590 :M f4_12 sf (G)S 192 590 :M f0_12 sf .36 .036( are at all times a subset of the vertices p)J 392 590 :M .349 .035(-adjacent to A and B)J 60 608 :M .076 .008(in )J f3_12 sf (Y)S 82 605 :M f0_9 sf (1)S 86 605 :M (7)S 90 608 :M f0_12 sf .125 .012( it follows from )J 169 608 :M .109 .011(Lemma 6)J 215 608 :M .121 .012( that step \246A is guaranteed to find a set which d)J 445 608 :M (-separates)S 60 626 :M (A and B, if any set d-separates A and B in )S f4_12 sf (G)S 274 626 :M f0_12 sf (.)S 60 662 :M ( )S 60 659.48 -.48 .48 204.48 659 .48 60 659 @a 60 673 :M f0_9 sf (1)S 64 673 :M (7)S 68 676 :M f0_10 sf .23 .023(This is because if a pair of vertices X,Y are p)J 253 676 :M .212 .021(-adjacent in )J f4_10 sf .102(G)A f0_10 sf .206 .021( then no set is found which d-separates them,)J 60 687 :M (and hence the edge between X and Y in )S f3_10 sf (Y)S 229 687 :M f0_10 sf ( is never deleted.)S endp %%Page: 42 42 %%BeginPageSetup initializepage (peter; page: 42 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (42)S gR gS 60 0 7 730 rC 60 50 :M f2_12 sf 12 f8_1 :p 6 :m ( )S 64 50 :M 6 :m ( )S gR gS 0 0 552 730 rC 60 50 :M f2_12 sf 12 f8_1 :p 54.803 :m (Section \246B)S 107 0 8 730 rC 107 50 :M 6 :m ( )S 112 50 :M 6 :m ( )S gR gS 0 0 552 730 rC 60 74 :M f0_12 sf .198 .02(The next lemma gives an important property of d-separating sets that are found through a)J 60 92 :M .472 .047(search which never tests a set unless it has already tested every proper subset of that set)J 60 110 :M (\(as in the CCD algorithm.\))S 60 134 :M f2_12 sf (Lemma 7:)S 112 134 :M f0_12 sf .019 .002( Suppose that in a graph )J 232 134 :M f4_12 sf (G)S 241 134 :M f0_12 sf .019 .002(, Y is not an ancestor of X or Z or )J f2_12 sf (R)S 415 134 :M f0_12 sf .02 .002(. If there is a set)J 60 152 :M f2_12 sf (S)S 67 152 :M f0_12 sf 2.38 .238(, )J 76 152 :M f2_12 sf (R)S 85 152 :M f1_12 sf S 94 152 :M f2_12 sf (S)S 101 152 :M f0_12 sf 1.87 .187(, such that Y)J 170 152 :M f1_12 sf S 179 152 :M f2_12 sf (S)S 186 152 :M f0_12 sf 1.872 .187( and every proper subset )J f2_12 sf .935(T)A f0_12 sf .898 .09( s.t. )J f2_12 sf (R)S 361 152 :M f1_12 sf S 370 152 :M f2_12 sf (T)S f1_12 sf S 387 152 :M f2_12 sf (S)S 394 152 :M f0_12 sf 1.636 .164(, not containing Y,)J 60 170 :M (d-connects X and Z, in )S 172 170 :M f4_12 sf (G)S 181 170 :M f0_12 sf ( , then )S 214 170 :M f2_12 sf (S)S 221 170 :M f0_12 sf ( d-connects X and Z in )S 334 170 :M f4_12 sf (G)S 343 170 :M f0_12 sf (.)S 60 194 :M f2_12 sf .69 .069(Proof. )J 96 194 :M f0_12 sf .849 .085(Let )J 117 194 :M f2_12 sf .57(T)A f0_10 sf 0 -3 rm .486 .049(* )J 0 3 rm 134 194 :M f0_12 sf .423 .042(= An\({X,Z})J f1_12 sf .132A f2_12 sf (R)S 212 194 :M f0_12 sf <29>S 216 194 :M f1_12 sf S f2_12 sf (S)S 232 194 :M f0_12 sf .657 .066(. Now, )J f2_12 sf (R)S 278 194 :M f1_12 sf S 287 194 :M f2_12 sf .456(T)A f0_10 sf 0 -3 rm .285(*)A 0 3 rm f0_12 sf .577 .058(, and )J f2_12 sf .456(T)A f0_10 sf 0 -3 rm .285(*)A 0 3 rm f0_12 sf .713 .071( is a proper subset of )J 451 194 :M f2_12 sf (S)S 458 194 :M f0_12 sf .849 .085(, so by)J 60 212 :M 1.356 .136(hypothesis there is a d-connecting path, )J 267 212 :M f4_12 sf .583(P)A f0_12 sf 1.347 .135(, conditional on )J 359 212 :M f2_12 sf .744(T)A f0_10 sf 0 -3 rm .464(*)A 0 3 rm f0_12 sf 1.382 .138(. By the definition of a)J 60 230 :M .794 .079(d-connecting path, every element on )J 244 230 :M f4_12 sf .42(P)A f0_12 sf .877 .088( is either an ancestor of one of the endpoints, or)J 60 248 :M f2_12 sf .511(T)A f0_10 sf 0 -3 rm .319(*)A 0 3 rm f0_12 sf 1.176 .118(. Moreover, by definition, every element in )J f2_12 sf .511(T)A f0_10 sf 0 -3 rm .319(*)A 0 3 rm f0_12 sf .716 .072( is an ancestor of X or Z or )J 452 248 :M f2_12 sf (R)S 461 248 :M f0_12 sf .98 .098(. Thus)J 60 266 :M 1.296 .13(every element on the path )J 197 266 :M f4_12 sf .865(P)A f0_12 sf 1.312 .131( is an ancestor of X or Z or )J f2_12 sf (R)S 362 266 :M f0_12 sf 1.314 .131(. Since neither Y nor any)J 60 284 :M .218 .022(element in )J f2_12 sf (S)S 121 284 :M f0_12 sf .073<5C>A f2_12 sf .175(T)A f0_10 sf 0 -3 rm .109(*)A 0 3 rm f0_12 sf .242 .024( is an ancestor of X or Z or )J f2_12 sf (R)S 281 284 :M f0_12 sf .267 .027(, it follows that no vertex in )J 420 284 :M f2_12 sf (S)S 427 284 :M f0_12 sf .071<5C>A f2_12 sf .169(T)A f2_10 sf 0 -3 rm .106(*)A 0 3 rm f0_12 sf .21 .021( lies on )J 482 284 :M f4_12 sf (P)S f0_12 sf (.)S 60 302 :M .48 .048(Since )J 91 302 :M f2_12 sf .24(T)A f2_10 sf 0 -3 rm .187 .019(* )J 0 3 rm f1_12 sf S 116 302 :M f2_12 sf .655 .066(S )J 127 302 :M f0_12 sf .508 .051(the only way in which )J 241 302 :M f4_12 sf .23(P)A f0_12 sf .496 .05( could fail to d-connect given )J f2_12 sf .276 .028(S )J 407 302 :M f0_12 sf .48 .048(would be if some)J 60 320 :M (element of )S 114 320 :M f2_12 sf (S)S 121 320 :M f0_12 sf <5C>S f2_12 sf (T)S f2_10 sf 0 -3 rm (*)S 0 3 rm f0_12 sf ( lay on the path. Hence )S 251 320 :M f4_12 sf (P)S f0_12 sf ( still d-connects X and Z given )S 409 320 :M f2_12 sf (S)S 416 320 :M f0_12 sf (.)S f1_12 sf <5C>S 60 344 :M f0_12 sf .371 .037(In a graph )J 114 344 :M f4_12 sf (G)S 123 344 :M f0_12 sf .356 .036( , if X and Y are d-separated given )J 295 344 :M f2_12 sf (S)S 302 344 :M f0_12 sf .306 .031(, and are d-connected given any proper)J 60 362 :M (subset of )S 106 362 :M f2_12 sf (S)S 113 362 :M f0_12 sf (, then )S 143 362 :M f2_12 sf (S)S 150 362 :M f0_12 sf ( is a )S f2_12 sf (minimal)S 215 362 :M f4_12 sf ( )S f2_12 sf (d-separating)S 283 362 :M f4_12 sf ( )S f0_12 sf (set for X and Y in )S f4_12 sf (G)S 384 362 :M f0_12 sf (.)S 60 386 :M (The following corollary is useful here:)S 60 410 :M f2_12 sf (Corollary )S 113 410 :M (3:)S 123 410 :M f0_12 sf -.008( In a graph )A 178 410 :M f4_12 sf (G)S 187 410 :M f0_12 sf (, if )S f2_12 sf (S)S 210 410 :M f0_12 sf -.005( is a minimal d-separating set for X and Y, then any vertex)A 60 428 :M (in )S f2_12 sf (S)S 79 428 :M f0_12 sf ( is an ancestor of X or Y in )S 212 428 :M f4_12 sf (G)S 221 428 :M f0_12 sf (.)S 60 452 :M f2_12 sf (Proof.)S 92 452 :M f0_12 sf 4.505 .451( The corollary follows immediately from Lemma 7)J 390 452 :M 4.302 .43(, with )J f2_12 sf (R)S 444 452 :M f0_12 sf S 457 452 :M f1_12 sf S 467 452 :M f0_12 sf 5.743 .574( via)J 60 470 :M (contraposition.)S 132 470 :M f1_12 sf <5C>S 60 494 :M f0_12 sf .073 .007(This shows that the non-collider orientation rule in \246B is correct. If A and B, and B and C)J 60 512 :M .139 .014(are p)J 85 512 :M .075 .008(-adjacent, but )J f2_12 sf .025(Sepset)A 185 512 :M f0_12 sf .122 .012(\(A,C\) contains B, then we know from the search procedure that)J 60 530 :M 1.288 .129(A and C are not d-separated given any subset of )J 311 530 :M f2_12 sf (Sepset)S 344 530 :M f0_12 sf 1.276 .128(\(A,C\). It follows that B is an)J 60 548 :M (ancestor of A or C. Hence A)S 197 548 :M f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (B)S f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (C should be oriented as\312A)S f1_12 sf (*)S f0_18 sf (-)S 388 0 6 730 rC 388 548 :M f1_12 sf 12 f7_1 :p 6 :m ( )S 391 548 :M 6 :m ( )S gR gS 0 0 552 730 rC 388 548 :M f1_12 sf 12 f7_1 :p 6 :m (*)S 388 0 6 730 rC 388 548 :M 6 :m ( )S 391 548 :M 6 :m ( )S gR gS 394 0 8 730 rC 394 548 :M f0_12 sf 12 f6_1 :p 6 :m ( )S 399 548 :M 6 :m ( )S gR gS 0 0 552 730 rC 394 548 :M f0_12 sf 12 f6_1 :p 8.004 :m (B)S 394 0 8 730 rC 394 548 :M 6 :m ( )S 399 548 :M 6 :m ( )S gR gS 402 0 6 730 rC 402 548 :M f1_12 sf 12 f7_1 :p 6 :m ( )S 405 548 :M 6 :m ( )S gR gS 0 0 552 730 rC 402 548 :M f1_12 sf 12 f7_1 :p 6 :m (*)S 402 0 6 730 rC 402 548 :M 6 :m ( )S 405 548 :M 6 :m ( )S gR gS 0 0 552 730 rC 408 548 :M f0_18 sf (-)S f1_12 sf (*)S f0_12 sf (C in the PAG.)S 60 572 :M (The following Lemma shows the correctness of the orientation rule in \246B:)S 60 596 :M f2_12 sf .43 .043(Lemma 8:)J 113 596 :M f0_12 sf .574 .057( In a graph )J 171 596 :M f4_12 sf (G)S 180 596 :M f0_12 sf .533 .053(, if A and B are p-adjacent, B and C are p)J 386 596 :M .48 .048(-adjacent, and B is an)J 60 614 :M (ancestor of A or C then A and C are d-connected given any set )S 364 614 :M f2_12 sf (S)S 371 614 :M f0_12 sf (, s.t. A,B,C )S 428 614 :M f1_12 sf S 437 614 :M f0_12 sf ( )S f2_12 sf (S)S 447 614 :M f0_12 sf (.)S 60 638 :M f2_12 sf (Proof)S 89 638 :M f0_12 sf .086 .009(. Without loss of generality, let us suppose that B is an ancestor of C. It is sufficient)J 60 656 :M .275 .028(to prove that A and C are d-connected conditional on )J f2_12 sf (S)S 329 656 :M f0_12 sf .279 .028(. There are two cases to consider,)J 60 674 :M (depending upon whether or not some \(proper\) descendant of B is in )S f2_12 sf (S)S 394 674 :M f0_12 sf (.)S endp %%Page: 43 43 %%BeginPageSetup initializepage (peter; page: 43 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (43)S gR gS 0 0 552 730 rC 73 54 :M f2_12 sf 1.208 .121(Case 1)J 109 54 :M f0_12 sf 1.185 .118(: Some \(proper\) descendant of B is in )J f2_12 sf (S)S 313 54 :M f0_12 sf 1.271 .127(. It follows from )J 403 54 :M 1.235 .124(Lemma 4 and the)J 73 70 :M 1.662 .166(p-adjacency of A and B that given any set )J f2_12 sf (S)S 305 70 :M f0_12 sf 1.878 .188( such that A, B, C)J 404 70 :M f1_12 sf S 413 70 :M f2_12 sf (S)S 420 70 :M f0_12 sf 1.864 .186(, there is a d-)J 73 86 :M 2.734 .273(connecting path from A to B and likewise a d-connecting path from B to C,)J 73 102 :M .051 .005(conditional on )J 146 102 :M f2_12 sf (S)S 153 102 :M f0_12 sf .058 .006(. Since some descendant of B is in )J f2_12 sf (S)S 328 102 :M f0_12 sf .065 .006(, but B itself is not in )J 433 102 :M f2_12 sf (S)S 440 102 :M f0_12 sf .056 .006(, it follows)J 73 118 :M .065 .007(again by a simple application )J 218 118 :M .064 .006(Lemma 1)J 264 118 :M .079 .008( that A and C are d)J 356 118 :M (-connected)S 408 118 :M .071 .007(, since it does not)J 73 134 :M (matter whether or not the path from A to B and the path from B to C collide at B.)S 73 154 :M f2_12 sf .63 .063(Case 2)J 108 154 :M f0_12 sf .689 .069(: No descendant of B is in )J 243 154 :M f2_12 sf (S)S 250 154 :M f0_12 sf .472(.)A f2_12 sf .472 .047( )J 257 154 :M f0_12 sf .609 .061(It follows from Lemma 4)J 382 154 :M .683 .068( that there is a path d-)J 73 170 :M .782 .078(connecting A and B. Since no descendant of B has been conditioned on the directed)J 73 186 :M .761 .076(path B)J 106 186 :M f1_12 sf S 118 186 :M f0_12 sf S f1_12 sf S 142 186 :M f0_12 sf .703 .07(C is d-connecting. Since B)J 275 186 :M f1_12 sf S 284 186 :M f2_12 sf (S)S 291 186 :M f0_12 sf .771 .077(, it follows from Lemma 1)J 423 186 :M .878 .088( that A and C)J 73 202 :M (are d-connected given )S 182 202 :M f2_12 sf (S)S 189 202 :M f0_12 sf (. )S f1_12 sf <5C>S 60 226 :M f0_12 sf .196 .02(It follows by contraposition that if A and B are p)J 297 226 :M .187 .019(-adjacent, B and C are p-adjacent, A and)J 60 244 :M .17 .017(C are d-separated given )J 177 244 :M f2_12 sf (Sepset)S 210 244 :M f0_12 sf .149 .015(, and B)J f1_12 sf S 287 244 :M f2_12 sf (Sepset)S 320 244 :M f0_12 sf .181 .018(, then B is not an ancestor of)J 60 262 :M (A or C, hence A)S f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (B)S f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (C should be oriented as A\321)S f1_12 sf (>)S 337 262 :M f0_12 sf (B)S f1_12 sf (<)S 352 262 :M f0_12 sf (\321C in the PAG.)S 60 0 7 730 rC 60 292 :M f2_12 sf 12 f8_1 :p 6 :m ( )S 64 292 :M 6 :m ( )S gR gS 0 0 552 730 rC 60 292 :M f2_12 sf 12 f8_1 :p 55.462 :m (Section \246C)S 60 316 :M .053 .005(Lemma 9:)J 112 316 :M f0_12 sf .07 .007( In a graph )J 167 316 :M f4_12 sf (G)S 176 316 :M f0_12 sf .065 .006(, suppose X is an ancestor of Y. If there is a set )J 407 316 :M f2_12 sf (S)S 414 316 :M f0_12 sf .065 .007( such that A and)J 60 334 :M .14 .014(Y are d-separated given )J 177 334 :M f2_12 sf (S)S 184 334 :M f0_12 sf .145 .014(, X and Y are d-connected given )J f2_12 sf (S)S 351 334 :M f0_12 sf .169 .017(, and X)J 387 334 :M f1_12 sf S 396 334 :M f2_12 sf (S)S 403 334 :M f0_12 sf .164 .016(, then A and X are)J 60 352 :M (d-separated given )S 147 352 :M f2_12 sf (S)S 154 352 :M f0_12 sf (.)S 60 376 :M f2_12 sf (Proof.)S 92 376 :M f0_12 sf .818 .082( Let X be an ancestor of Y. Let )J 254 376 :M f2_12 sf (S)S 261 376 :M f0_12 sf .764 .076( be any set such that X and Y are d-connected)J 60 394 :M (given )S 90 394 :M f2_12 sf (S)S 97 394 :M f0_12 sf (, X)S 112 394 :M f1_12 sf S 121 394 :M f2_12 sf (S)S 128 394 :M f0_12 sf .012 .001(, and A and Y are d)J 222 394 :M (-separated by )S f2_12 sf (S)S 295 394 :M f0_12 sf .01 .001(. Suppose, for a contradiction, that A and)J 60 412 :M .394 .039(X are d-connected given )J f2_12 sf (S)S 190 412 :M f0_12 sf .441 .044(. It then follows that there is a d-connecting path )J 433 412 :M f4_12 sf .254(P)A f0_12 sf .447 .045( from A to)J 60 430 :M (X. There are now two cases:)S 60 454 :M f2_12 sf .374 .037(Case 1)J 95 454 :M f0_12 sf .401 .04(: Some descendant of X is in )J 240 454 :M f2_12 sf (S)S 247 454 :M f0_12 sf .325 .032(. Since X)J f1_12 sf S 301 454 :M f2_12 sf (S)S 308 454 :M f0_12 sf .381 .038(, and some descendant of X is in )J f2_12 sf (S)S 478 454 :M f0_12 sf .432 .043(, it)J 60 472 :M .49 .049(follows from )J 127 472 :M .49 .049(Lemma 1)J 174 472 :M .531 .053( that we can put together the d-connecting path from A to X and)J 60 490 :M .999 .1(the d-connecting path from X to Y given )J 271 490 :M f2_12 sf (S)S 278 490 :M f0_12 sf 1.029 .103(, to form a d-connecting path from A to Y)J 60 508 :M (given )S 90 508 :M f2_12 sf (S)S 97 508 :M f0_12 sf (. This is a contradiction since we assumed that A and Y were d)S 399 508 :M (-separated given )S 480 508 :M f2_12 sf (S)S 487 508 :M f0_12 sf (.)S 60 532 :M f2_12 sf .186 .019(Case 2:)J 99 532 :M f0_12 sf .206 .021( No descendant of X is in )J f2_12 sf (S)S 233 532 :M f0_12 sf .212 .021(. In this case since X is an ancestor of Y, there is a d-)J 60 550 :M 1.13 .113(connecting directed path X)J 196 550 :M f1_12 sf S 208 550 :M f0_12 sf S f1_12 sf S 232 550 :M f0_12 sf 1.405 .141(Y. Again, by Lemma 1)J 352 550 :M 1.446 .145( we can put together the d-)J 60 568 :M .972 .097(connecting path from A to X and the d-connecting directed path from X to Y. This is)J 60 586 :M (again a contradiction since we assumed that A and Y were d)S 350 586 :M (-separated given )S 431 586 :M f2_12 sf (S)S 438 586 :M f0_12 sf (.)S 60 610 :M .595 .059(We have now shown that under the conditions in the antecedent, )J 383 610 :M f2_12 sf (S)S 390 610 :M f0_12 sf .61 .061( is a d-separating set)J 60 628 :M (for A and X. )S 124 628 :M f1_12 sf <5C>S 60 652 :M f2_12 sf 1.236 .124(Lemma 10:)J 120 652 :M f0_12 sf 1.628 .163( Let A, X and Y be three vertices in a graph )J 360 652 :M f4_12 sf (G)S 369 652 :M f0_12 sf 1.628 .163(, such that X and Y are)J 60 670 :M (p-adjacent. If there is a set )S 189 670 :M f2_12 sf (S)S 196 670 :M f0_12 sf ( such that)S 78 686 :M (\(i\) X)S 101 686 :M f1_12 sf S 110 686 :M f2_12 sf (S)S 117 686 :M f0_12 sf (,)S endp %%Page: 44 44 %%BeginPageSetup initializepage (peter; page: 44 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (44)S gR gS 0 0 552 730 rC 78 54 :M f0_12 sf (\(ii\) A and Yare d)S 160 54 :M (-separated given )S 241 54 :M f2_12 sf (S)S 248 54 :M f0_12 sf (, and)S 78 70 :M (\(iii\) A and X are d-connected given )S 252 70 :M f2_12 sf (S)S 259 70 :M f0_12 sf (,)S 60 86 :M (then X is not an ancestor of Y.)S 60 110 :M f2_12 sf (Proof.)S 92 110 :M f0_12 sf .918 .092( Suppose that there is such a set )J f2_12 sf (S)S 265 110 :M f0_12 sf .981 .098(. If X and Y are p-adjacent then X and Y are)J 60 128 :M .166 .017(d-connected by every subset of the other variables. In particular X and Y are d-connected)J 60 146 :M .21 .021(given )J f2_12 sf (S)S 97 146 :M f0_12 sf .305 .031(. Since )J 134 146 :M f2_12 sf (S)S 141 146 :M f0_12 sf .282 .028( d-separates A and Y but d-connects A and X, it follows from )J 446 146 :M .257 .026(Lemma 9)J 60 164 :M (that X is not an ancestor of Y.)S 204 164 :M f1_12 sf <5C>S 60 188 :M f0_12 sf (Step \246C simply applies Lemma 10)S 225 188 :M (. Suppose that is a triple such that:)S 78 206 :M (\(i\) A is not p-adjacent to X or Y,)S 78 223 :M (\(ii\) X and Y are p)S 163 223 :M (-adjacent in )S f3_12 sf (Y)S 231 223 :M f0_12 sf (,)S 78 241 :M (\(iii\) X )S 111 241 :M f1_12 sf S 120 241 :M f2_12 sf (Sepset)S 153 241 :M f0_12 sf (.)S 60 266 :M 1.651 .165(\246C\(i\) is justified in the following way. Suppose that )J 335 266 :M f2_12 sf (Sepset)S 368 266 :M f0_12 sf 2.497 .25( )J f1_12 sf 1.138 .114J 422 266 :M f2_12 sf (Sepset)S 455 266 :M f0_12 sf (.)S 60 284 :M .89 .089(Recall that the search procedure used in \246A to find )J 320 284 :M f2_12 sf (Sepset)S 353 284 :M f0_12 sf .815 .081( tests every subset of)J 60 302 :M f2_12 sf (Sepset)S 93 302 :M f0_12 sf .213 .021( to see if it d)J 189 302 :M .199 .02(-separates A and X, )J 287 302 :M .114 .011(before testing )J f2_12 sf .038(Sepset)A 389 302 :M f0_12 sf .147 .015(. In particular,)J 60 320 :M .453 .045(if )J 71 320 :M f2_12 sf (Sepset)S 104 320 :M f0_12 sf .363 .036( )J 142 320 :M f1_12 sf .118 .012J f2_12 sf .08(Sepset)A 187 320 :M f0_12 sf .31 .031(, then A and X are d-connected given )J f2_12 sf .114(Sepset)A 440 320 :M f0_12 sf .302 .03(, so)J 60 338 :M (taking )S 93 338 :M f2_12 sf (S)S 100 338 :M f0_12 sf ( = )S 113 338 :M f2_12 sf (Sepset)S 146 338 :M f0_12 sf (, we can apply Lemma 10)S 304 338 :M ( to orient X )S 362 338 :M f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (Y as X<\321Y.)S 60 362 :M 1.269 .127(\246C\(ii\) is justified in the following way. Suppose that A and X are d-connected given)J 60 380 :M f2_12 sf (Sepset)S 93 380 :M f0_12 sf .755 .076(. Since X\312)J f1_12 sf S 186 380 :M f0_12 sf S f2_12 sf (Sepset)S 222 380 :M f0_12 sf .859 .086(, setting )J 300 380 :M f2_12 sf (S)S 307 380 :M f0_12 sf .316 .032J f2_12 sf .21(Sepset)A 354 380 :M f0_12 sf .913 .091(, we can again apply)J 60 398 :M (Lemma 10)S 112 398 :M ( to orient X )S 170 398 :M f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (Y as X<\321Y.)S 60 422 :M .376 .038(The condition in \246C\(ii\) that )J f2_12 sf .144(Sepset)A 231 422 :M f0_12 sf .491 .049( )J 269 422 :M f1_12 sf S 278 422 :M f0_12 sf .737 .074( )J 282 422 :M f2_12 sf (Sepset)S 315 422 :M f0_12 sf .476 .048( is not needed to make \246C\(ii\))J 60 440 :M .357 .036(correct \(as evidenced by the fact that it plays no role in the justification of the rule\); it is)J 60 458 :M 4.117 .412(included in order to avoid carrying out a redundant test of d-separation. If)J 60 467 180 18 rC 240 485 :M psb currentpoint pse 60 466 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5760 div 608 3 -1 roll exch div scale currentpoint translate 64 56 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Bold f1 (Sepset) -16 264 sh (Sepset) 3165 264 sh 384 /Symbol f1 (<) 1140 264 sh (>) 2267 264 sh (\314) 2741 264 sh (\271) 2774 506 sh (<) 4321 264 sh (>) 5441 264 sh 384 /Times-Roman f1 (A,) 1463 264 sh (X) 1877 264 sh ( ) 2595 264 sh ( ) 3069 264 sh (A) 4644 264 sh (Y) 5053 264 sh 384 /Times-Roman f1 (,) 4920 264 sh end MTsave restore pse gR gS 0 0 552 730 rC 240 476 :M f0_12 sf 4.503 .45(, then A and X are not d-connected given)J 60 494 :M f2_12 sf (Sepset)S 93 494 :M f0_12 sf 1.943 .194(. \(This is because Y)J f1_12 sf S 244 494 :M f2_12 sf (Sepset)S 277 494 :M f0_12 sf 2.041 .204(. Hence X)J 366 494 :M f1_12 sf .521(*)A f0_12 sf .521A f1_12 sf .521(*)A f0_12 sf 1.743 .174(Y will eventually by)J 60 512 :M .949 .095(another application of \246C\(i\) be oriented as X\321>Y in the PAG. It follows that X is an)J 60 530 :M 1.042 .104(ancestor of Y in )J 146 530 :M f4_12 sf (G)S 155 530 :M f0_12 sf 1.216 .122(. By )J 181 530 :M .912 .091(Lemma 9)J 229 530 :M 1.076 .108(, since X is an ancestor of Y in )J 393 530 :M f4_12 sf (G)S 402 530 :M f0_12 sf 1.105 .11(, A and X are not)J 60 548 :M .182 .018(d-connected given )J f2_12 sf .049(Sepset)A 184 548 :M f0_12 sf .312 .031(.\) If )J 240 548 :M f2_12 sf (Sepset)S 273 548 :M f0_12 sf .347 .035( = )J 321 548 :M f2_12 sf (Sepset)S 354 548 :M f0_12 sf .308 .031( then there is no need)J 60 566 :M .07 .007(to test whether A and X are d-connected given )J 286 566 :M f2_12 sf (Sepset)S 319 566 :M f0_12 sf .063 .006(, because it is already known)J 60 584 :M (that they are not d-connected \(by definition of )S 284 584 :M f2_12 sf (Sepset)S 317 584 :M f0_12 sf (\).)S 60 608 :M .182 .018(It is a feature of this orientation rule that X and Y may be arbitrarily far from A. Rules of)J 60 626 :M 2.617 .262(this type are needed by a cyclic discovery algorithm, because, as was shown in)J 60 644 :M .377 .038(Richardson \(1994b\), two cyclic graphs may agree \324locally\325 on d)J 373 644 :M .281 .028(-separation relations, but)J 60 662 :M .813 .081(disagree on some d-separation relation between distant variables. \(Whether or not such)J endp %%Page: 45 45 %%BeginPageSetup initializepage (peter; page: 45 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (45)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf .191 .019(rules will ever be used on real data, in which \324distant\325 variables are generally found to be)J 60 74 :M (independent by statistical tests is another question.\))S 60 0 7 730 rC 60 106 :M f2_12 sf 12 f8_1 :p 6 :m ( )S 64 106 :M 6 :m ( )S gR gS 0 0 552 730 rC 60 106 :M f2_12 sf 12 f8_1 :p 55.462 :m (Section \246D)S 107 0 8 730 rC 107 106 :M 6 :m ( )S 112 106 :M 6 :m ( )S gR gS 0 0 552 730 rC 60 130 :M f0_12 sf 1.295 .13(This section searches to find \324extra\325 d-separating sets for unshielded colliders. In the)J 60 148 :M .48 .048(acyclic case a triple of vertices X)J f1_12 sf .172(*)A f0_12 sf (-)S 234 148 :M f1_12 sf (*)S f0_12 sf (Y)S 249 148 :M f1_12 sf (*)S f0_12 sf (-)S 259 148 :M f1_12 sf .245(*)A f0_12 sf .569 .057(Z, where X and Y are p)J 383 148 :M .53 .053(-adjacent, Y and Z are)J 60 166 :M .321 .032(p-adjacent, but X and Z are not p)J 222 166 :M (-adjacent)S 265 166 :M .28 .028( either has the property that every d-separating)J 60 184 :M .733 .073(set for X and Z contains Y, or that every d-separating set for X and Z does not contain)J 60 202 :M (Y.)S 72 199 :M f0_9 sf (1)S 76 199 :M (8)S 80 202 :M f0_12 sf .721 .072( However, in the cyclic case it is possible for X and Z to be d-separated by one set)J 60 220 :M .692 .069(containing Y, and one set not containing Y. We already know from )J 399 220 :M .663 .066(Lemma 8)J 446 220 :M .816 .082( that if X)J 60 238 :M .207 .021(and Z are d-separated by some set which does not contain Y, then Y is not an ancestor of)J 60 256 :M 1.113 .111(X or Z. What can we infer if in addition X and Z are also d)J 366 256 :M .965 .096(-separated by a set which)J 60 274 :M (contains Y? This is answered by the next Lemma and Corollary.)S 60 298 :M f2_12 sf .145 .014(Lemma )J 103 298 :M (11:)S 119 298 :M f0_12 sf .147 .015( If in a graph )J f4_12 sf (G)S 194 298 :M f0_12 sf .157 .016(, Y is a descendant of a common child of X and Z then X and)J 60 316 :M (Z are d-connected by any set containing Y.)S 60 340 :M f2_12 sf (Proof.)S 92 340 :M f0_12 sf .608 .061( Suppose that Y is a descendant of a common child C of X and Z. Then the path)J 60 358 :M (X)S 69 358 :M f1_12 sf S 81 358 :M f0_12 sf (C)S f1_12 sf S 101 358 :M f0_12 sf (Z d-connects X and Z given any set containing Y.)S f1_12 sf ( \\)S 60 382 :M f2_12 sf .283 .028(Corollary 4:)J 123 382 :M f0_12 sf .464 .046( If in a graph )J 191 382 :M f4_12 sf (G)S 200 382 :M f0_12 sf .431 .043(, X and Y are p-adjacent, Y and Z are p)J 394 382 :M .364 .036(-adjacent, but X and)J 60 400 :M .869 .087(Z are not p-adjacent, Y is not an an)J 238 400 :M .899 .09(cestor of X or Z, and there is some set )J 437 400 :M f2_12 sf (S)S 444 400 :M f0_12 sf .862 .086( such that)J 60 418 :M (Y)S 69 418 :M f1_12 sf S 78 418 :M f2_12 sf (S)S 85 418 :M f0_12 sf .025 .003(, and X and Z are d)J 178 418 :M .016 .002(-separated given )J f2_12 sf (S)S 266 418 :M f0_12 sf .023 .002(, then Y is not a descendant of a common child)J 60 436 :M (of X and Z.)S 60 460 :M .382 .038(It follows from )J 138 460 :M .337 .034(Lemma 12)J 191 460 :M .39 .039( that if is a triple such that X and Z are d-connected)J 60 478 :M .912 .091(given any set containing Y, and d-separated by some set not containing Y, then Y is a)J 60 496 :M (descendant of a common child of X and Z.)S 60 520 :M f2_12 sf .386 .039(Lemma )J 103 520 :M (12:)S 119 520 :M f0_12 sf .445 .044( If in graph )J 178 520 :M f4_12 sf (G)S 187 520 :M f0_12 sf .415 .042(, Y is not a descendant of a common child of X and Z, then X)J 60 538 :M (and Z are d-separated by the set )S f2_12 sf (T)S f0_12 sf (, defined as follows:)S 60 562 :M f2_12 sf (S )S 70 562 :M f0_12 sf (= Children\(X\) )S f1_12 sf S f0_12 sf ( An\({X,Y,Z}\))S 60 578 :M f2_12 sf (T = )S 81 578 :M f0_12 sf (\(Parents\()S f2_12 sf (S )S 134 578 :M f1_12 sf S 152 578 :M f0_12 sf (X}\) )S f1_12 sf S f2_12 sf (S)S 192 578 :M f0_12 sf (\)\\\(Descendants\(Children\(X\))S 327 578 :M f1_12 sf S f0_12 sf (Children\(Z\)\) )S f1_12 sf S f0_12 sf ({X,Z}\))S 60 602 :M (Further, if X and Y, and Y and Z are p)S 245 602 :M (-adjacent then Y)S f1_12 sf S 332 602 :M f2_12 sf (T)S f0_12 sf (.)S 60 626 :M f2_12 sf (Proof. )S 95 626 :M f0_12 sf (It follows from )S 171 626 :M (Lemma 5)S 217 626 :M (, with )S f2_12 sf (Q)S f0_12 sf (={Y} that X and Z are d)S 373 626 :M (-separated given )S 454 626 :M f2_12 sf (T)S f0_12 sf (.)S 60 642 :M (All that remains is to show that Y)S 222 642 :M f1_12 sf S 231 642 :M f2_12 sf (T)S f0_12 sf (. There are three cases to consider here:)S 60 673 :M ( )S 60 670.48 -.48 .48 204.48 670 .48 60 670 @a 60 684 :M f0_9 sf (1)S 64 684 :M (8)S 68 687 :M f0_10 sf (This is also true even in the acyclic case with latent variables.)S endp %%Page: 46 46 %%BeginPageSetup initializepage (peter; page: 46 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (46)S gR gS 0 0 552 730 rC 60 56 :M f2_12 sf .15 .015(Case 1:)J 98 56 :M f0_12 sf .178 .018( Y is a child of X. If Y is a child of X, then since Y is an ancestor of Y, Y)J 457 56 :M f1_12 sf S 466 56 :M f0_12 sf .067 .007( )J f2_12 sf (S)S 476 56 :M f0_12 sf .184 .018(. In)J 60 74 :M (both cases since Y is not a descendant of a common child of X and Z, Y)S f1_12 sf S 415 74 :M f2_12 sf (T)S f0_12 sf (.)S 60 98 :M f2_12 sf 1.056 .106(Case 2:)J 100 98 :M f0_12 sf 1.245 .125( Y is a parent of X. Since Y is a parent of X and Y is not a descendant of a)J 60 116 :M (common child of X and Z, Y)S 199 116 :M f1_12 sf S 208 116 :M f2_12 sf (T)S f0_12 sf (.)S 60 140 :M f2_12 sf .177 .018(Case 3: )J f0_12 sf .17 .017(X and Y have a common child C that is an ancestor of X or Y. Since C is a child)J 60 158 :M .086 .009(of X and an ancestor of X or Y, C)J 224 158 :M f1_12 sf S 233 158 :M f2_12 sf (S)S 240 158 :M f0_12 sf .084 .008(. Since Y is a parent of C, and Y is not a descendant)J 60 176 :M (of a common child of X and Z then Y)S 241 176 :M f1_12 sf S 250 176 :M f2_12 sf (T)S f0_12 sf (. )S f1_12 sf <5C>S 60 200 :M f2_12 sf .599 .06(Lemma 13: )J 123 200 :M f0_12 sf .555 .055(In directed graph )J 211 200 :M f4_12 sf (G)S 220 200 :M f0_12 sf .59 .059(, if X and Z are d-separated by some set )J f2_12 sf (R)S 430 200 :M f0_12 sf .606 .061(, then for all)J 60 218 :M (sets )S 81 218 :M f2_12 sf (Q)S f0_12 sf ( )S f1_12 sf S 102 218 :M f0_12 sf ( An\()S 124 218 :M f2_12 sf (R )S 136 218 :M f1_12 sf S f0_12 sf ({X,Z}\)\\{X,Z}, X and Z are d-separated by )S f2_12 sf (R)S 363 218 :M f0_12 sf ( )S f1_12 sf S f0_12 sf ( )S f2_12 sf (Q)S f0_12 sf (.)S 60 242 :M f2_12 sf .078 .008(Proof. )J 95 242 :M f0_12 sf .081 .008(Suppose for a contradiction that there is a path )J f4_12 sf (P)S f0_12 sf .081 .008( d-connecting X and Z given )J 471 242 :M f2_12 sf (R)S 480 242 :M f0_12 sf S f1_12 sf S 60 260 :M f2_12 sf .835(Q)A f0_12 sf 1.337 .134(. It follows that every vertex on )J 238 260 :M f4_12 sf .845(P)A f0_12 sf 1.447 .145( is an ancestor of either X, Z, or )J 420 260 :M f2_12 sf (R)S 429 260 :M f0_12 sf .298 .03( )J f1_12 sf 1.114 .111J f2_12 sf 1.022(Q)A f0_12 sf 2.234 .223(. Since)J 60 278 :M f2_12 sf (Q)S f0_12 sf S f1_12 sf S 81 278 :M f0_12 sf S 103 278 :M f2_12 sf (R)S 112 278 :M f0_12 sf S f1_12 sf .145A f0_12 sf .271 .027(\312{X,Z}\) it follows that every vertex on )J 315 278 :M f4_12 sf .204(P)A f0_12 sf .32 .032( is an ancestor of X, Z or )J 448 278 :M f2_12 sf (R)S 457 278 :M f0_12 sf .346 .035(. Let A)J 60 296 :M .21 .021(be the collider furthest from X on )J f4_12 sf .099(P)A f0_12 sf .181 .018( which is an ancestor of X and not )J 403 296 :M f2_12 sf (R)S 412 296 :M f0_12 sf .215 .022( \(or X if no such)J 60 314 :M .022 .002(collider exists\), let B be the first collider after A on )J 310 314 :M f4_12 sf (P)S f0_12 sf .023 .002( which is an ancestor of Z and not )J 483 314 :M f2_12 sf (R)S 60 332 :M f0_12 sf 2.211 .221(\(or Z if no such collider exists\). Clearly the paths X)J 342 332 :M f1_12 sf S 354 332 :M f0_12 sf S f1_12 sf S 378 332 :M f0_12 sf 2.506 .251(A, and B)J 428 332 :M f1_12 sf S 440 332 :M f0_12 sf S f1_12 sf S 464 332 :M f0_12 sf 2.416 .242(Z are)J 60 350 :M .422 .042(d-connecting given )J f2_12 sf (R)S 165 350 :M f0_12 sf .62 .062(, since by the definition of A and B, no vertex on these paths is in)J 60 368 :M f2_12 sf (R)S 69 368 :M f0_12 sf .011 .001(. In addition the subpath of )J f2_12 sf (P)S f0_12 sf .012 .001( between A and B is also d-connecting given )J 426 368 :M f2_12 sf (R)S 435 368 :M f0_12 sf ( since every)S 60 386 :M 1.974 .197(collider is an ancestor of )J 196 386 :M f2_12 sf (R)S 205 386 :M f0_12 sf 2.023 .202(, and no non-collider lies in )J 358 386 :M f2_12 sf (R)S 367 386 :M f0_12 sf 1.935 .194(, since, by hypothesis )J 485 386 :M f4_12 sf (P)S 60 404 :M f0_12 sf .151 .015(d-connects given )J 145 404 :M f2_12 sf (R)S 154 404 :M f0_12 sf .045 .005( )J f1_12 sf .154A f0_12 sf .045 .005( )J f2_12 sf .156(Q)A f0_12 sf .219 .022(. It follows, by )J 253 404 :M .17 .017(Lemma 1)J 299 404 :M .182 .018(, that there is a path d-connecting X and)J 60 422 :M (Z given )S 100 422 :M f2_12 sf (R)S 109 422 :M f0_12 sf (. This is a contradiction.)S 225 422 :M f1_12 sf <5C>S 60 446 :M f0_12 sf .15 .015(The search in section \246D considers in turn each triple A\321)J 341 446 :M f1_12 sf (>)S 348 446 :M f0_12 sf (B)S f1_12 sf (<)S 363 446 :M f0_12 sf .187 .019(\321C in )J 399 446 :M f3_12 sf (Y)S 409 446 :M f0_12 sf .177 .018(, A and C are not)J 60 464 :M .198 .02(p-adjacent, and attempts to find a set )J f2_12 sf (R)S 249 464 :M f0_12 sf .239 .024( which is a subset of )J 352 464 :M f2_12 sf (Local)S 381 464 :M f0_12 sf <28>S 385 464 :M f3_12 sf (Y)S 395 464 :M f0_12 sf .202 .02(,A\)\\{C} such that A)J 60 482 :M .135 .013(and C are d-separated given )J 197 482 :M f2_12 sf (R)S 206 482 :M f0_12 sf ( )S f1_12 sf .152A f0_12 sf .175 .018( {B} )J 244 482 :M f1_12 sf .157A f0_12 sf .051 .005( )J 257 482 :M f2_12 sf (Sepset)S 290 482 :M f0_12 sf .13 .013(. It follows from Lemma 11)J 458 482 :M .148 .015(, that if)J 60 500 :M .424 .042(there is some set which d-separates A and C, and contains B, then B is not a descendant)J 60 518 :M .494 .049(of a common child of A and C. It then follows from )J 320 518 :M .411 .041(Lemma 12)J 373 518 :M .488 .049( that in this case there is)J 60 536 :M .083 .008(some subset, the set )J f2_12 sf (T)S f0_12 sf .087 .009( given in the Lemma, which contains B, d-separates A and C and in)J 60 554 :M .208 .021(which every vertex is either a parent of A, a child of A, or a parent of a child of A and so)J 60 572 :M f2_12 sf <54CA>S f1_12 sf S 80 572 :M f0_12 sf S f2_12 sf (Local)S 112 572 :M f0_12 sf <28>S 116 572 :M f3_12 sf (Y)S 126 572 :M f0_12 sf 1.143 .114(,X\). Since )J f2_12 sf .451(Sepset)A 216 572 :M f0_12 sf 1.933 .193( is a minimal d-separating set for A and C, it)J 60 590 :M .916 .092(follows that )J 123 590 :M f2_12 sf (Sepset)S 156 590 :M f0_12 sf .269()A f2_12 sf .121 .012( )J 194 590 :M f1_12 sf S 203 590 :M f0_12 sf .709 .071( An\({A,C}\)\\{A,C} \()J f1_12 sf S 313 590 :M f0_12 sf .841 .084( An\()J f2_12 sf .435(T)A f0_12 sf .163A f1_12 sf .501A f0_12 sf .885 .089({A,C}\) . Hence by )J 455 590 :M (Lemma)S 60 608 :M (13, )S f2_12 sf (T)S f0_12 sf ( )S f1_12 sf S f0_12 sf ( )S f2_12 sf (Sepset)S 134 608 :M f0_12 sf ( also d-separates A and C.)S 60 632 :M 2.074 .207(The reader may wonder why \246D tests sets of the form )J 354 632 :M f2_12 sf .87(T)A f0_12 sf .296 .03( )J f1_12 sf 1.001A f0_12 sf .296 .03( )J f2_12 sf .71(Sepset)A 416 632 :M f0_12 sf 1.38 .138(, \(where)J 60 650 :M f2_12 sf <54CA>S f1_12 sf S 80 650 :M f0_12 sf S f2_12 sf (Local)S 112 650 :M f0_12 sf <28>S 116 650 :M f3_12 sf (Y)S 126 650 :M f0_12 sf .782 .078(,A\)\), instead of just testing sets of the form )J 346 650 :M f2_12 sf <54CA>S f1_12 sf S 366 650 :M f0_12 sf S f2_12 sf (Local)S 398 650 :M f0_12 sf <28>S 402 650 :M f3_12 sf (Y)S 412 650 :M f0_12 sf .716 .072(,A\)\); Lemma 12)J 60 668 :M .694 .069(shows that a search of the latter kind would succeed in finding a d-separating set for A)J 60 686 :M 1.338 .134(and C which contained B. The answer is that from Lemma 13)J 378 686 :M 1.442 .144( we know that any set)J endp %%Page: 47 47 %%BeginPageSetup initializepage (peter; page: 47 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (47)S gR gS 0 0 552 730 rC 60 56 :M f2_12 sf <54CA>S f1_12 sf S 80 56 :M f0_12 sf S f2_12 sf (Local)S 112 56 :M f0_12 sf <28>S 116 56 :M f3_12 sf (Y)S 126 56 :M f0_12 sf 2.587 .259(,A\) which d-separates A and C is such that )J 369 56 :M f2_12 sf 2.616(T)A f0_12 sf .891 .089( )J f1_12 sf 3.011A f0_12 sf .98 .098( )J 399 56 :M f2_12 sf (Sepset)S 432 56 :M f0_12 sf 1.997 .2( also)J 60 74 :M .383 .038(d-separates A and C, but the reverse is not true. In particular the smallest set )J 437 74 :M f2_12 sf .174(T)A f0_12 sf .345 .035( such that)J 60 92 :M f2_12 sf .085(T)A f0_12 sf ( )S f1_12 sf .098A f0_12 sf ( )S 84 92 :M f2_12 sf (Sepset)S 117 92 :M f0_12 sf .076 .008( d-separates A and C may be considerably smaller than the smallest set)J 60 110 :M f2_12 sf (T)S f0_12 sf ( which d-separates A and C alone, hence the search is significantly faster.)S 421 107 :M f0_9 sf (1)S 425 107 :M (9)S 60 134 :M f0_12 sf (We require one more lemma to explain why we initialize m = 1, and do not test )S 444 134 :M f2_12 sf (T )S f0_12 sf (= )S 465 134 :M f1_12 sf S 475 134 :M f0_12 sf (:)S 60 158 :M f2_12 sf .77 .077(Lemma )J 104 158 :M (14:)S 120 158 :M f0_12 sf .713 .071( In directed graph )J f4_12 sf (G)S 221 158 :M f0_12 sf .9 .09(, if X and Y are p)J 312 158 :M .791 .079(-adjacent, Y and Z are p)J 433 158 :M .55 .055(-adjacent, X)J 60 176 :M .297 .03(and Z are not p)J 135 176 :M .283 .028(-adjacent, Y is not an ancestor of X or Z, and )J 358 176 :M f2_12 sf (S)S 365 176 :M f0_12 sf .247 .025( is a minimal d-separating)J 60 194 :M (set for X and Z then X and Z are d-connected given )S 311 194 :M f2_12 sf (S)S 318 194 :M f0_12 sf ( )S f1_12 sf S f0_12 sf ( {Y}.)S 60 218 :M f2_12 sf .569 .057(Proof. )J 96 218 :M f0_12 sf .569 .057(According to )J 164 218 :M .455 .046(Corollary 1)J 220 218 :M .63 .063(, if X and Y are p-adjacent then either X)J 421 218 :M f1_12 sf S 433 218 :M f0_12 sf .431 .043(Y, Y)J f1_12 sf S 469 218 :M f0_12 sf .7 .07(X or)J 60 236 :M (X)S 69 236 :M f1_12 sf S 81 236 :M f0_12 sf (C)S f1_12 sf S 101 236 :M f0_12 sf .396 .04(Y, where C is an ancestor of X or Y. Thus under the hypothesis that Y is not an)J 60 254 :M .79 .079(ancestor of X it follows that X is an ancestor of Y. Moreover, it follows that there is a)J 60 272 :M .489 .049(directed path )J 127 272 :M f4_12 sf .269(P)A f0_12 sf .552 .055( from X to Y, on which every vertex except X is a descendant of Y, and)J 60 290 :M .62 .062(hence on which every vertex except X is not an ancestor of X or Z. \(In the case X)J f1_12 sf S 480 290 :M f0_12 sf (Y,)S 60 308 :M 1.761 .176(the last assertion is trivial. In the other case it merely states a property of the path)J 60 326 :M (X)S 69 326 :M f1_12 sf S 81 326 :M f0_12 sf (C)S f1_12 sf S 101 326 :M f0_12 sf .104 .01(\311Y, where C is a common child of X and Y.\) Likewise there is a path )J 446 326 :M f2_12 sf .06(Q)A f0_12 sf .09 .009( from Z)J 60 344 :M (to Y on which every vertex except Z is not an ancestor of X or Z.)S 60 368 :M 2.018 .202(If )J 74 368 :M f2_12 sf (S)S 81 368 :M f0_12 sf 1.603 .16( is a minimal d-separating set every vertex in )J 322 368 :M f2_12 sf 1.039 .104(S )J f0_12 sf 1.831 .183(is an ancestor of X or Z, \(and)J 60 386 :M (X,Z\312)S 82 386 :M f1_12 sf S 91 386 :M f2_12 sf (S)S 98 386 :M f0_12 sf .418 .042(\). Hence no vertex on )J 208 386 :M f4_12 sf .272(P)A f0_12 sf .247 .025( or )J f4_12 sf (Q)S 241 386 :M f0_12 sf .384 .038( is in )J f2_12 sf (S)S 276 386 :M f0_12 sf .332 .033(. It follows that )J f4_12 sf .185(P)A f0_12 sf .437 .044( d-connects X and Y given)J 60 404 :M f2_12 sf (S)S 67 404 :M f0_12 sf .206 .021(, and )J 94 404 :M f4_12 sf (Q)S 103 404 :M f0_12 sf .171 .017( d-connects Y and Z given )J f2_12 sf (S)S 241 404 :M f0_12 sf .178 .018(. It then follows from )J 348 404 :M .161 .016(Lemma 1)J 394 404 :M .171 .017( that these paths can)J 60 422 :M .517 .052(be joined to form a single d-connecting path, hence X and Z are d-connected given )J f2_12 sf (S)S 479 422 :M f0_12 sf .812 .081( )J 483 422 :M f1_12 sf S 60 440 :M f0_12 sf ({Y}.)S f1_12 sf <5C>S 60 464 :M f0_12 sf .079 .008(This completes the proof that step \246D of the algorithm will succeed in finding a set which)J 60 482 :M .107 .011(d-separates A and C, and contains B, for each triple A\321)J 333 482 :M f1_12 sf (>)S 340 482 :M f0_12 sf (B)S f1_12 sf (<)S 355 482 :M f0_12 sf .117 .012(\321C in the PAG, if any such)J 60 500 :M (set exists.)S 60 0 7 730 rC 60 524 :M f2_12 sf 12 f8_1 :p 6 :m ( )S 64 524 :M 6 :m ( )S gR gS 0 0 552 730 rC 60 524 :M f2_12 sf 12 f8_1 :p 54.803 :m (Section \246E)S 107 0 8 730 rC 107 524 :M 6 :m ( )S 112 524 :M 6 :m ( )S gR gS 0 0 552 730 rC 60 548 :M f0_12 sf .182 .018(The following Lemma provides the justification of \246E)J f2_12 sf ( )S f0_12 sf .267 .027(where A\321)J 378 0 7 730 rC 378 548 :M .286 .029( )J 382 548 :M .286 .029( )J gR gS 0 0 552 730 rC 378 548 :M f0_12 sf (>B<)S 393 0 7 730 rC 393 548 :M .286 .029( )J 397 548 :M .286 .029( )J gR gS 0 0 552 730 rC 400 548 :M f0_12 sf .2 .02J 447 0 7 730 rC 447 548 :M .286 .029( )J 451 548 :M .286 .029( )J gR gS 0 0 552 730 rC 447 548 :M f0_12 sf (>D<)S 462 0 7 730 rC 462 548 :M .286 .029( )J 466 548 :M .286 .029( )J gR 1 G gS 0 0 552 730 rC 0 0 1 1 rF 472 548 :M psb /wp$x1 378 def /wp$x2 399 def /wp$y 550 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 469 548 :M 0 G f0_12 sf S 1 G 0 0 1 1 rF 492 548 :M psb /wp$x1 447 def /wp$x2 468 def /wp$y 550 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 566 :M 0 G (and D is not in )S 134 566 :M f2_12 sf (SupSepset)S 187 566 :M f0_12 sf (, in which case B)S 314 566 :M f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (D is oriented as B\321)S 430 566 :M f1_12 sf (>)S 437 566 :M f0_12 sf (D.)S 60 640 :M ( )S 60 637.48 -.48 .48 204.48 637 .48 60 637 @a 60 651 :M f0_9 sf (1)S 64 651 :M (9)S 68 654 :M f0_10 sf .268 .027(In some cases the cardinality of the smallest set \()J 268 654 :M f2_10 sf (T)S 275 654 :M f0_10 sf .417 .042( )J 278 654 :M f1_10 sf S 286 654 :M f0_10 sf .417 .042( )J 289 654 :M f2_10 sf .057(Sepset)A f0_10 sf .214 .021(\) may be greater than the cardinality)J 60 665 :M .28 .028(of the smallest )J f2_10 sf (T)S 129 665 :M f0_10 sf .315 .032(; but this is not true in general, and since we only intend to discover linear models this is)J 60 676 :M .364 .036(insignificant. \(With discrete models conditioning on a large set of variables in a conditional independence)J 60 687 :M (test may reduce dramatically the power of the test.\))S endp %%Page: 48 48 %%BeginPageSetup initializepage (peter; page: 48 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (48)S gR gS 0 0 552 730 rC 60 54 :M f2_12 sf 2.295 .229(Lemma )J 106 54 :M (15:)S 122 54 :M f0_12 sf 2.776 .278( If in a PAG )J 201 54 :M f3_12 sf (Y)S 211 54 :M f0_12 sf 2.868 .287( for )J 238 54 :M f4_12 sf (G)S 247 54 :M f0_12 sf 2.648 .265(, X\321)J 277 0 7 730 rC 277 54 :M 3.442 .344( )J 278 54 :M 3.278 .328( )J gR gS 0 0 552 730 rC 277 54 :M f0_12 sf (>V<)S 292 0 7 730 rC 292 54 :M 3.442 .344( )J 293 54 :M 3.278 .328( )J gR gS 0 0 552 730 rC 299 54 :M f0_12 sf 2.294 .229J 349 0 7 730 rC 349 54 :M 3.442 .344( )J 350 54 :M 3.278 .328( )J gR gS 0 0 552 730 rC 349 54 :M f0_12 sf (>W<)S 367 0 7 730 rC 367 54 :M 3.442 .344( )J 368 54 :M 3.278 .328( )J gR 1 G gS 0 0 552 730 rC 0 0 1 1 rF 380 54 :M psb /wp$x1 277 def /wp$x2 298 def /wp$y 56 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 374 54 :M 0 G f0_12 sf 2.531 .253(\321Z, X and Z are not)J 1 G 0 0 1 1 rF 491 54 :M psb /wp$x1 349 def /wp$x2 373 def /wp$y 56 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 70 :M 0 G .507 .051(p-adjacent, and W is an ancestor of V in )J 262 70 :M f4_12 sf (G)S 271 70 :M f0_12 sf .484 .048(, then any set )J f2_12 sf (S)S 347 70 :M f0_12 sf .536 .054( such that V)J 407 70 :M f1_12 sf S 416 70 :M f2_12 sf (S)S 423 70 :M f0_12 sf .563 .056(, and X and Z)J 60 86 :M (are d)S 84 86 :M (-separated by )S f2_12 sf (S)S 157 86 :M f0_12 sf (,)S f2_12 sf ( )S f0_12 sf (also contains W.)S 60 110 :M f2_12 sf .553 .055(Proof. )J 96 110 :M f0_12 sf .509 .051(Suppose there were some d-separating set )J f2_12 sf (S)S 312 110 :M f0_12 sf .626 .063( for X and Z which contained V and)J 60 128 :M .373 .037(did not contain W. Then, since W is an ancestor of V and V)J 354 128 :M f1_12 sf S 363 128 :M f2_12 sf (S)S 370 128 :M f0_12 sf .413 .041(, but W)J 407 128 :M f1_12 sf S 416 128 :M f2_12 sf (S)S 423 128 :M f0_12 sf .366 .037(, it follows by)J 60 146 :M .467 .047(Lemma 1 that we could put together a d-connecting path from X to W and from W to Z)J 60 164 :M .075 .008(to form a new d-connecting path \(irrespective of whether or not these paths collide at W\).)J 60 182 :M .311 .031(Such d-connecting paths between X and W, and between W and Z exist \(by )J 433 182 :M .222 .022(Corollary 1\))J 60 200 :M (since X is p-adjacent to W and W is p-adjacent to Z. This is a contradiction.)S 422 200 :M f1_12 sf <5C>S 60 224 :M f0_12 sf 1.439 .144(In the case in which A\321)J 190 0 7 730 rC 190 224 :M 2.072 .207( )J 191 224 :M 1.973 .197( )J gR gS 0 0 552 730 rC 190 224 :M 0 G f0_12 sf (>B<)S 205 0 7 730 rC 205 224 :M 2.072 .207( )J 206 224 :M 1.973 .197( )J gR gS 0 0 552 730 rC 212 224 :M 0 G f0_12 sf 1.381 .138J 262 0 7 730 rC 262 224 :M 2.072 .207( )J 263 224 :M 1.973 .197( )J gR gS 0 0 552 730 rC 262 224 :M 0 G f0_12 sf (>D<)S 277 0 7 730 rC 277 224 :M 2.072 .207( )J 278 224 :M 1.973 .197( )J gR gS 0 0 552 730 rC 0 0 1 1 rF 288 224 :M psb /wp$x1 190 def /wp$x2 211 def /wp$y 226 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 284 224 :M 0 G f0_12 sf .953 .095(\321C, and D is in )J f2_12 sf .509(SupSepset)A 428 224 :M f0_12 sf 1.036 .104( the)J 1 G 0 0 1 1 rF 491 224 :M psb /wp$x1 262 def /wp$x2 283 def /wp$y 226 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 242 :M 0 G .571 .057(algorithm orients B)J f1_12 sf .143(*)A f0_12 sf .143A f1_12 sf .143(*)A f0_12 sf .312 .031(D as B)J 208 242 :M f1_12 sf .206(*)A f0_12 sf .554 .055(\321D, this inference can be justified as follows. If D is in)J 60 260 :M f2_12 sf (SupSepset)S 113 260 :M f0_12 sf 1.935 .194( then it follows from )J 274 260 :M 1.887 .189(Lemma 7)J 323 260 :M 2.092 .209( and the nature of the search of)J 60 278 :M f2_12 sf (SupSepset)S 113 278 :M f0_12 sf ()S 157 275 :M f0_9 sf (2)S 161 275 :M (0)S 165 278 :M f0_12 sf .603 .06( that D is an ancestor of {B}\312)J f1_12 sf .48 .048J 322 278 :M f2_12 sf (Sepset)S 355 278 :M f0_12 sf .513 .051(. Since )J 426 278 :M f2_12 sf (Sepset)S 459 278 :M f0_12 sf ()S 60 296 :M .244 .024(is a minimal d-separating set for A and C, every vertex in )J f2_12 sf .102(Sepset)A 376 296 :M f0_12 sf .262 .026( is an ancestor of)J 60 314 :M .158 .016(A or C. Thus if D is in )J f2_12 sf .098(SupSepset)A 226 314 :M f0_12 sf .201 .02(, then D is an ancestor of A,C or B. However,)J 60 332 :M .05 .005(since there are arrowheads at D on the edges from A to D, and C to D, it follows that D is)J 60 350 :M .761 .076(not an ancestor of A or C, and hence D is an ancestor of B. Thus it is correct to orient)J 60 368 :M (B)S f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (D as B)S 119 368 :M f1_12 sf (*)S f0_12 sf S 60 388 :M .046 .005(In the case in which A\321>D<\321C in )J f3_12 sf (Y)S 247 388 :M f0_12 sf .054 .005(, \(A and C are not p)J 343 388 :M .043 .004(-adjacent and there is no dotted)J 60 404 :M .667 .067(line A\321)J 103 0 7 730 rC 103 404 :M 1.067 .107( )J 104 404 :M 1.016 .102( )J gR gS 0 0 552 730 rC 103 404 :M 0 G f0_12 sf (>D<)S 118 0 7 730 rC 118 404 :M 1.067 .107( )J 119 404 :M 1.016 .102( )J gR gS 0 0 552 730 rC 125 404 :M 0 G f0_12 sf .684 .068(\321C\), it follows from Lemma 12)J 287 404 :M .711 .071( by contraposition that since A and C are)J 1 G 0 0 1 1 rF 491 404 :M psb /wp$x1 103 def /wp$x2 124 def /wp$y 406 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 420 :M 0 G 1.949 .195(d-connected by any set )J f2_12 sf (S)S 192 420 :M f0_12 sf 2.296 .23( that contains D, \(and does not contain A or C\), D is a)J 60 436 :M .676 .068(descendant of a common child of A and C. Moreover since A and C are d-separated by)J 60 452 :M .373 .037(some set containing B, B is not a descendant of a common child of A and C. Hence B is)J 60 468 :M .083 .008(not a descendant of D. Thus in the case where in )J f3_12 sf (Y)S 307 468 :M f0_12 sf .096 .01(, A\321)J 335 0 7 730 rC 335 468 :M .125 .012( )J 336 468 :M .119 .012( )J gR gS 0 0 552 730 rC 335 468 :M 0 G f0_12 sf (>B<)S 350 0 7 730 rC 350 468 :M .125 .012( )J 351 468 :M .119 .012( )J gR gS 0 0 552 730 rC 357 468 :M 0 G f0_12 sf .063 .006(\321C, A\321>D)J f1_12 sf (<)S 426 468 :M f0_12 sf .094 .009(\321C, B and D)J 1 G 0 0 1 1 rF 492 468 :M psb /wp$x1 335 def /wp$x2 356 def /wp$y 470 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 484 :M 0 G (are p)S 84 484 :M (-adjacent, B*\320*D should be oriented as B<\321D.)S 60 0 7 730 rC 60 508 :M 12 f6_1 :p 6 :m ( )S 64 508 :M 6 :m ( )S gR gS 0 0 552 730 rC 60 508 :M 0 G f0_12 sf 12 f6_1 :p 35.991 :m (Section)S 90 0 6 730 rC 90 508 :M 6 :m ( )S 93 508 :M 6 :m ( )S gR gS 96 0 3 730 rC 96 508 :M 0 G f2_12 sf 12 f8_1 :p 3 :m ( )S 96 508 :M 6 :m ( )S gR gS 0 0 552 730 rC 96 508 :M 0 G f2_12 sf 12 f8_1 :p 16.808 :m <20A646>S 105 0 8 730 rC 105 508 :M 6 :m ( )S 110 508 :M 6 :m ( )S gR gS 0 0 552 730 rC 60 532 :M 0 G f0_12 sf .681 .068(A and C are d)J 131 532 :M .536 .054(-separated by )J 198 532 :M f2_12 sf (SupSepset)S 251 532 :M f0_12 sf .536 .054(, and B)J 332 532 :M f1_12 sf S 341 532 :M f2_12 sf (SupSepset)S 394 532 :M f0_12 sf .491 .049(. Hence by)J 60 550 :M 4.051 .405(Lemma 13)J 119 550 :M 4.869 .487(, if D is an ancestor of B, then A and C are d-separated by)J 60 568 :M f2_12 sf (SupSepset)S 113 568 :M f0_12 sf .172(\312)A f1_12 sf .269A f0_12 sf .561 .056(\312{D}. Hence by contraposition, if A and C are d-connected given)J 60 586 :M f2_12 sf (SupSepset)S 113 586 :M f0_12 sf .3(\312)A f1_12 sf .469A f0_12 sf .72 .072(\312{D} then D is not an ancestor of B. \(In fact, it follows that D is)J 60 604 :M .242 .024(not an ancestor of A,B or C.\) Since D is not an ancestor of B, but B and D are p-adjacent)J 60 622 :M (it follows that B is an ancestor of D. Thus B)S 272 622 :M f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (D should be oriented as B\321>D in )S f3_12 sf (Y)S 473 622 :M f14_13 sf (.)S 60 662 :M f0_12 sf ( )S 60 659.48 -.48 .48 204.48 659 .48 60 659 @a 60 673 :M f0_9 sf (2)S 64 673 :M (0)S 68 676 :M f0_10 sf 2.134 .213(Namely the fact that section )J 200 676 :M f2_10 sf .806A f0_10 sf 2.19 .219(D looks for the smallest superset of {B}\312)J f1_10 sf 1.381 .138J 405 676 :M f2_10 sf .179(Sepset)A f0_10 sf 1.073 .107(, which)J 60 687 :M (d-separates A and C.)S endp %%Page: 49 49 %%BeginPageSetup initializepage (peter; page: 49 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (49)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf (This completes the proof of the correctness of the CCD algorithm. )S f1_12 sf <5C>S 60 104 :M f2_12 sf (7)S 66 104 :M (.)S 69 104 :M (6)S 75 104 :M (.)S 78 104 :M ( )S 96 104 :M (Proof of )S 141 104 :M (Theorem 8)S 60 131 :M f0_12 sf .143 .014(In order to prove the d)J 169 131 :M .119 .012(-separation completeness of the CCD algorithm, all that is required)J 60 149 :M .279 .028(is to show that whenever the first input to the CCD algorithm is a d)J 389 149 :M .201 .02(-separation oracle for)J 60 167 :M f4_12 sf (G)S 69 169 :M f0_9 sf (1)S 74 167 :M f0_12 sf .244 .024( that results in output )J f3_12 sf (Y)S 190 169 :M f0_9 sf (1)S 195 167 :M f0_12 sf .273 .027(, and the second input to the CCD algorithm is a d)J 440 167 :M (-separation)S 60 185 :M .294 .029(oracle for )J f4_12 sf (G)S 119 187 :M f0_9 sf (2)S 124 185 :M f0_12 sf .367 .037( that results in output )J 231 185 :M f3_12 sf (Y)S 241 187 :M f0_9 sf (2)S 246 185 :M f0_12 sf .302 .03(, and )J f3_12 sf (Y)S 283 187 :M f0_9 sf (1)S 288 185 :M f0_12 sf .304 .03( and )J f3_12 sf (Y)S 322 187 :M f0_9 sf (2)S 327 185 :M f0_12 sf .357 .036( are identical, then )J 421 185 :M f4_12 sf (G)S 430 187 :M f0_9 sf (1)S 435 185 :M f0_12 sf .313 .031( and )J f4_12 sf (G)S 468 187 :M f0_9 sf (2)S 473 185 :M f0_12 sf .412 .041( are)J 60 203 :M .838 .084(d-separation equivalent. We shall do this by proving that when d-separation oracles for)J 60 221 :M f4_12 sf (G)S 69 223 :M f0_9 sf (1)S 74 221 :M f0_12 sf .273 .027( and )J 98 221 :M f4_12 sf (G)S 107 223 :M f0_9 sf (2)S 112 221 :M f0_12 sf .219 .022( are used as input to the CCD algorithm and produce the same PAG as output,)J 60 239 :M .651 .065(then )J 85 239 :M f4_12 sf (G)S 94 241 :M f0_9 sf (1)S 99 239 :M f0_12 sf .528 .053(, and )J f4_12 sf (G)S 136 241 :M f0_9 sf (2)S 141 239 :M f0_12 sf .539 .054( satisfy the five conditions of the Cyclic Equivalence Theorem CET\(I\)-)J 60 257 :M 3.358 .336(\(V\) \(given below\) with respect to one another. It has already been shown in)J 60 275 :M .179 .018(Richardson\(1994b\) that two graphs )J 234 275 :M f4_12 sf (G)S 243 277 :M f0_9 sf (1)S 248 275 :M f0_12 sf .273 .027( and )J 272 275 :M f4_12 sf (G)S 281 277 :M f0_9 sf (2)S 286 275 :M f0_12 sf .262 .026( are d)J 314 275 :M .172 .017(-separation equivalent to one another)J 60 293 :M (if and only if they satisfy these 5 conditions.)S 60 317 :M .266 .027(Before stating the Cyclic Equivalence Theorem we require a number of extra definitions.)J 60 335 :M .625 .062(In a cyclic graph )J f4_12 sf (G)S 156 335 :M f0_12 sf .49 .049(, we say triple of vertices forms an )J f2_12 sf 1.166 .117(unshielded conductor)J 60 353 :M f0_12 sf (if:)S 99 371 :M (\(i\))S 118 371 :M (A and B are p)S 185 371 :M (-adjacent, B and C are p)S 300 371 :M (-adjacent, A and C are not p)S 434 371 :M (-adjacent)S 97 389 :M (\(ii\))S 118 389 :M (B is an ancestor of A or C)S 60 413 :M 1.621 .162(If satisfies \(i\), but B is not an ancestor of A or C, we say is an)J 60 431 :M f2_12 sf (unshielded non-conductor.)S 60 455 :M f0_12 sf .943 .094(In a cyclic graph )J f4_12 sf (G)S 158 455 :M f0_12 sf 1.037 .104(, we say triple of vertices is an )J 368 455 :M f2_12 sf .719 .072(unshielded perfect non-)J 60 473 :M (conductor)S 112 473 :M f4_12 sf ( )S f0_12 sf (if)S 96 491 :M (\(i\) A and B are p)S 177 491 :M (-adjacent, B and C are p)S 292 491 :M (-adjacent, but A and C are not)S 96 509 :M (p-adjacent,)S 96 527 :M (\(ii\) B is not an ancestor of A or C, and)S 92 545 :M ( \(iii\) B is a descendant of a common child of A and C.)S 60 569 :M .242 .024(If satisfies \(i\) and \(ii\) but B is not a descendant of a common child of A and C,)J 60 587 :M (we say is an )S 169 587 :M f2_12 sf (unshielded imperfect non-conductor)S 355 587 :M f4_12 sf (.)S 60 611 :M f0_12 sf .591 .059(If S 133 613 :M (n+)S 146 613 :M f0_9 sf (1)S 151 611 :M f0_12 sf .686 .069(> is a sequence of distinct vertices s.t. )J 345 611 :M f1_12 sf (")S 354 611 :M f0_12 sf .824 .082(i 0 )J cF f1_12 sf .082A sf .824 .082( i )J cF f1_12 sf .082A sf .824 .082(\312n, X)J 421 613 :M .139(i)A 0 -2 rm .511 .051( and X)J 0 2 rm .21(i+)A f0_9 sf (1)S 473 611 :M f0_12 sf .798 .08( are)J 60 629 :M (p-adjacent then we will refer to S 274 631 :M ( n+)S 290 631 :M f0_9 sf (1)S 295 629 :M f0_12 sf (> as an )S f2_12 sf (itinerary)S f0_12 sf (.)S endp %%Page: 50 50 %%BeginPageSetup initializepage (peter; page: 50 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (50)S gR gS 0 0 552 730 rC 60 66 :M f0_12 sf (If S 115 68 :M f0_10 sf (n)S f0_12 sf (+)S 127 68 :M f0_9 sf (1)S 132 66 :M f0_12 sf (> is an itinerary such that:)S 78 84 :M (\(i\) )S f1_12 sf (")S 101 84 :M f0_12 sf (t 1)S cF f1_12 sf S sf ( t )S cF f1_12 sf S sf ( n, is an unshielded conductor,)S 78 104 :M (\(ii\) )S 96 104 :M f1_12 sf (")S 105 104 :M f0_12 sf (k 1)S cF f1_12 sf S sf S cF f1_12 sf S sf (\312n, X)S 169 106 :M (k-1)S 185 104 :M ( is an ancestor of X)S 278 106 :M (k)S 0 -2 rm (, and X)S 0 2 rm 319 106 :M (k+1)S 338 104 :M ( is an ancestor of X)S 431 106 :M (k)S 0 -2 rm (, and)S 0 2 rm 78 122 :M (\(iii\) X)S 108 124 :M f0_9 sf (0)S 113 122 :M f0_12 sf ( is )S f4_12 sf (not )S f0_12 sf (a descendant of X)S f0_9 sf 0 2 rm (1)S 0 -2 rm 236 122 :M f0_12 sf (, and X)S 271 124 :M (n)S 0 -2 rm ( is )S 0 2 rm f4_12 sf 0 -2 rm (not )S 0 2 rm f0_12 sf 0 -2 rm (an ancestor of X)S 0 2 rm 388 124 :M (n+)S 401 124 :M f0_9 sf (1)S 406 122 :M f0_12 sf (,)S 60 140 :M 2.086 .209(then and are )J f2_12 sf 2.65 .265(mutually exclusive \(m.e.\) unshielded)J 60 158 :M (conductors on the itinerary )S 203 158 :M f0_12 sf (S 247 160 :M (n+)S 260 160 :M f0_9 sf (1)S 265 158 :M f0_12 sf (>.)S 60 182 :M 1.106 .111(If S 117 184 :M (n+)S 130 184 :M f0_9 sf (1)S 135 182 :M f0_12 sf .988 .099(> is an itinerary such that )J f1_12 sf (")S 278 182 :M f0_12 sf 1.249 .125(i,j 0 )J cF f1_12 sf .125A sf 1.249 .125( i < j)J 337 182 :M f1_12 sf (-)S 344 182 :M f0_12 sf 1.082 .108(1< j )J cF f1_12 sf .108A sf 1.082 .108( n+1 X)J 0 2 rm .302(i)A 0 -2 rm 1.115 .111( and X)J 0 2 rm .302(j)A 0 -2 rm 1.207 .121( are not)J 60 200 :M .09 .009(p-adjacent in the graph then we say that S 298 202 :M (n+)S 311 202 :M f0_9 sf (1)S 316 200 :M f0_12 sf .115 .012(> is an )J 352 200 :M f2_12 sf .049 .005(uncovered itinerary.)J 457 200 :M f0_12 sf .101 .01(. i.e. an)J 60 218 :M .716 .072(itinerary is uncovered if the only vertices on the itinerary which are p)J 407 218 :M .616 .062(-adjacent to other)J 60 236 :M (vertices on the itinerary, are those that occur consecutively on the itinerary.)S 60 260 :M f2_12 sf .569 .057(Theorem )J 111 260 :M .806 .081(9: )J 125 260 :M f0_12 sf <28>S 129 260 :M f2_12 sf .631 .063(Cyclic Equivalence Theorem,)J f0_12 sf .404 .04( Richardson 1994b\) Graphs )J f4_12 sf (G)S 429 262 :M f0_9 sf (1)S 434 260 :M f0_12 sf .564 .056( and )J f4_12 sf (G)S 468 262 :M f0_9 sf (2)S 473 260 :M f0_12 sf .744 .074( are)J 60 278 :M (d-separation equivalent if and only the following five conditions hold:)S 96 298 :M (CET\(I\) )S 134 298 :M f4_12 sf (G)S 143 300 :M f0_9 sf (1)S 148 298 :M f0_12 sf ( and )S f4_12 sf (G)S 180 300 :M f0_9 sf (2)S 185 298 :M f0_12 sf ( have the same p)S 265 298 :M (-adjacencies,)S 96 318 :M (CET\(II\) )S 138 318 :M f4_12 sf (G)S 147 320 :M f0_9 sf (1)S 152 318 :M f0_12 sf ( and )S f4_12 sf (G)S 184 320 :M f0_9 sf (2)S 189 320 :M f2_9 sf ( )S f0_12 sf 0 -2 rm (have the same unshielded elements i.e.)S 0 2 rm 168 335 :M (\(IIa\) the same unshielded conductors, and)S 168 352 :M (\(IIb\) the same unshielded perfect non-conductors,)S 96 372 :M .657 .066(CET\(III\) For all triples and , and are m.e.)J 96 388 :M .015 .002(conductors on some uncovered itinerary P\312)J f1_12 sf S 308 388 :M f0_12 sf .014 .001( in )J f4_12 sf (G)S 422 390 :M f0_9 sf (1)S 427 388 :M f0_12 sf .021 .002( if and only if)J 96 404 :M 2.988 .299( and are m.e. conductors on some uncovered itinerary)J 96 420 :M (Q)S 105 420 :M f2_12 sf S f1_12 sf S 115 420 :M f0_12 sf ( in )S f4_12 sf (G)S 229 422 :M f0_9 sf (2)S 234 422 :M f2_9 sf ( )S f0_12 sf 0 -2 rm (,)S 0 2 rm 96 440 :M .608 .061(CET\(IV\) If and are unshielded imperfect non-conductors \(in)J 96 456 :M f4_12 sf (G)S 105 458 :M f0_9 sf (1)S 110 456 :M f0_12 sf ( and )S f4_12 sf (G)S 142 458 :M f0_9 sf (2)S 147 456 :M f0_12 sf (\), then X is an ancestor of Y in )S 298 456 :M f4_12 sf (G)S 307 458 :M f0_9 sf (1)S 312 456 :M f0_12 sf ( iff X is an ancestor of Y in )S f4_12 sf (G)S 455 458 :M f0_9 sf (2)S 460 456 :M f0_12 sf (,)S 96 476 :M 1.154 .115(CET\(V\) If and are mutually exclusive conductors on some)J 96 492 :M 2.673 .267(uncovered itinerary )J 204 492 :M f2_12 sf 2.259(P)A f0_12 sf .84 .084( )J f1_12 sf S 226 492 :M f0_12 sf 2.832 .283( and is an unshielded)J 96 508 :M .182 .018(imperfect non-conductor \(in )J f4_12 sf (G)S 244 510 :M f0_9 sf (1)S 249 508 :M f0_12 sf .218 .022( and )J f4_12 sf (G)S 282 510 :M f0_9 sf (2)S 287 508 :M f0_12 sf .257 .026(\), then M is a descendant of B in )J f4_12 sf (G)S 458 510 :M f0_9 sf (1)S 463 508 :M f0_12 sf .299 .03( iff M)J 96 524 :M (is a descendant of B in )S 208 524 :M f4_12 sf (G)S 217 526 :M f0_9 sf (2)S 222 524 :M f0_12 sf (.)S 60 548 :M f2_12 sf .251 .025(Lemma 16:)J 118 548 :M f0_12 sf .286 .029( Given a sequence of vertices S 309 550 :M f0_10 sf (n+1)S 325 548 :M f0_12 sf .289 .029(> in a directed graph )J f4_12 sf (G)S 438 548 :M f14_13 sf .455 .045( )J 442 548 :M f0_12 sf .238 .024(having the)J 60 566 :M .264 .026(property that )J 126 566 :M f1_12 sf (")S 135 566 :M f0_12 sf .257 .026(k, 0\312)J cF f1_12 sf .026A sf .257 .026J cF f1_12 sf .026A sf .257 .026(\312n, X)J 206 568 :M f0_10 sf .097(k)A f0_12 sf 0 -2 rm .257 .026( is an ancestor of X)J 0 2 rm f0_10 sf .097(k)A f0_9 sf .186(+1)A 321 566 :M f0_12 sf .335 .034(, and X)J 357 568 :M f0_10 sf .174(k)A f0_12 sf 0 -2 rm .291 .029( is p)J 0 2 rm 383 566 :M .264 .026(-adjacent to X)J 451 568 :M f0_10 sf .053(k)A f0_9 sf .097 .01(+1 )J f0_12 sf 0 -2 rm .064(there)A 0 2 rm 60 584 :M (is a subsequence of the X)S f0_10 sf 0 2 rm (i)S 0 -2 rm 185 584 :M f0_12 sf (\325s, which we label the Y)S 302 586 :M f0_10 sf (j)S 305 584 :M f0_12 sf (\325s having the following properties:)S 96 602 :M (\(a\) X)S 121 604 :M f0_9 sf (0)S 126 602 :M f1_12 sf S 133 602 :M f0_12 sf (Y)S 142 604 :M f0_9 sf (0)S 96 620 :M f0_12 sf (\(b\) )S 113 620 :M f1_12 sf (")S 122 620 :M f0_12 sf (j, Y)S 140 622 :M f0_9 sf (j)S 143 620 :M f0_12 sf ( is an ancestor of Y)S 236 622 :M f0_10 sf (j)S 239 622 :M f0_9 sf (+1)S 96 638 :M f0_12 sf .348 .035(\(c\) )J f1_12 sf (")S 122 638 :M f0_12 sf .475 .047(j,k If j\312<\312k, Y)J f0_10 sf 0 2 rm (j)S 0 -2 rm 190 638 :M f0_12 sf .575 .058( and Y)J f0_10 sf 0 2 rm .234(k)A 0 -2 rm f0_12 sf .498 .05( are p)J 257 638 :M .511 .051(-adjacent in the graph if and only if k = j+1. i.e.)J 96 656 :M (the only Y)S 147 658 :M f0_10 sf (k)S f0_12 sf 0 -2 rm (\325s which are p)S 0 2 rm 220 656 :M (-adjacent are those that occur consecutively.)S 60 680 :M f2_12 sf (Proof)S 89 680 :M f0_12 sf (. The Y)S f0_10 sf 0 2 rm (k)S 0 -2 rm f0_12 sf (\325s can be constructed as follows:)S endp %%Page: 51 51 %%BeginPageSetup initializepage (peter; page: 51 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (51)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf (Let Y)S 88 58 :M f0_9 sf (0)S 93 56 :M f0_12 sf ( )S f1_12 sf S 103 56 :M f0_12 sf ( X)S 115 58 :M f0_9 sf (0)S 120 56 :M f0_12 sf (.)S 60 80 :M (Let Y)S 88 82 :M f0_10 sf (k)S f0_9 sf (+1)S 103 80 :M f0_12 sf ( )S f1_12 sf S 113 80 :M f0_12 sf ( X)S 125 82 :M f1_9 sf (h)S f0_12 sf 0 -2 rm ( where )S 0 2 rm f1_12 sf 0 -2 rm (h)S 0 2 rm f0_12 sf 0 -2 rm ( is the greatest h\312> j such that X)S 0 2 rm 324 82 :M f0_10 sf (h)S f0_12 sf 0 -2 rm ( is p-adjacent to X)S 0 2 rm 416 82 :M f0_10 sf (j)S 419 80 :M f0_12 sf ( where X)S 463 82 :M f0_9 sf (j)S 466 80 :M f1_12 sf S 473 80 :M f0_12 sf (Y)S 482 82 :M f0_10 sf (k)S f0_12 sf 0 -2 rm (.)S 0 2 rm 60 104 :M .04 .004(Property \(a\) is immediate from the construction. Property \(b\) follows from the transitivity)J 60 122 :M .192 .019(of the ancestor relation, and the fact that the Y)J f0_10 sf 0 2 rm .06(k)A 0 -2 rm f0_12 sf .192 .019(\325s are a subsequence of the X)J 433 124 :M f0_10 sf (i)S 436 122 :M f0_12 sf .206 .021(\325s. It is also)J 60 140 :M .775 .078(clear, from the construction that if k = j+1 then Y)J 309 142 :M f0_10 sf .31 .031(j )J f0_12 sf 0 -2 rm 1.364 .136(and Y)J 0 2 rm f0_10 sf .353(k)A f0_12 sf 0 -2 rm .752 .075( are p)J 0 2 rm 380 140 :M .554 .055(-adjacent. Moreover, if)J 60 158 :M (Y)S 69 160 :M f0_10 sf (j)S 72 158 :M f1_12 sf S 79 158 :M f0_12 sf (X)S 88 160 :M f1_9 sf (a)S 94 155 :M f1_8 sf (2)S 98 155 :M (1)S 102 160 :M f0_10 sf .231 .023( )J 105 158 :M f0_12 sf .11 .011(and Y)J f0_10 sf 0 2 rm (k)S 0 -2 rm f1_12 sf S 146 158 :M f0_12 sf (X)S 155 160 :M f1_9 sf (b)S 160 158 :M f0_12 sf .185 .018( are p)J 188 158 :M .135 .013(-adjacent, and j\312<\312k, then it follows again from the construction)J 60 176 :M .118 .012(that if Y)J f0_10 sf 0 2 rm (j)S 0 -2 rm 103 178 :M f0_9 sf (+1)S 113 176 :M f1_12 sf S 120 176 :M f0_12 sf (X)S 129 178 :M f1_9 sf (g)S 133 176 :M f0_12 sf .141 .014(, then )J 164 176 :M f1_12 sf (b)S 171 176 :M f0_12 sf S cF f1_12 sf S sf S 184 176 :M f1_12 sf (g)S 189 176 :M f0_12 sf .115 .011(, so k\312)J cF f1_12 sf .011A sf .115 .011(\312j+1. \(This is because the Y)J f0_10 sf 0 2 rm (k)S 0 -2 rm f0_12 sf .124 .012(\325s are a subsequence of the)J 60 194 :M (X)S 69 196 :M f0_10 sf (i)S 72 194 :M f0_12 sf (\325s.\) Hence Y)S 133 196 :M f0_10 sf (j)S 136 196 :M f0_9 sf (+1)S 146 194 :M f1_12 sf S 153 194 :M f0_12 sf (Y)S 162 196 :M f0_10 sf (k)S f0_12 sf 0 -2 rm (.)S 0 2 rm f1_12 sf 0 -2 rm <5C>S 0 2 rm 60 218 :M f2_12 sf 1.053 .105(Lemma 17:)J 120 218 :M f0_12 sf 1.58 .158( Let )J 146 218 :M f4_12 sf (G)S 155 220 :M f0_9 sf (1)S 160 218 :M f0_12 sf 1.106 .111( and )J f4_12 sf (G)S 196 220 :M f0_9 sf (2)S 201 218 :M f0_12 sf 1.169 .117( be two graphs satisfying CET\(I\)\320\(III\) Suppose there is a)J 60 236 :M 1.059 .106(directed path D)J 138 238 :M f0_9 sf (1)S 143 236 :M f1_12 sf S 155 236 :M f0_12 sf S 176 238 :M f0_10 sf .563(n)A f0_12 sf 0 -2 rm .825 .083(, in )J 0 2 rm f4_12 sf 0 -2 rm (G)S 0 2 rm 211 238 :M f0_9 sf (1)S 216 236 :M f0_12 sf 1.385 .138(. Let D)J 254 238 :M f0_9 sf (0)S 259 236 :M f0_12 sf 1.241 .124( be a vertex distinct from D)J 402 238 :M f0_9 sf (1)S 407 236 :M f1_12 sf (,)S f0_12 sf S 434 238 :M f0_10 sf .437(n)A f0_12 sf 0 -2 rm .989 .099(, s.t. D)J 0 2 rm f0_9 sf (0)S 479 236 :M f0_12 sf 1.5 .15( is)J 60 254 :M .117 .012(p-adjacent to D)J f0_9 sf 0 2 rm (1)S 0 -2 rm 138 254 :M f0_12 sf .188 .019( in )J 154 254 :M f4_12 sf (G)S 163 256 :M f0_9 sf (1)S 168 254 :M f0_12 sf .126 .013( and )J f4_12 sf (G)S 200 256 :M f0_9 sf (2)S 205 254 :M f0_12 sf .181 .018(, D)J 220 256 :M f0_9 sf (0)S 225 254 :M f0_12 sf .171 .017( is not p)J 264 254 :M .119 .012(-adjacent to D)J f0_9 sf 0 2 rm (2)S 0 -2 rm 336 254 :M f0_12 sf <2CC944>S 360 256 :M f0_10 sf .107(n)A f0_12 sf 0 -2 rm .142 .014( in )J 0 2 rm 381 254 :M f4_12 sf (G)S 390 256 :M f0_9 sf (1)S 395 254 :M f0_12 sf .117 .012( or )J f4_12 sf (G)S 420 256 :M f0_9 sf (2)S 425 254 :M f0_12 sf .173 .017( and D)J 458 256 :M f0_9 sf (0)S 463 254 :M f0_12 sf .167 .017( is not)J 60 272 :M (a descendant of D)S f0_9 sf 0 2 rm (1)S 0 -2 rm 151 272 :M f0_12 sf ( in )S f4_12 sf (G)S 175 274 :M f0_9 sf (1)S 180 272 :M f0_12 sf ( or )S 196 272 :M f4_12 sf (G)S 205 274 :M f0_9 sf (2)S 210 272 :M f0_12 sf (. It then follows that D)S 319 274 :M f0_9 sf (1)S 324 272 :M f0_12 sf ( is an ancestor of D)S 417 274 :M f0_10 sf (n)S f0_12 sf 0 -2 rm ( in )S 0 2 rm f4_12 sf 0 -2 rm (G)S 0 2 rm 446 274 :M f0_9 sf (2)S 451 272 :M f0_12 sf (.)S 60 296 :M f2_12 sf (Proof)S 89 296 :M f0_12 sf (. By induction on n.)S 60 320 :M f2_12 sf (Base)S 84 320 :M f0_12 sf 2.541 .254( )J 90 320 :M f2_12 sf (Case)S 115 320 :M f0_12 sf 1.761 .176(: n = 2. Since, by hypothesis, D)J 284 322 :M f0_9 sf (0)S 289 320 :M f0_12 sf 1.669 .167( is not p-adjacent to D)J f0_9 sf 0 2 rm (2)S 0 -2 rm 411 320 :M f0_12 sf 1.657 .166(, it follows that)J 60 338 :M (S 95 340 :M f0_9 sf (1)S 100 338 :M f0_12 sf <2CCA44>S 115 340 :M f0_9 sf (2)S 120 338 :M f0_12 sf -.004(> forms an unshielded conductor in )A f4_12 sf (G)S 302 340 :M f0_9 sf (1)S 307 338 :M f0_12 sf ( \(since D)S f0_9 sf 0 2 rm (1)S 0 -2 rm 355 338 :M f0_12 sf -.007( is an ancestor of D)A 448 340 :M f0_9 sf (2)S 453 338 :M f0_12 sf (\). Hence)S 60 356 :M .041 .004(this triple of vertices also forms an unshielded conductor in )J 349 356 :M f4_12 sf (G)S 358 358 :M f0_9 sf (2)S 363 356 :M f0_12 sf .042 .004(, by CET\(IIa\). Hence D)J f0_9 sf 0 2 rm (1)S 0 -2 rm 481 356 :M f0_12 sf .056 .006( is)J 60 374 :M .289 .029(an ancestor of D)J 141 376 :M f0_9 sf (0)S 146 374 :M f0_12 sf .293 .029( or D)J f0_9 sf 0 2 rm (2)S 0 -2 rm 176 374 :M f0_12 sf .231 .023( in )J f4_12 sf (G)S 201 376 :M f0_9 sf (2)S 206 374 :M f0_12 sf .275 .028(. Since, by hypothesis D)J 325 376 :M f0_9 sf (1)S 330 374 :M f0_12 sf .313 .031( is not an ancestor of D)J 444 376 :M f0_9 sf (0)S 449 374 :M f0_12 sf .231 .023( in )J f4_12 sf (G)S 474 376 :M f0_9 sf (2)S 479 374 :M f0_12 sf .333 .033(, it)J 60 392 :M (follows that D)S 129 394 :M f0_9 sf (1)S 134 392 :M f0_12 sf ( is an ancestor of D)S 227 394 :M f0_9 sf (2)S 232 392 :M f0_12 sf ( in )S f4_12 sf (G)S 256 394 :M f0_9 sf (2)S 261 392 :M f0_12 sf (.)S 60 416 :M f2_12 sf .062 .006(Induction Case:)J 142 416 :M f0_12 sf .098 .01( Suppose that the hypothesis is true for paths of length n. It follows from)J 60 434 :M .671 .067(Lemma 16)J 113 434 :M .725 .072( that there is a subsequence S 0 2 rm 292 434 :M f0_12 sf (D)S 301 436 :M f0_9 sf (0)S 306 434 :M f0_12 sf (,D)S 318 436 :M f1_9 sf (a)S 324 436 :M f0_9 sf (\(1\))S f0_12 sf 0 -2 rm (,D)S 0 2 rm 346 436 :M f1_9 sf (a)S 352 436 :M f0_9 sf (\(2\))S f0_12 sf 0 -2 rm S 0 2 rm 383 436 :M f1_9 sf (a)S 389 436 :M f0_9 sf (\(r\))S 398 434 :M f1_12 sf S 405 434 :M f0_12 sf (D)S 414 436 :M f0_10 sf .227(n)A f0_12 sf 0 -2 rm .727 .073(> such that the)J 0 2 rm 60 452 :M .782 .078(only p-adjacent vertices are those that occur consecutively, and in )J 391 452 :M f4_12 sf (G)S 400 454 :M f0_9 sf (1)S 405 454 :M f2_9 sf .129 .013( )J f0_12 sf 0 -2 rm 1.105 .11(each vertex is an)J 0 2 rm 60 470 :M .998 .1(ancestor of the next vertex in the sequence. Moreover, since, by hypothesis, D)J f0_9 sf 0 2 rm (0)S 0 -2 rm 460 470 :M f0_12 sf 1.282 .128( is not)J 60 488 :M .843 .084(p-adjacent to D)J 136 490 :M f0_9 sf (2)S 141 488 :M f0_12 sf <2CC944>S 165 490 :M f0_10 sf .302(n)A f0_12 sf 0 -2 rm .862 .086(, it follows that D)J 0 2 rm f1_9 sf (a)S 266 490 :M f0_9 sf (\(1\))S f1_12 sf 0 -2 rm S 0 2 rm 283 488 :M f0_12 sf (D)S 292 490 :M f0_9 sf (1)S 297 488 :M f0_12 sf 1.061 .106(. Since )J 336 488 :M f4_12 sf (G)S 345 490 :M f0_9 sf (1)S 350 488 :M f0_12 sf .836 .084( and )J f4_12 sf (G)S 385 490 :M f0_9 sf (2)S 390 490 :M f2_9 sf .09 .009( )J f0_12 sf 0 -2 rm 1.077 .108(satisfy CET\(I\), they)J 0 2 rm 60 506 :M .66 .066(have the same p-adjacencies, hence in )J 251 506 :M f4_12 sf (G)S 260 508 :M f0_9 sf (2)S 265 508 :M f2_9 sf 1.068 .107( )J 269 506 :M f0_12 sf .644 .064(the only vertices that are p-adjacent are those)J 60 524 :M .607 .061(that occur consecutively in the sequence. Suppose, for a contradiction that D)J 439 526 :M f1_9 sf (a)S 445 526 :M f0_9 sf .148(\(r-1\))A f0_12 sf 0 -2 rm .505 .051( is not)J 0 2 rm 60 542 :M .934 .093(an ancestor of D)J 144 544 :M f1_9 sf (a)S 150 544 :M f0_9 sf (\(r\))S 159 542 :M f0_12 sf 1.219 .122( in )J 176 542 :M f4_12 sf (G)S 185 544 :M f0_9 sf (2)S 190 542 :M f0_12 sf 1.001 .1(. Let s be the smallest j such that D)J 371 544 :M f1_9 sf (a)S 377 544 :M f0_9 sf .183(\(j\))A f0_12 sf 0 -2 rm .909 .091( is not an ancestor of)J 0 2 rm 60 560 :M (D)S 69 562 :M f1_9 sf (a)S 75 562 :M f0_9 sf .409 .041(\(j-1\) )J f0_12 sf 0 -2 rm .369 .037(in )J 0 2 rm 105 560 :M f4_12 sf (G)S 114 562 :M f0_9 sf (2)S 119 560 :M f0_12 sf .365 .036(. \(Such a j exists since D)J f1_9 sf 0 2 rm (a)S 0 -2 rm 246 562 :M f0_9 sf (\(1\))S f1_12 sf 0 -2 rm S 0 2 rm 263 560 :M f0_12 sf (D)S 272 562 :M f0_9 sf (1)S 277 560 :M f0_12 sf .358 .036( and D)J f1_9 sf 0 2 rm (a)S 0 -2 rm 316 562 :M f0_9 sf (\(0\))S f1_12 sf 0 -2 rm S 0 2 rm 333 560 :M f0_12 sf (D)S 342 562 :M f0_9 sf (0)S 347 560 :M f0_12 sf .391 .039( is not a descendant of D)J 469 562 :M f0_9 sf (1)S 474 560 :M f0_12 sf .391 .039(.\) It)J 60 578 :M .027 .003(then follows that and are mutually exclusive)J 60 596 :M .428 .043(conductors on the unshielded itinerary S 313 598 :M f1_9 sf (a)S 319 598 :M f0_9 sf (\(r\))S 328 596 :M f0_12 sf .668 .067(> in )J 352 596 :M f4_12 sf (G)S 361 599 :M f0_7 sf (2)S 365 596 :M f0_12 sf .527 .053(. But these two triples are)J 60 614 :M .253 .025(not mutually exclusive in )J 187 614 :M f4_12 sf (G)S 196 616 :M f0_9 sf (1)S 201 614 :M f0_12 sf .305 .031( since D)J 240 616 :M f1_9 sf (a)S 246 616 :M f0_9 sf .062(\(r-1\))A f0_12 sf 0 -2 rm .253 .025( is an ancestor of D)J 0 2 rm 358 616 :M f1_9 sf (a)S 364 616 :M f0_9 sf (\(r\))S 373 614 :M f0_12 sf .358 .036( in )J 388 614 :M f4_12 sf (G)S 397 616 :M f0_9 sf (1)S 402 614 :M f0_12 sf .239 .024(; hence )J f4_12 sf (G)S 449 616 :M f0_9 sf (1)S 454 614 :M f0_12 sf .24 .024( and )J f4_12 sf (G)S 487 616 :M f0_9 sf (2)S 60 632 :M f0_12 sf (fail to satisfy CET\(III\), which is a contradiction.)S 60 675 :M ( )S 60 672.48 -.48 .48 204.48 672 .48 60 672 @a 60 684 :M f0_8 sf (2)S 64 684 :M (1)S 68 687 :M f0_10 sf ( That is, the j)S f0_6 sf 0 -4 rm (th)S 0 4 rm 125 687 :M f0_10 sf ( vertex in the sequence of Y vertices is the )S f1_10 sf (a)S f0_6 sf 0 -4 rm (th)S 0 4 rm 308 687 :M f0_10 sf ( vertex in the sequence of X vertices.)S endp %%Page: 52 52 %%BeginPageSetup initializepage (peter; page: 52 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (52)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf 1.071 .107(It follows that D)J 144 58 :M f1_9 sf (a)S 150 58 :M f0_9 sf .235(\(r-1\))A f0_12 sf 0 -2 rm .948 .095( is an ancestor of D)J 0 2 rm f1_9 sf (a)S 273 58 :M f0_9 sf (\(r\))S 282 56 :M f0_12 sf .875 .087( in )J f4_12 sf (G)S 309 58 :M f0_9 sf (2)S 314 56 :M f0_12 sf 1.038 .104(. It then follows from the induction)J 60 74 :M (hypothesis that D)S 144 76 :M f0_9 sf (1)S 149 74 :M f0_12 sf ( is an ancestor of D)S 242 76 :M f1_9 sf (a)S 248 76 :M f0_9 sf (\(r\))S 257 74 :M f1_12 sf S 264 74 :M f0_12 sf (D)S 273 76 :M f0_10 sf (n)S f0_12 sf 0 -2 rm (. )S 0 2 rm f1_12 sf 0 -2 rm <5C>S 0 2 rm 60 98 :M f2_12 sf 1.257 .126(Theorem 8)J 118 98 :M f0_12 sf 1.741 .174(: \(d)J 137 98 :M 1.306 .131(-separation Completeness\) If the CCD algorithm, when given as input)J 60 116 :M .089 .009(d-separation oracles for the graphs )J 229 116 :M f4_12 sf (G)S 238 118 :M f0_9 sf (1)S 243 116 :M f0_12 sf .052 .005(, )J f4_12 sf (G)S 258 118 :M f0_9 sf (2)S 263 116 :M f0_12 sf .102 .01( produces as output PAGs )J 392 116 :M f3_12 sf (Y)S 402 118 :M f3_9 sf (1)S 407 116 :M f0_12 sf .049 .005(, )J f3_12 sf (Y)S 423 118 :M f3_9 sf (2)S 428 116 :M f0_12 sf .066 .007( respectively,)J 60 134 :M .378 .038(then )J 85 134 :M f3_12 sf (Y)S 95 136 :M f3_9 sf (1)S 100 134 :M f0_12 sf .378 .038( is identical to )J 173 134 :M f3_12 sf (Y)S 183 136 :M f3_9 sf (2)S 188 134 :M f0_12 sf .356 .036( if and only if )J f4_12 sf (G)S 267 137 :M f0_7 sf (1)S 271 134 :M f0_12 sf .309 .031( and )J f4_12 sf (G)S 304 137 :M f0_7 sf (2)S 308 134 :M f0_12 sf .423 .042( are d)J 336 134 :M .274 .027(-separation equivalent, i.e. )J 467 134 :M f4_12 sf (G)S 476 137 :M f0_7 sf (2)S 480 134 :M f0_12 sf S f1_12 sf S 60 152 :M f2_12 sf (Equiv)S 91 152 :M f0_12 sf <28>S 95 152 :M f4_12 sf (G)S 104 155 :M f0_7 sf (1)S 108 152 :M f0_12 sf (\) and vice versa.)S 60 176 :M f2_12 sf (Proof.)S 92 176 :M f0_12 sf .422 .042( We will show that if two graphs, )J 260 176 :M f4_12 sf (G)S 269 178 :M f0_9 sf (1)S 274 176 :M f0_12 sf .348 .035( and )J f4_12 sf (G)S 307 178 :M f0_9 sf (2)S 312 176 :M f0_12 sf .497 .05( are )J 334 176 :M f4_12 sf .125(not)A f0_12 sf .201 .02( d)J 359 176 :M .254 .025(-separation equivalent, then)J 60 194 :M 1.113 .111(the PAGs output by the CCD algorithm, given d-separation oracles for )J f4_12 sf (G)S 431 196 :M f0_9 sf (1)S 436 194 :M f0_12 sf 1.083 .108( and )J f4_12 sf (G)S 472 196 :M f0_9 sf (2)S 477 194 :M f0_12 sf 1.547 .155( as)J 60 212 :M (input, would differ in some respect.)S 60 236 :M .7 .07(It follows from the Cyclic Equivalence Theorem that if )J f4_12 sf (G)S 347 238 :M f0_9 sf (1)S 352 236 :M f0_12 sf .971 .097( and )J 378 236 :M f4_12 sf (G)S 387 238 :M f0_9 sf (2)S 392 236 :M f0_12 sf .699 .07( are not d-separation)J 60 254 :M .082 .008(equivalent, then they fail to satisfy one or more of the five conditions CET\(I\)-\(V\). Let )J f3_12 sf (Y)S 487 256 :M f0_10 sf (1)S 60 272 :M f0_12 sf 2.729 .273(and )J 84 272 :M f3_12 sf (Y)S 94 274 :M f0_9 sf (2)S 99 272 :M f0_12 sf 2.181 .218( denote, respectively, the PAGs output by the CCD algorithm when given)J 60 290 :M (d-separation oracles for )S f4_12 sf (G)S 184 292 :M f0_9 sf (1)S 189 290 :M f0_12 sf ( and )S f4_12 sf (G)S 221 292 :M f0_9 sf (2)S 226 290 :M f0_12 sf ( as input.)S 60 314 :M f2_12 sf 1.411 .141(Case 1)J 96 314 :M f0_12 sf .745 .074(: )J f4_12 sf (G)S 113 316 :M f0_9 sf (1)S 118 314 :M f0_12 sf 1.763 .176( and )J 146 314 :M f4_12 sf (G)S 155 316 :M f0_9 sf (2)S 160 314 :M f0_12 sf 1.365 .137( fail to satisfy CET\(I\). In this case the two graphs have different)J 60 332 :M 2.022 .202(p-adjacencies. Let us suppose without loss of generality that there is some pair of)J 60 350 :M .562 .056(variables, X and Y which are p-adjacent in )J f4_12 sf (G)S 283 352 :M f0_9 sf (1)S 288 350 :M f0_12 sf .693 .069( and not p)J 339 350 :M .563 .056(-adjacent in )J 398 350 :M f4_12 sf (G)S 407 352 :M f0_9 sf (2)S 412 350 :M f0_12 sf .667 .067(. Since X and Y)J 60 368 :M .645 .064(are p)J 85 368 :M .477 .048(-adjacent in )J f4_12 sf (G)S 154 370 :M f0_9 sf (1)S 159 368 :M f0_12 sf .586 .059(, X and Y are d-connected conditional upon any subset of the other)J 60 386 :M (vertices. Hence there is an edge between X and Y in )S 314 386 :M f3_12 sf (Y)S 324 388 :M f0_10 sf (1)S f0_12 sf 0 -2 rm (.)S 0 2 rm 60 410 :M .202 .02(Since X and Y are not p)J 177 410 :M .146 .015(-adjacent in )J f4_12 sf (G)S 244 412 :M f0_9 sf (2)S 249 410 :M f0_12 sf .192 .019(, there is some subset )J 356 410 :M f2_12 sf (S)S 363 410 :M f0_12 sf .134 .013(, \(X,Y)J f1_12 sf S 402 410 :M f2_12 sf (S)S 409 410 :M f0_12 sf .197 .02(\) such that X and)J 60 428 :M .521 .052(Y are d-separated in )J 162 428 :M f4_12 sf (G)S 171 430 :M f0_9 sf .711 .071(2 )J 179 428 :M f0_12 sf .399 .04(given )J f2_12 sf (S)S 216 428 :M f0_12 sf .549 .055(. It follows from )J 302 428 :M .576 .058(Lemma 6 that X and Y are d)J 444 428 :M (-separated)S 60 446 :M .297 .03(by a set of variables )J f2_12 sf .173(T)A f0_12 sf .301 .03(, such that either )J f2_12 sf .173(T)A f0_12 sf .3 .03( is a subset of the vertices p-adjacent to X, or )J 484 446 :M f2_12 sf (T)S 60 464 :M f0_12 sf .111 .011(is a subset of the vertices p-adjacent to Y. It follows that in step \246A of the CCD algorithm)J 60 482 :M .062 .006(the edge between X and Y in )J 203 482 :M f3_12 sf (Y)S 213 484 :M f0_9 sf (2)S 218 482 :M f14_13 sf .09 .009( )J 221 482 :M f0_12 sf .056 .006(would be removed. Since edges are not added back in at)J 60 500 :M .334 .033(any later stage of the algorithm, there is no edge in )J f3_12 sf (Y)S 322 502 :M f0_9 sf (2)S 327 500 :M f0_12 sf .344 .034( between X and Y. Hence )J f3_12 sf (Y)S 466 502 :M f0_10 sf .102(1)A f0_12 sf 0 -2 rm .32 .032( and)J 0 2 rm 60 518 :M f3_12 sf (Y)S 70 520 :M f0_9 sf (2)S 75 518 :M f0_12 sf ( are different.)S 60 542 :M f2_12 sf .155 .016(Case 2)J 94 542 :M f0_12 sf .212 .021(: )J 101 542 :M f4_12 sf (G)S 110 544 :M f0_9 sf (1)S 115 542 :M f0_12 sf .136 .014( and )J f4_12 sf (G)S 147 544 :M f0_9 sf (2)S 152 542 :M f0_12 sf .154 .015( fail to satisfy CET\(IIa\). We assume that )J 352 542 :M f4_12 sf (G)S 361 544 :M f0_9 sf (1)S 366 542 :M f0_12 sf .136 .014( and )J f4_12 sf (G)S 398 544 :M f0_9 sf (2)S 403 542 :M f0_12 sf .145 .015( satisfy CET\(I\). In)J 60 560 :M .155 .015(this case the two graphs have different unshielded non-conductors. Thus we may assume,)J 60 578 :M -.004(without loss of generality, that there is some triple of vertices such that in )A 463 578 :M f4_12 sf (G)S 472 580 :M f0_9 sf (1)S 477 578 :M f0_12 sf (, Y)S 60 596 :M (is an ancestor of X or Z, while Y is not an ancestor of either X or Z)S 383 596 :M f4_12 sf ( )S f0_12 sf (in )S f4_12 sf (G)S 407 598 :M f0_9 sf (2)S 412 596 :M f0_12 sf (.)S 60 620 :M 1.946 .195(If Y is an ancestor of X or Z then it follows from Lemma 8)J 382 620 :M 1.826 .183( that every set which)J 60 638 :M .702 .07(d-separates X and Z includes Y. Hence Y)J f1_12 sf S 275 638 :M f2_12 sf (Sepset)S 308 638 :M f0_12 sf .652 .065(\(X,Z\) in )J f4_12 sf (G)S 361 640 :M f0_9 sf (1)S 366 638 :M f0_12 sf .76 .076(. It then follows from the)J 60 656 :M .852 .085(correctness of the algorithm that in )J 239 656 :M f3_12 sf (Y)S 249 658 :M f0_9 sf (1)S 254 656 :M f0_12 sf .903 .09(, either X\321>Y\321)J f1_12 sf .502 .05(*Z, )J f0_12 sf (X)S 370 656 :M f1_12 sf .204(*)A f0_12 sf .334A f1_12 sf .249(Z)A f0_12 sf .375 .038(, or X)J 453 656 :M f1_12 sf (*)S f0_12 sf S 465 0 6 730 rC 465 656 :M f1_12 sf 12 f7_1 :p 4.364 :m 1.364 .136( )J 467 656 :M 8.727 :m 1.299 .13( )J gR gS 0 0 552 730 rC 465 656 :M f1_12 sf 12 f7_1 :p 6 :m (*)S 465 0 6 730 rC 465 656 :M 4.364 :m 1.364 .136( )J 467 656 :M 8.727 :m 1.299 .13( )J gR gS 471 0 9 730 rC 471 656 :M f0_12 sf 12 f6_1 :p 8.727 :m 1.299 .13( )J 476 656 :M 8.727 :m 1.299 .13( )J gR gS 0 0 552 730 rC 471 656 :M f0_12 sf 12 f6_1 :p 8.663 :m (Y)S 471 0 9 730 rC 471 656 :M 8.727 :m 1.299 .13( )J 476 656 :M 8.727 :m 1.299 .13( )J gR gS 480 0 6 730 rC 480 656 :M f1_12 sf 12 f7_1 :p 4.364 :m 1.364 .136( )J 482 656 :M 8.727 :m 1.299 .13( )J gR gS 0 0 552 730 rC 480 656 :M f1_12 sf 12 f7_1 :p 6 :m (*)S 480 0 6 730 rC 480 656 :M 4.364 :m 1.364 .136( )J 482 656 :M 8.727 :m 1.299 .13( )J gR gS 0 0 552 730 rC 486 656 :M f0_12 sf S 60 674 :M f1_12 sf (*Z)S f0_12 sf (.)S endp %%Page: 53 53 %%BeginPageSetup initializepage (peter; page: 53 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (53)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf .32 .032(If Y is not an ancestor of X or Z in )J 235 56 :M f4_12 sf (G)S 244 58 :M f0_9 sf (2)S 249 56 :M f0_12 sf .283 .028(, then Y is not in any minimal d-separating set for)J 60 74 :M .67 .067(X and Z. In particular Y)J f1_12 sf S 190 74 :M f2_12 sf (Sepset)S 223 74 :M f0_12 sf .735 .073(\(X,Z\) for )J 273 74 :M f4_12 sf (G)S 282 76 :M f0_9 sf (2)S 287 74 :M f0_12 sf .672 .067(. Again it follows from the correctness of)J 60 92 :M .536 .054(the algorithm that is oriented as X)J f1_12 sf .173(*)A f0_12 sf .197A f1_12 sf .173(*)A f0_12 sf .368 .037(Z or X)J 356 92 :M f1_12 sf (*)S f0_12 sf S 368 0 7 730 rC 368 92 :M .923 .092( )J 371 92 :M .879 .088( )J gR gS 0 0 552 730 rC 368 92 :M f0_12 sf (>Y<)S 383 0 7 730 rC 383 92 :M .923 .092( )J 386 92 :M .879 .088( )J gR gS 0 0 552 730 rC 390 92 :M f0_12 sf .32A f1_12 sf .32(*)A f0_12 sf .503 .05(Z in )J 427 92 :M f3_12 sf (Y)S 437 94 :M f0_9 sf (2)S 442 92 :M f0_12 sf .71 .071(. Thus )J 477 92 :M f3_12 sf (Y)S 487 94 :M f0_9 sf (1)S 1 G 0 0 1 1 rF 491 94 :M psb /wp$x1 368 def /wp$x2 389 def /wp$y 94 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 110 :M 0 G f0_12 sf (and )S f3_12 sf (Y)S 90 112 :M f0_9 sf (2)S 95 110 :M f0_12 sf ( are different.)S 60 134 :M f2_12 sf .707 .071(Case 3)J 95 134 :M f0_12 sf .373 .037(: )J f4_12 sf (G)S 111 136 :M f0_9 sf (1)S 116 134 :M f0_12 sf .618 .062( and )J f4_12 sf (G)S 150 136 :M f0_9 sf (2)S 155 134 :M f0_12 sf .701 .07( fail to satisfy CET\(IIb\). We assume that )J 362 134 :M f4_12 sf (G)S 371 136 :M f0_9 sf (1)S 376 134 :M f0_12 sf .618 .062( and )J f4_12 sf (G)S 410 136 :M f0_9 sf (2)S 415 134 :M f0_12 sf .606 .061( satisfy CET\(I\),)J 60 152 :M 4.737 .474(CET\(IIa\). In this case the two graphs have different unshielded imperfect)J 60 170 :M 1.961 .196(non-conductors, i.e. there is some triple such tha)J 366 170 :M 2.132 .213(t it forms an unshielded)J 60 188 :M .239 .024(non-conductor in both )J 170 188 :M f4_12 sf (G)S 179 190 :M f0_9 sf (1)S 184 188 :M f0_12 sf .237 .024( and )J f4_12 sf (G)S 217 190 :M f0_9 sf (2)S 222 188 :M f0_12 sf .282 .028(, but in one graph Y is a descendant of a common child)J 60 206 :M .521 .052(of X and Z, while in the other graph it is not. Let us assume that Y is a descendant of a)J 60 224 :M (common child of X and Z in )S 200 224 :M f4_12 sf (G)S 209 226 :M f0_9 sf (1)S 214 224 :M f0_12 sf (, while in )S 262 224 :M f4_12 sf (G)S 271 226 :M f0_9 sf (2)S 276 224 :M f0_12 sf ( it is not.)S 60 248 :M 2.026 .203(It follows from )J 146 248 :M 1.787 .179(Lemma 11)J 201 248 :M 1.887 .189( that in )J f4_12 sf (G)S 255 250 :M f0_9 sf (1)S 260 248 :M f0_12 sf 2.009 .201(, X and Z are d-connected given any subset)J 60 266 :M 2.152 .215(containing Y. In this case the search in CCD section \246D will fail to find any set)J 60 284 :M f2_12 sf (Supsepset)S 111 284 :M f0_12 sf .538 .054(. Hence will be oriented as X)J f1_12 sf .079 .008( )J 351 284 :M f0_12 sf .153A f1_12 sf .192 .019( Z)J f0_12 sf .422 .042( \(i.e. without dotted)J 60 302 :M 1.161 .116(underlining\) in )J 139 302 :M f3_12 sf (Y)S 149 304 :M f0_9 sf (1)S 154 302 :M f0_12 sf 1.458 .146(. If Y is not a descendant of a common child of X and Z, then it)J 60 320 :M .243 .024(follows from Lemma 12 and Lemma 13)J 255 320 :M .29 .029( that there is some subset )J f2_12 sf .164(T)A f0_12 sf .142 .014( of )J 405 320 :M f2_12 sf (Local)S 434 320 :M f0_12 sf <28>S 438 320 :M f3_12 sf (Y)S 448 320 :M f0_12 sf .209 .021(,X\), such)J 60 338 :M .155 .016(that X and Z are d-separated given )J 230 338 :M f2_12 sf .126(T)A f0_12 sf ( )S f1_12 sf .145A f0_12 sf .165 .017( {Y} )J f1_12 sf .145A f0_12 sf ( )S 289 338 :M f2_12 sf (Sepset)S 322 338 :M f0_12 sf .149 .015(. Section \246D will find such a)J 60 356 :M .529 .053(set )J f2_12 sf .363(T)A f0_12 sf .766 .077(, and hence will be oriented as X)J f1_12 sf .272(*)A f0_12 sf S 311 0 7 730 rC 311 356 :M 1.213 .121( )J 314 356 :M 1.155 .115( )J gR gS 0 0 552 730 rC 311 356 :M f0_12 sf (>Y<)S 326 0 7 730 rC 326 356 :M 1.213 .121( )J 329 356 :M 1.155 .115( )J gR gS 0 0 552 730 rC 333 356 :M f0_12 sf .329A f1_12 sf .366(*Z)A f0_12 sf .35 .035( in )J f3_12 sf (Y)S 379 358 :M f0_9 sf (2)S 384 356 :M f0_12 sf .728 .073(. Since no subsequent)J 1 G 0 0 1 1 rF 491 356 :M psb /wp$x1 311 def /wp$x2 332 def /wp$y 358 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 374 :M 0 G 1.769 .177(orientation rule removes or adds dotted underlining, it follows that )J 412 374 :M f3_12 sf (Y)S 422 376 :M f0_9 sf (1)S 427 374 :M f0_12 sf 1.688 .169( and )J f3_12 sf (Y)S 466 376 :M f0_9 sf (2)S 471 374 :M f0_12 sf 2.287 .229( are)J 60 392 :M (different.)S 60 416 :M f2_12 sf .79 .079(Case 4)J 95 416 :M f0_12 sf .417 .042(: )J f4_12 sf (G)S 111 418 :M f0_9 sf (1)S 116 416 :M f0_12 sf .988 .099( and )J 142 416 :M f4_12 sf (G)S 151 418 :M f0_9 sf (2)S 156 416 :M f0_12 sf .744 .074( fail to satisfy CET\(III\). We assume that )J f4_12 sf (G)S 370 418 :M f0_9 sf (1)S 375 416 :M f0_12 sf .988 .099( and )J 401 416 :M f4_12 sf (G)S 410 418 :M f0_9 sf (2)S 415 416 :M f0_12 sf .677 .068( satisfy CET\(I\),)J 60 434 :M 1.187 .119(CET\(IIa\), CET\(IIb\). In this case the two graphs have the same p-adjacencies, and the)J 60 452 :M 1.81 .181(same unshielded conductors, perfect non-conductors, and imperfect non-conductors.)J 60 470 :M .096 .01(However, the two graphs have different mutually exclusive conductors. Hence in both )J f4_12 sf (G)S 487 472 :M f0_9 sf (1)S 60 488 :M f0_12 sf 5.015 .502(and )J 87 488 :M f4_12 sf (G)S 96 490 :M f0_9 sf (2)S 101 488 :M f0_12 sf 3.97 .397( there is an uncovered itinerary, S 339 490 :M f0_10 sf (n+)S 350 490 :M f0_9 sf (1)S 355 488 :M f0_12 sf 4.14 .414(> such that every triple)J 60 506 :M ( \(1)J cF f1_12 sf .07A sf .701 .07(k)J cF f1_12 sf .07A sf .701 .07(n\) on this itinerary is a conductor, but in one graph )S 60 524 :M (and are mutually exclusive, i.e. X)J 304 526 :M f0_9 sf (1)S 309 524 :M f0_12 sf .014 .001( is not an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 425 524 :M f0_12 sf (, and X)S f0_10 sf 0 2 rm (n)S 0 -2 rm f0_12 sf ( is not)S 60 542 :M .622 .062(an ancestor of X)J f0_10 sf 0 2 rm .386(n+)A 0 -2 rm 153 544 :M f0_9 sf (1)S 158 542 :M f0_12 sf .695 .069(, while in the other they are not mutually exclusive. Let us suppose)J 60 560 :M -.004(without loss of generality that and are mutually exclusive in)A 60 578 :M f4_12 sf (G)S 69 580 :M f0_9 sf (1)S 74 578 :M f0_12 sf (, while in )S 122 578 :M f4_12 sf (G)S 131 580 :M f0_9 sf (2)S 136 578 :M f0_12 sf ( they are not.)S 60 602 :M .379 .038(It follows from the definition of m.e. conductors that the vertices X)J 390 604 :M f0_9 sf (1)S 395 602 :M f0_12 sf <2CC958>S 419 604 :M f0_10 sf .085(n)A f0_12 sf 0 -2 rm .335 .034(, inclusive are)J 0 2 rm 60 620 :M f4_12 sf .374(not)A f0_12 sf 1.202 .12( ancestors of X)J 154 622 :M f0_9 sf (0)S 159 620 :M f0_12 sf 1.781 .178( or X)J 188 622 :M f0_10 sf (n)S f0_9 sf (+1)S 203 620 :M f0_12 sf 1.138 .114( in )J f4_12 sf (G)S 231 622 :M f0_9 sf (1)S 236 620 :M f0_12 sf 1.474 .147(. Hence {X)J 295 622 :M f0_9 sf (1)S 300 620 :M f0_12 sf <2CC958>S 324 622 :M f0_10 sf .777(n)A f0_12 sf 0 -2 rm 1.237 .124(} )J 0 2 rm 340 620 :M f1_12 sf .439A f0_12 sf .13 .013( )J f2_12 sf .311(Sepset)A 387 620 :M f0_12 sf (\(X)S 400 622 :M f0_9 sf (0)S 405 620 :M f0_12 sf (,X)S 417 622 :M f0_10 sf (n)S f0_9 sf (+1)S 432 620 :M f0_12 sf <29CA3DCA>S 449 620 :M f1_12 sf S 459 620 :M f0_12 sf 1.336 .134(, since)J 60 638 :M f2_12 sf (Sepset)S 93 638 :M f0_12 sf (\(X)S 106 640 :M f0_9 sf (0)S 111 638 :M f0_12 sf (,X)S 123 640 :M f0_10 sf (n)S f0_9 sf (+1)S 138 638 :M f0_12 sf 1.522 .152(\) is minimal, and so is a subset of An\(X)J 349 640 :M f0_9 sf (0)S 354 638 :M f0_12 sf (,X)S 366 640 :M f0_10 sf (n)S f0_9 sf (+1)S 381 638 :M f0_12 sf .467 .047(\). \()J f2_12 sf .305(Sepset)A 430 638 :M f0_12 sf (\(X)S 443 640 :M f0_9 sf (0)S 448 638 :M f0_12 sf (,X)S 460 640 :M f0_10 sf (n)S f0_9 sf (+1)S 475 638 :M f0_12 sf 1.678 .168(\) is)J 60 656 :M 8.907 .891(calculated for )J 159 656 :M f4_12 sf (G)S 168 659 :M f0_7 sf (1)S 172 656 :M f0_12 sf 8.5 .85(.\) For the same reason Descendants\({X)J f0_9 sf 0 2 rm (1)S 0 -2 rm 441 656 :M f0_12 sf <2CC958>S 465 658 :M f0_10 sf (n)S f0_12 sf 0 -2 rm <7D29CA>S 0 2 rm 483 656 :M f1_12 sf S 60 674 :M f2_12 sf (Sepset)S 93 674 :M f0_12 sf (\(X)S 106 676 :M f0_9 sf (0)S 111 674 :M f0_12 sf (,X)S 123 676 :M f0_10 sf (n)S f0_9 sf (+1)S 138 674 :M f0_12 sf <29CA3DCA>S 155 674 :M f1_12 sf S 165 674 :M f0_12 sf .793 .079(. It follows from the definition of a pair of m.e. conductors on an)J endp %%Page: 54 54 %%BeginPageSetup initializepage (peter; page: 54 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (54)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf 1.016 .102(itinerary that X)J 137 58 :M f0_10 sf .409(k)A f0_12 sf 0 -2 rm 1.087 .109( is an ancestor of X)J 0 2 rm f0_10 sf .871(k+)A 254 58 :M f0_9 sf (1)S 259 56 :M f0_12 sf 1.278 .128( \(1 )J cF f1_12 sf .128A sf 1.278 .128( k\312< n\), thus there is a directed path )J 474 56 :M f4_12 sf (P)S f4_10 sf 0 2 rm (k)S 0 -2 rm f1_12 sf S 60 74 :M f0_12 sf (X)S 69 76 :M f0_9 sf (k)S 74 74 :M f1_12 sf S 86 74 :M f0_12 sf S f1_12 sf S 110 74 :M f0_12 sf (X)S 119 76 :M f0_9 sf .383(k+1)A f0_12 sf 0 -2 rm 1.4 .14(. Since no descendant of X)J 0 2 rm 274 76 :M f0_9 sf (1)S 279 74 :M f0_12 sf (,\311,X)S 306 76 :M f0_10 sf .546(n)A f0_12 sf 0 -2 rm .777 .078( is in )J 0 2 rm f2_12 sf 0 -2 rm .713(Sepset)A 0 2 rm 377 74 :M f0_12 sf (\(X)S 390 76 :M f0_9 sf (0)S 395 74 :M f0_12 sf (,X)S 407 76 :M f0_10 sf (n)S f0_9 sf (+1)S 422 74 :M f0_12 sf 1.625 .163(\), each of the)J 60 92 :M .261 .026(directed paths )J 131 92 :M f4_12 sf .107(P)A f4_10 sf 0 2 rm .065(k)A 0 -2 rm f0_12 sf .28 .028( d-connects each vertex X)J f0_10 sf 0 2 rm .073(k)A 0 -2 rm f0_12 sf .216 .022( to its successor X)J f0_10 sf 0 2 rm .073(k)A 0 -2 rm f0_9 sf 0 2 rm .14(+1)A 0 -2 rm 377 92 :M f0_12 sf .202 .02<202831CA>J cF f1_12 sf .02A sf .202 .02(\312k\312<\312n\), conditional)J 60 110 :M .113 .011(on )J f2_12 sf .064(Sepset)A 108 110 :M f0_12 sf (\(X)S 121 112 :M f0_9 sf (0)S 126 110 :M f0_12 sf (,X)S 138 112 :M f0_10 sf (n)S f0_9 sf (+1)S 153 110 :M f0_12 sf .301 .03(\). In addition, since X)J 260 112 :M f0_9 sf (0)S 265 110 :M f0_12 sf .373 .037( and X)J 298 112 :M f0_9 sf (1)S 303 110 :M f0_12 sf .373 .037( are p)J 331 110 :M .203 .02(-adjacent ther)J 397 110 :M .302 .03(e is some path )J f4_12 sf (Q)S 479 110 :M f0_12 sf .388 .039( d-)J 60 128 :M .565 .057(connecting X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 130 128 :M f0_12 sf 1.065 .107( and X)J 165 130 :M f0_9 sf (1)S 170 128 :M f0_12 sf 1.024 .102( given )J 206 128 :M f2_12 sf (Sepset)S 239 128 :M f0_12 sf (\(X)S 252 130 :M f0_9 sf (0)S 257 128 :M f0_12 sf (,X)S 269 130 :M f0_10 sf (n)S f0_9 sf (+1)S 284 128 :M f0_12 sf .929 .093(\). Since each )J 353 128 :M f4_12 sf (P)S f4_9 sf 0 2 rm (i)S 0 -2 rm 363 128 :M f0_12 sf .979 .098( is out of X)J f0_9 sf 0 2 rm (i)S 0 -2 rm 425 128 :M f0_12 sf .908 .091( \(i.e. the path)J 60 146 :M (goes X)S 94 148 :M f0_9 sf (i)S 97 146 :M f1_12 sf S 109 146 :M f0_12 sf S f1_12 sf S 133 146 :M f0_12 sf (X)S 142 148 :M f0_9 sf (2)S 147 146 :M f0_12 sf -.005(\), by applying )A 217 146 :M (Lemma 1)S 263 146 :M (, with )S f14_13 sf (T)S 300 146 :M f15_12 sf ( )S f0_12 sf -.019(= { )A 322 146 :M f4_12 sf (Q)S 331 146 :M f0_12 sf (,)S f4_12 sf (P)S f0_9 sf 0 2 rm (1)S 0 -2 rm 346 146 :M f4_12 sf <2CC950>S 367 148 :M f4_10 sf (n)S f0_12 sf 0 -2 rm (}, )S 0 2 rm 384 146 :M f4_12 sf (R)S f0_12 sf ( = S 448 148 :M f0_10 sf (n)S f0_12 sf 0 -2 rm (>, and )S 0 2 rm f2_12 sf 0 -2 rm (S)S 0 2 rm 60 164 :M f4_12 sf 1.136(=)A f0_12 sf .383 .038( )J f2_12 sf .916(Sepset)A 110 164 :M f0_12 sf (\(X)S 123 166 :M f0_9 sf (0)S 128 164 :M f0_12 sf (,X)S 140 166 :M f0_10 sf (n)S f0_9 sf (+1)S 155 164 :M f0_12 sf 4.127 .413(\) that we can form a path d-connecting X)J 402 166 :M f0_9 sf (0)S 407 164 :M f0_12 sf 3.399 .34( and X)J f0_10 sf 0 2 rm 1.381(n)A 0 -2 rm f0_12 sf 5.463 .546( given)J 60 182 :M f2_12 sf (Sepset)S 93 182 :M f0_12 sf (\(X)S 106 184 :M f0_9 sf (0)S 111 182 :M f0_12 sf (,X)S 123 184 :M f0_10 sf (n)S f0_9 sf (+1)S 138 182 :M f0_12 sf .781 .078(\). A symmetric argument shows that X)J 332 184 :M f0_9 sf (1)S 337 182 :M f0_12 sf .653 .065( and X)J f0_10 sf 0 2 rm .266(n)A 0 -2 rm f0_9 sf 0 2 rm .509(+1)A 0 -2 rm 386 182 :M f0_12 sf .742 .074( are also d-connected)J 60 200 :M .948 .095(given )J 92 200 :M f2_12 sf (Sepset)S 125 200 :M f0_12 sf (\(X)S 138 202 :M f0_9 sf (0)S 143 200 :M f0_12 sf (,X)S 155 202 :M f0_10 sf (n)S f0_9 sf (+1)S 170 200 :M f0_12 sf .957 .096(\). It then follows that the edges X)J 341 202 :M f0_9 sf (0)S 346 200 :M f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (X)S 379 202 :M f0_9 sf (1)S 384 200 :M f0_12 sf 1.138 .114( and X)J 419 202 :M f0_10 sf (n)S f1_12 sf 0 -2 rm (*)S 0 2 rm f0_12 sf 0 -2 rm S 0 2 rm f1_12 sf 0 -2 rm (*)S 0 2 rm f0_12 sf 0 -2 rm (X)S 0 2 rm 457 202 :M f0_10 sf (n+)S 468 202 :M f0_9 sf (1)S 473 200 :M f0_12 sf 1.094 .109( are)J 60 218 :M 1.092 .109(oriented as X)J 128 220 :M f0_9 sf (0)S 133 218 :M f0_12 sf S f0_9 sf 0 2 rm (1)S 0 -2 rm 165 218 :M f0_12 sf .573 .057( and X)J f0_10 sf 0 2 rm .233(n)A 0 -2 rm f0_12 sf .426<3CD158>A f0_10 sf 0 2 rm .495(n+)A 0 -2 rm 243 220 :M f0_9 sf (1)S 248 218 :M f0_12 sf 1.165 .117( by stage \246C of the CCD algorithm \(unless they)J 60 236 :M .259 .026(have already been oriented this way in a previous stage of the algorithm\). Thus again, by)J 60 254 :M 1.26 .126(the correctness of the algorithm these arrowheads will be present in )J f3_12 sf (Y)S 420 256 :M f0_9 sf (1)S 425 254 :M f0_12 sf .959 .096(. \(Subsequent)J 60 272 :M .864 .086(stages of the algorithm only add '\320' and '>' endpoints, not ')J f0_10 sf .266(o)A f0_12 sf .863 .086(' endpoints. If either of the)J 60 290 :M (arrowhead at X)S 134 292 :M f0_9 sf (1)S 139 290 :M f0_12 sf ( or X)S 164 292 :M f0_10 sf (n)S f0_12 sf 0 -2 rm ( were replaced with a '\320' the algorithm would be incorrect.\))S 0 2 rm 60 314 :M .198 .02(Since by hypothesis, and are not mutually exclusive in )J f4_12 sf (G)S 484 316 :M f0_9 sf (2)S 489 314 :M f0_12 sf (,)S 60 332 :M 1.628 .163(either X)J f0_9 sf 0 2 rm (1)S 0 -2 rm 107 332 :M f0_12 sf 2.227 .223( is an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 221 332 :M f0_12 sf 2.615 .261(, or X)J 256 334 :M f0_10 sf <6ECA>S 264 332 :M f0_12 sf 1.982 .198(is an ancestor of X)J f0_10 sf 0 2 rm 1.383(n+)A 0 -2 rm 378 334 :M f0_9 sf (1)S 383 332 :M f0_12 sf 2.179 .218(. It follows from the)J 60 350 :M .72 .072(correctness of the orientation rules in the CCD algorithm that the edges X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 433 350 :M f1_12 sf (*)S f0_12 sf S f1_12 sf (*)S f0_12 sf (X)S 466 352 :M f0_9 sf (1)S 471 350 :M f0_12 sf .907 .091( and)J 60 368 :M (X)S 69 370 :M f0_10 sf (n)S f1_12 sf 0 -2 rm (*)S 0 2 rm f0_12 sf 0 -2 rm S 0 2 rm f1_12 sf 0 -2 rm (*)S 0 2 rm f0_12 sf 0 -2 rm (X)S 0 2 rm 107 370 :M f0_10 sf (n+)S 118 370 :M f0_9 sf (1)S 123 368 :M f0_12 sf .016 .002( will not both be oriented as X)J 269 370 :M f0_9 sf (0)S 274 368 :M f1_12 sf (*)S f0_12 sf S f0_9 sf 0 2 rm (1)S 0 -2 rm 312 368 :M f0_12 sf ( and X)S f0_10 sf 0 2 rm (n)S 0 -2 rm f0_12 sf <3CD1>S 368 368 :M f1_12 sf (*)S f0_12 sf (X)S 383 370 :M f0_10 sf (n+)S 394 370 :M f0_9 sf (1)S 399 368 :M f0_12 sf ( in )S f3_12 sf (Y)S 424 370 :M f0_9 sf (2)S 429 368 :M f0_12 sf (. Thus )S f3_12 sf (Y)S 472 370 :M f0_9 sf (1)S 477 368 :M f0_12 sf ( and)S 60 386 :M f3_12 sf (Y)S 70 388 :M f0_9 sf (2)S 75 386 :M f0_12 sf ( will once again be different.)S 60 410 :M f2_12 sf .433 .043(Case 5)J f0_12 sf .119 .012(: )J 101 410 :M f4_12 sf (G)S 110 412 :M f0_9 sf (1)S 115 410 :M f0_12 sf .24 .024( and )J f4_12 sf (G)S 148 412 :M f0_9 sf (2)S 153 410 :M f0_12 sf .261 .026( fail to satisfy either CET\(IV\) or CET\(V\). We assume that )J f4_12 sf (G)S 449 412 :M f0_9 sf (1)S 454 410 :M f0_12 sf .24 .024( and )J f4_12 sf (G)S 487 412 :M f0_9 sf (2)S 60 428 :M f0_12 sf .561 .056(satisfy CET\(I\)\320\(III\).)J 160 425 :M f0_9 sf (2)S 164 425 :M (2)S 168 428 :M f0_12 sf .866 .087( If )J f4_12 sf (G)S 194 430 :M f0_9 sf (1)S 199 428 :M f0_12 sf 1.403 .14( and )J 226 428 :M f4_12 sf (G)S 235 430 :M f0_9 sf (2)S 240 428 :M f0_12 sf 1.09 .109( fail to satisfy either CET\(IV\) or CET\(V\), then in)J 60 446 :M .177 .018(either case we have the following situation: There is some sequence of vertices in )J f4_12 sf (G)S 467 448 :M f0_9 sf (1)S 472 446 :M f0_12 sf .222 .022( and)J 60 464 :M f4_12 sf (G)S 69 466 :M f0_9 sf (2 )S 76 464 :M f0_12 sf (S 137 466 :M f0_10 sf (n)S f0_12 sf 0 -2 rm (, X)S 0 2 rm 157 466 :M f0_10 sf (n)S f0_9 sf (+1)S 172 464 :M f0_12 sf (>, )S 185 461 :M f0_9 sf (2)S 189 461 :M (3)S 193 464 :M f0_12 sf ( satisfying the following:)S 96 482 :M (\(a\) if i\312>\312j then X)S f0_10 sf 0 2 rm (i)S 0 -2 rm 180 482 :M f0_12 sf ( and X)S 212 484 :M f0_10 sf (j)S 215 482 :M f0_12 sf ( are p)S 242 482 :M (-adjacent if and only if i = j+1,)S 96 500 :M (\(b\) X)S 122 502 :M f0_9 sf (1)S 127 500 :M f0_12 sf ( is not an ancestor of X)S f0_9 sf 0 2 rm (0)S 0 -2 rm 243 500 :M f0_12 sf (, and X)S 278 502 :M f0_10 sf (n)S f0_12 sf 0 -2 rm ( is not an ancestor of X)S 0 2 rm f0_10 sf (n)S f0_9 sf (+1)S 409 500 :M f0_12 sf (, and)S 96 518 :M (\(c\))S f1_12 sf (")S 118 518 :M f0_12 sf (k, 1\312)S cF f1_12 sf S sf S cF f1_12 sf S sf (\312n, X)S 188 520 :M f0_10 sf (k)S f0_9 sf (-1)S f0_12 sf 0 -2 rm (, and X)S 0 2 rm 235 520 :M f0_10 sf (k)S f0_9 sf (+1)S 250 518 :M f0_12 sf ( are ancestors of X)S 340 520 :M f0_10 sf (k)S f0_12 sf 0 -2 rm (.)S 0 2 rm 60 542 :M 1.272 .127(In addition there is some vertex V, p)J 249 542 :M 1.167 .117(-adjacent to X)J 320 544 :M f0_9 sf (0)S 325 542 :M f0_12 sf 1.54 .154( and X)J 361 544 :M f0_10 sf .509(n)A f0_9 sf .926 .093(+1 )J f0_12 sf 0 -2 rm .966 .097(in )J 0 2 rm f4_12 sf 0 -2 rm (G)S 0 2 rm 402 544 :M f0_9 sf (1)S 407 542 :M f0_12 sf 1.123 .112( and )J f4_12 sf (G)S 443 544 :M f0_9 sf (2)S 448 544 :M f2_9 sf .312 .031( )J f0_12 sf 0 -2 rm 2.012 .201(, not an)J 0 2 rm 60 560 :M .517 .052(ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 131 560 :M f0_12 sf .76 .076( or X)J 158 562 :M f0_10 sf (n)S f0_9 sf (+1)S 173 560 :M f0_12 sf .486 .049( in )J f4_12 sf (G)S 199 562 :M f0_9 sf (1)S 204 560 :M f0_12 sf .793 .079( or )J 222 560 :M f4_12 sf (G)S 231 562 :M f0_9 sf (2)S 236 560 :M f0_12 sf .621 .062( and not a descendant of a common child of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 471 560 :M f0_12 sf .701 .07( and)J 60 578 :M (X)S 69 580 :M f0_10 sf (n)S f0_9 sf (+1)S 84 578 :M f0_12 sf .43 .043( in )J f4_12 sf (G)S 109 580 :M f0_9 sf (1)S 114 578 :M f0_12 sf .701 .07( or )J 132 578 :M f4_12 sf (G)S 141 580 :M f0_9 sf (2)S 146 578 :M f0_12 sf .556 .056(. As explained in case 3, this implies that in both of the PAGs )J 456 578 :M f3_12 sf (Y)S 466 580 :M f0_9 sf (1)S 471 578 :M f0_12 sf .62 .062( and)J 60 596 :M f3_12 sf (Y)S 70 598 :M f0_9 sf (2)S 75 596 :M f0_12 sf (, X)S 90 598 :M f0_9 sf (0)S 95 596 :M f0_12 sf S 107 0 7 730 rC 107 596 :M ( )S 111 596 :M ( )S gR gS 0 0 552 730 rC 107 596 :M f0_12 sf (>V<)S 122 0 7 730 rC 122 596 :M ( )S 126 596 :M ( )S gR gS 0 0 552 730 rC 129 596 :M f0_12 sf S 150 598 :M f0_10 sf (n)S f0_9 sf (+1)S 165 596 :M f0_12 sf (.)S 1 G 0 0 1 1 rF 168 596 :M psb /wp$x1 107 def /wp$x2 128 def /wp$y 598 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 649 :M 0 G ( )S 60 646.48 -.48 .48 204.48 646 .48 60 646 @a 60 660 :M f0_9 sf (2)S 64 660 :M (2)S 68 663 :M f0_10 sf .16 .016(The conditions under which CET\(IV\) or CET\(V\) fail are quite intricate precisely because the assumption)J 60 674 :M (that CET\(I\)-\(III\) are satisfied implies that the graphs agree in many respects.)S 60 684 :M f0_9 sf (2)S 64 684 :M (3)S 68 687 :M f0_10 sf ( In the case where CET\(IV\) fails n=1, while if CET\(V\) fails, n>1.)S endp %%Page: 55 55 %%BeginPageSetup initializepage (peter; page: 55 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (55)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf (Since )S 90 56 :M f4_12 sf (G)S 99 58 :M f0_9 sf (1)S 104 56 :M f0_12 sf ( and )S f4_12 sf (G)S 136 58 :M f0_9 sf (2)S 141 56 :M f0_12 sf ( fail to satisfy CET\(IV\) or CET\(V\), in one graph V is a descendant of X)S f0_9 sf 0 2 rm (1)S 0 -2 rm 491 56 :M f0_12 sf (,)S 60 74 :M 1.199 .12(while in the other graph V is not a descendant of X)J 325 76 :M f0_9 sf (1)S 330 74 :M f0_12 sf 1.15 .115(. Let us suppose without loss of)J 60 92 :M .327 .033(generality that V is a descendant of X)J f0_9 sf 0 2 rm (1)S 0 -2 rm 249 92 :M f0_12 sf .275 .027( in )J f4_12 sf (G)S 274 94 :M f0_9 sf (1)S 279 92 :M f0_12 sf .372 .037(, and V is not a descendant of X)J 437 94 :M f0_9 sf (1)S 442 92 :M f0_12 sf .275 .027( in )J f4_12 sf (G)S 467 94 :M f0_9 sf (2)S 472 92 :M f0_12 sf .397 .04(. As)J 60 110 :M 2.402 .24(in previous cases it is sufficient to show that if )J 322 110 :M f3_12 sf (Y)S 332 112 :M f0_9 sf (1)S 337 110 :M f0_12 sf 3.002 .3( and )J 368 110 :M f3_12 sf (Y)S 378 112 :M f0_9 sf (2)S 383 110 :M f0_12 sf 2.573 .257( are the CCD PAGs)J 60 128 :M (corresponding to )S 144 128 :M f4_12 sf (G)S 153 130 :M f0_9 sf (1)S 158 128 :M f0_12 sf ( and )S f4_12 sf (G)S 190 130 :M f0_9 sf (2)S 195 128 :M f0_12 sf ( respectively, then )S 286 128 :M f3_12 sf (Y)S 296 130 :M f0_9 sf (1)S 301 128 :M f0_12 sf ( and )S f3_12 sf (Y)S 334 130 :M f0_9 sf (2)S 339 128 :M f0_12 sf ( are different. We may suppose,)S 60 146 :M .525 .052(again without loss of generality that V is the closest such vertex to any X)J f0_10 sf 0 2 rm .161(k)A 0 -2 rm f0_12 sf .619 .062( \(1)J cF f1_12 sf .062A sf .619 .062J cF f1_12 sf .062A sf .619 .062(\312n\) in)J 60 164 :M f4_12 sf (G)S 69 166 :M f0_9 sf (1)S 74 164 :M f0_12 sf .337 .034(, in the sense that the shortest directed path )J 288 164 :M f4_12 sf (P)S f1_12 sf S 302 164 :M f0_12 sf (X)S 311 166 :M f0_10 sf (k)S f1_12 sf 0 -2 rm S 0 2 rm 328 164 :M f0_12 sf S f1_12 sf S 352 164 :M f0_12 sf .299 .03(V in )J f4_12 sf (G)S 386 166 :M f0_9 sf (1)S 391 164 :M f0_12 sf .34 .034( contains at most the)J 60 182 :M .966 .097(same number of vertices as the shortest directed path in )J f4_12 sf (G)S 353 184 :M f0_9 sf (1)S 358 182 :M f0_12 sf .878 .088( from any X)J f0_10 sf 0 2 rm .309(k)A 0 -2 rm f0_12 sf 1.188 .119( \(1)J cF f1_12 sf .119A sf 1.188 .119J cF f1_12 sf .119A sf 1.188 .119(\312n\) to)J 60 200 :M (some other vertex V\325 satisfying the conditions on V.)S 60 224 :M f2_12 sf .081(Claim:)A f0_12 sf .174 .017( Let W be the first vertex on )J 237 224 :M f4_12 sf .095(P)A f0_12 sf .187 .019( which is p-adjacent to V, \(both in )J f4_12 sf (G)S 421 226 :M f0_9 sf (1)S 426 224 :M f0_12 sf .248 .025( and )J 450 224 :M f4_12 sf (G)S 459 226 :M f0_9 sf (2)S 464 224 :M f0_12 sf .198 .02( since)J 60 242 :M .8 .08(by CET\(I\) )J 115 242 :M f4_12 sf (G)S 124 244 :M f0_9 sf (1)S 129 242 :M f0_12 sf .726 .073( and G)J 0 2 rm .354(2)A 0 -2 rm .858 .086( have the same p)J 254 242 :M .657 .066(-adjacencies\). We will show that the assumption)J 60 260 :M .611 .061(that V is the closest such vertex to any X)J f0_10 sf 0 2 rm .205(k)A 0 -2 rm f0_12 sf .331 .033( \(in )J 289 260 :M f4_12 sf (G)S 298 262 :M f0_9 sf (1)S 303 260 :M f0_12 sf .52 .052(\) together with the assumption that )J 478 260 :M f4_12 sf (G)S 487 262 :M f0_9 sf (1)S 60 278 :M f0_12 sf .602 .06(and )J 80 278 :M f4_12 sf (G)S 89 280 :M f0_9 sf (2)S 94 278 :M f0_12 sf .508 .051( satisfy CET\(I\)-\(III\) imply that W is a descendant of X)J 365 280 :M f0_9 sf (1)S 370 278 :M f0_12 sf .681 .068( in )J 387 278 :M f4_12 sf (G)S 396 280 :M f0_9 sf (2)S 401 278 :M f0_12 sf .549 .055(. We prove this by)J 60 296 :M .826 .083(showing that every vertex in the directed subpath )J 310 296 :M f4_12 sf (P)S f0_12 sf (\(X)S 330 298 :M f0_10 sf .202(k)A f0_12 sf 0 -2 rm .617 .062(, W\))J 0 2 rm f1_12 sf 0 -2 rm S 0 2 rm 364 296 :M f0_12 sf (X)S 373 298 :M f0_10 sf (k)S f1_12 sf 0 -2 rm S 0 2 rm 390 296 :M f0_12 sf 1.066 .107(\311W in )J 432 296 :M f4_12 sf (G)S 441 298 :M f0_9 sf (1)S 446 296 :M f0_12 sf 1.025 .102( is also a)J 60 314 :M (descendant of X)S 138 316 :M f0_9 sf (1)S 143 314 :M f0_12 sf ( in )S f4_12 sf (G)S 167 316 :M f0_9 sf (2)S 172 314 :M f0_12 sf (.)S 60 338 :M f2_12 sf (Proof of Claim:)S 140 338 :M f0_12 sf ( By induction on the vertices occurring on the path )S 387 338 :M f4_12 sf (P)S f0_12 sf (\(X)S 407 340 :M f0_10 sf (k)S f0_12 sf 0 -2 rm (, W\).)S 0 2 rm 60 362 :M f2_12 sf (Base Case:)S 116 362 :M f0_12 sf ( X)S 128 364 :M f0_10 sf (k)S f0_12 sf 0 -2 rm (. By hypothesis X)S 0 2 rm 219 364 :M f0_10 sf (k)S f0_12 sf 0 -2 rm ( is a descendant of X)S 0 2 rm f0_9 sf (1)S 329 362 :M f0_12 sf ( in both )S 369 362 :M f4_12 sf (G)S 378 364 :M f0_9 sf (1)S 383 362 :M f0_12 sf ( and )S f4_12 sf (G)S 415 364 :M f0_9 sf (2)S 420 362 :M f0_12 sf (.)S 60 386 :M f2_12 sf .937 .094(Induction Case:)J 144 386 :M f0_12 sf 1.499 .15( Consider Y)J 207 388 :M f0_10 sf .243(r)A f0_12 sf 0 -2 rm .998 .1(, where )J 0 2 rm f4_12 sf 0 -2 rm .535(P)A 0 2 rm f0_12 sf 0 -2 rm .924(\(X)A 0 2 rm 272 388 :M f0_10 sf .427(k)A f0_12 sf 0 -2 rm 1.401 .14(, W\))J 0 2 rm 301 386 :M f1_12 sf S 308 386 :M f0_12 sf (X)S 317 388 :M f0_10 sf (k)S f1_12 sf 0 -2 rm S 0 2 rm 334 386 :M f0_12 sf (Y)S 343 388 :M f0_9 sf (1)S 348 386 :M f1_12 sf S 360 386 :M f0_12 sf S f1_12 sf S 384 386 :M f0_12 sf (Y)S 393 388 :M f0_10 sf (r)S f1_12 sf 0 -2 rm S 0 2 rm 408 386 :M f0_12 sf S 429 388 :M f0_10 sf (t)S 432 386 :M f1_12 sf S 439 386 :M f0_12 sf 1.606 .161(W. By the)J 60 404 :M 1.459 .146(induction hypothesis, for s\312<\312r, Y)J 230 406 :M f0_10 sf (s)S 234 404 :M f0_12 sf 1.795 .179( is a descendant of X)J f0_9 sf 0 2 rm (1)S 0 -2 rm 352 404 :M f0_12 sf 1.418 .142( in )J f4_12 sf (G)S 381 406 :M f0_9 sf (2)S 386 404 :M f0_12 sf 1.837 .184(. Now there are two)J 60 422 :M (subcases to consider:)S 60 446 :M f2_12 sf .915 .092(Subcase 1:)J f0_12 sf .48 .048( Not both X)J 175 448 :M f0_9 sf (0)S 180 446 :M f0_12 sf .75 .075( and X)J 214 448 :M f0_10 sf (n)S f0_9 sf (+1)S 229 446 :M f0_12 sf .636 .064( are p-adjacent to Y)J 326 448 :M f0_10 sf .108(r)A f0_12 sf 0 -2 rm .587 .059(. Suppose without loss that X)J 0 2 rm 475 448 :M f0_9 sf (0)S 480 446 :M f0_12 sf .781 .078( is)J 60 464 :M 1.442 .144(not p-adjacent to Y)J 158 466 :M f0_10 sf .449(r)A f0_12 sf 0 -2 rm 1.618 .162(. Since in )J 0 2 rm 216 464 :M f4_12 sf (G)S 225 466 :M f0_9 sf (1)S 230 464 :M f0_12 sf 1.535 .154( there is a directed path X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 372 464 :M f1_12 sf S 384 464 :M f0_12 sf S 405 466 :M f0_10 sf (k)S f1_12 sf 0 -2 rm S 0 2 rm 422 464 :M f0_12 sf (Y)S 431 466 :M f0_9 sf (1)S 436 464 :M f1_12 sf S 448 464 :M f0_12 sf S 469 466 :M f0_10 sf .36(r)A f0_12 sf 0 -2 rm 1.498 .15(, by)J 0 2 rm 60 482 :M .834 .083(Lemma 16)J 114 482 :M .895 .09( it then follows that there is some subsequence of this sequence of vertices,)J 60 500 :M f4_12 sf (Q)S 69 500 :M f1_12 sf S 76 500 :M f0_12 sf (S 120 502 :M f0_10 sf .074(r)A f0_12 sf 0 -2 rm .392 .039(> such that consecutive vertices in )J 0 2 rm 295 500 :M f4_12 sf (Q)S 304 500 :M f0_12 sf .507 .051( are p)J 332 500 :M .347 .035(-adjacent, but only these vertices)J 60 518 :M .362 .036(are p)J 85 518 :M .254 .025(-adjacent. Moreover, since X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 230 518 :M f0_12 sf .4 .04( is not p)J 270 518 :M .307 .031(-adjacent to Y)J 338 520 :M f0_10 sf .067(r)A f0_12 sf 0 -2 rm .323 .032(, this sequence of vertices is of)J 0 2 rm 60 536 :M .442 .044(length greater than 2, i.e. )J f4_12 sf (Q)S 195 536 :M f1_12 sf S 202 536 :M f0_12 sf ( where D is the first vertex in the subsequence)J 60 554 :M .3 .03(after X)J 95 556 :M f0_9 sf (0)S 100 554 :M f0_12 sf .288 .029(, hence either D)J f1_12 sf S 184 554 :M f0_12 sf (X)S 193 556 :M f1_9 sf (k)S 198 554 :M f0_12 sf .32 .032<202831CA>J cF f1_12 sf .032A sf .32 .032J 225 554 :M f1_12 sf (k)S 232 554 :M f0_12 sf .281 .028J cF f1_12 sf .028A sf .281 .028(\312k\) or D)J f1_12 sf S 287 554 :M f0_12 sf (Y)S 296 556 :M f1_9 sf .061(m)A f0_12 sf 0 -2 rm .199 .02(, \(1\312)J cF f1_12 sf .02A sf .199 .02J 0 2 rm f1_12 sf 0 -2 rm (m)S 0 2 rm 337 554 :M f0_12 sf .34 .034(\312< r\). Since in either case D is a)J 60 572 :M 1.451 .145(descendant of X)J f0_9 sf 0 2 rm (1)S 0 -2 rm 148 572 :M f0_12 sf 2.12 .212( in both )J 196 572 :M f4_12 sf (G)S 205 574 :M f0_9 sf (1)S 210 572 :M f0_12 sf 2.238 .224( and )J 239 572 :M f4_12 sf (G)S 248 574 :M f0_9 sf (2)S 253 572 :M f0_12 sf 1.705 .17(, \(either by the induction hypothesis or by the)J 60 590 :M -.003(hypothesis of case 5\), but X)A 194 592 :M f0_9 sf (0)S 199 590 :M f0_12 sf -.004( is not a descendant of X)A 318 592 :M f0_9 sf (1)S 323 590 :M f0_12 sf ( in )S f4_12 sf (G)S 347 592 :M f0_9 sf (1)S 352 590 :M f0_12 sf ( or )S 368 590 :M f4_12 sf (G)S 377 592 :M f0_9 sf (2)S 382 590 :M f0_12 sf -.004( it follows that D is not)A 60 608 :M .66 .066(an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 147 608 :M f0_12 sf .57 .057( in )J f4_12 sf (G)S 173 610 :M f0_9 sf (1)S 178 608 :M f0_12 sf .578 .058( or )J f4_12 sf (G)S 205 610 :M f0_9 sf (2)S 210 608 :M f0_12 sf .764 .076(. Hence we may apply )J 326 608 :M .629 .063(Lemma 17)J 379 608 :M .874 .087(, to deduce that Y)J f0_10 sf 0 2 rm .198(r)A 0 -2 rm f0_12 sf .48 .048( is a)J 60 626 :M (descendant of D. Hence Y)S f0_10 sf 0 2 rm (r)S 0 -2 rm f0_12 sf ( is a descendant of X)S f0_9 sf 0 2 rm (1)S 0 -2 rm 294 626 :M f0_12 sf (, since X)S f0_9 sf 0 2 rm (1)S 0 -2 rm 341 626 :M f0_12 sf ( is an ancestor of D.)S 60 650 :M f2_12 sf .34 .034(Subcase 2:)J f0_12 sf .183 .018( Both X)J 153 652 :M f0_9 sf .121 .012(0 )J f0_12 sf 0 -2 rm .447 .045(and X)J 0 2 rm 190 652 :M f0_10 sf (n)S f0_9 sf (+1)S 205 650 :M f0_12 sf .262 .026( are p-adjacent to Y)J 300 652 :M f0_10 sf .164(r)A f0_9 sf .202 .02(. )J 309 650 :M f0_12 sf .236 .024(First note that in )J f4_12 sf (G)S 401 652 :M f0_9 sf (1)S 406 650 :M f0_12 sf .329 .033( the vertex Y)J f0_10 sf 0 2 rm .076(r)A 0 -2 rm f0_12 sf .183 .018( is a)J 60 668 :M .639 .064(descendant of X)J 141 670 :M f0_10 sf .271(k)A f0_12 sf 0 -2 rm .701 .07(, and X)J 0 2 rm f0_10 sf .271(k)A f0_12 sf 0 -2 rm .717 .072( is not an ancestor of X)J 0 2 rm f0_9 sf (0)S 310 668 :M f0_12 sf .906 .091( or X)J 337 670 :M f0_10 sf (n)S f0_9 sf (+1)S 352 668 :M f0_12 sf .75 .075(. It follows that Y)J 442 670 :M f0_10 sf .231(r)A f0_12 sf 0 -2 rm .777 .078( is not an)J 0 2 rm 60 686 :M .331 .033(ancestor of X)J 126 688 :M f0_9 sf (0)S 131 686 :M f0_12 sf .441 .044( or X)J 157 688 :M f0_10 sf (n)S f0_9 sf (+1)S 172 686 :M f0_12 sf .282 .028( in )J f4_12 sf (G)S 197 688 :M f0_9 sf (1)S 202 686 :M f0_12 sf .359 .036( . Moreover, since X)J 303 688 :M f0_9 sf .481 .048(0 )J 311 686 :M f0_12 sf .447 .045(and X)J f0_10 sf 0 2 rm .116(n)A 0 -2 rm f0_9 sf 0 2 rm .228 .023(+1 )J 0 -2 rm 358 686 :M f0_12 sf .293 .029(are not p-adjacent, forms an unshielded non-conductor in )J f4_12 sf (G)S 311 58 :M f0_9 sf (1)S 316 56 :M f0_12 sf 2.638 .264(. Hence forms an)J 60 74 :M (unshielded non-conductor in )S f4_12 sf (G)S 209 76 :M f0_9 sf (2)S 214 74 :M f0_12 sf (, since by hypothesis )S 317 74 :M f4_12 sf (G)S 326 76 :M f0_9 sf (1 )S 333 74 :M f0_12 sf (and )S f4_12 sf (G)S 362 76 :M f0_9 sf (2)S 367 74 :M f0_12 sf ( satisfy CET\(IIa\). So Y)S 479 76 :M f0_10 sf (r)S f0_12 sf 0 -2 rm ( is)S 0 2 rm 60 92 :M .07 .007(not an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 162 92 :M f0_12 sf .092 .009( or X)J 187 94 :M f0_10 sf (n)S f0_9 sf (+1)S 202 92 :M f0_12 sf .096 .01( in )J 218 92 :M f4_12 sf (G)S 227 94 :M f0_9 sf (1)S 232 92 :M f0_12 sf .06 .006( or )J f4_12 sf (G)S 257 94 :M f0_9 sf (2)S 262 92 :M f0_12 sf .07 .007(. Further, since Y)J 346 94 :M f0_10 sf (r)S f0_12 sf 0 -2 rm .074 .007( is an ancestor of V in )J 0 2 rm f4_12 sf 0 -2 rm (G)S 0 2 rm 467 94 :M f0_9 sf (1)S 472 92 :M f0_12 sf .085 .009( and)J 60 110 :M .222 .022(V is not a descendant of a common child of X)J 285 112 :M f0_9 sf .11 .011(0 )J f0_12 sf 0 -2 rm .378 .038(and X)J 0 2 rm f0_10 sf .098(n)A f0_9 sf .178 .018(+1 )J f0_12 sf 0 -2 rm .201 .02(in )J 0 2 rm 351 110 :M f4_12 sf (G)S 360 112 :M f0_9 sf (1)S 365 110 :M f0_12 sf .253 .025(, it follows that Y)J f0_10 sf 0 2 rm .059(r)A 0 -2 rm f0_12 sf .175 .018( is not a)J 60 128 :M 1.549 .155(descendant of a common child of X)J f0_9 sf 0 2 rm .516 .052(0 )J 0 -2 rm 253 128 :M f0_12 sf 1.804 .18(and X)J f0_10 sf 0 2 rm .467(n)A 0 -2 rm f0_9 sf 0 2 rm .92 .092(+1 )J 0 -2 rm 303 128 :M f0_12 sf .966 .097(in )J f4_12 sf (G)S 326 130 :M f0_9 sf (1)S 331 128 :M f0_12 sf 1.528 .153(. Thus forms an)J 60 146 :M 1.511 .151(unshielded imperfect non-conductor in )J 261 146 :M f4_12 sf (G)S 270 148 :M f0_9 sf (1)S 275 146 :M f0_12 sf 1.672 .167(. Since )J f4_12 sf (G)S 325 148 :M f0_9 sf 2.645 .264(1 )J 335 146 :M f0_12 sf 1.457 .146(and )J f4_12 sf (G)S 367 148 :M f0_9 sf (2)S 372 146 :M f0_12 sf 1.64 .164( satisfy CET\(IIb\), forms an unshielded imperfect non-conductor in)J 352 164 :M f4_12 sf 1.155 .115( G)J f0_9 sf 0 2 rm (2)S 0 -2 rm 370 164 :M f0_12 sf 1.332 .133(. Now, if Y)J f0_9 sf 0 2 rm (r)S 0 -2 rm 433 164 :M f0_12 sf 1.44 .144( were not a)J 60 182 :M .523 .052(descendant of X)J 140 184 :M f0_9 sf (1)S 145 182 :M f0_12 sf .473 .047( in )J f4_12 sf (G)S 171 184 :M f0_9 sf (2)S 176 182 :M f0_12 sf .432 .043( , then Y)J f0_10 sf 0 2 rm .136(r)A 0 -2 rm f0_12 sf .635 .064( would satisfy the conditions on V, yet be closer to X)J f0_10 sf 0 2 rm (k)S 0 -2 rm 60 200 :M f0_12 sf .588 .059(than V \(Y)J 110 202 :M f0_10 sf .126(r)A f0_12 sf 0 -2 rm .635 .064( occurs before V on the shortest directed path from X)J 0 2 rm f0_10 sf .19(k)A f0_12 sf 0 -2 rm .311 .031( to V in )J 0 2 rm 425 200 :M f4_12 sf (G)S 434 202 :M f0_9 sf (1)S 439 200 :M f0_12 sf .603 .06(\). This is a)J 60 218 :M (contradiction, hence Y)S 169 220 :M f0_10 sf (r)S f0_12 sf 0 -2 rm ( is a descendant of X)S 0 2 rm f0_10 sf <6BCA>S 280 218 :M f0_12 sf (in)S f0_10 sf 0 2 rm ( )S 0 -2 rm 292 218 :M f4_12 sf (G)S 301 220 :M f0_9 sf (2)S 306 218 :M f0_12 sf (.)S 60 242 :M (This completes the proof of the claim. We now show that )S 339 242 :M f3_12 sf (Y)S 349 244 :M f0_9 sf (1)S 354 242 :M f0_12 sf ( and )S f3_12 sf (Y)S 387 244 :M f0_9 sf (2)S 392 242 :M f0_12 sf ( are different.)S 60 266 :M .135 .014(Consider the edge W)J 162 266 :M f1_12 sf .05(*)A f0_12 sf .05A f1_12 sf .05(*)A f0_12 sf .081 .008(V in )J f3_12 sf (Y)S 214 268 :M f0_9 sf (1)S 219 266 :M f0_12 sf .181 .018(. In )J 239 266 :M f4_12 sf (G)S 248 268 :M f0_9 sf (1)S 253 266 :M f0_12 sf .15 .015(, W is an ancestor of V, hence it follows from the)J 60 284 :M .988 .099(correctness of the algorithm in )J 218 284 :M f3_12 sf (Y)S 228 286 :M f0_9 sf (1)S 233 284 :M f0_12 sf 1.116 .112( this edge is oriented as W)J 370 284 :M f0_10 sf .324(o)A f0_12 sf .389A f1_12 sf .389(*)A f0_12 sf 1.153 .115(V or W\321)J f1_12 sf .389(*)A f0_12 sf .718 .072(V. In )J 475 284 :M f4_12 sf (G)S 484 286 :M f0_9 sf (2)S 489 284 :M f0_12 sf (,)S 60 302 :M (however, since X)S 144 304 :M f0_9 sf (1)S 149 302 :M f0_12 sf .011 .001( is not an ancestor of V, but, as we have just shown X)J 407 304 :M f0_9 sf (1)S 412 302 :M f0_12 sf .011 .001( is an ancestor of)J 60 320 :M (W, it follows that W is not an ancestor of V. There are now two cases to consider:)S 60 344 :M f2_12 sf 6.701 .67(Subcase 1:)J f0_12 sf 2.484 .248( n = 1 and W)J f1_12 sf S 213 344 :M f0_12 sf (X)S 222 346 :M f0_9 sf (1)S 227 344 :M f0_12 sf 3.688 .369(. In this case X)J 320 346 :M f0_9 sf (0)S 325 344 :M f0_12 sf S 337 0 7 730 rC 336 344 :M 4.829 .483( )J gR gS 0 0 552 730 rC 337 344 :M f0_12 sf (>X)S 344 0 8 730 rC 344 344 :M 4.829 .483( )J gR gS 352 0 5 730 rC 350 346 :M f0_9 sf 4.829 .483( )J gR gS 0 0 552 730 rC 352 346 :M f0_9 sf (1)S 352 0 5 730 rC 350 346 :M 4.829 .483( )J gR 1 G gS 0 0 552 730 rC 0 0 1 1 rF 364 346 :M psb /wp$x1 337 def /wp$x2 351 def /wp$y 346 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 357 0 7 730 rC 356 344 :M 0 G f0_12 sf 4.829 .483( )J gR gS 0 0 552 730 rC 357 344 :M 0 G f0_12 sf (<)S 357 0 7 730 rC 356 344 :M 4.829 .483( )J gR gS 0 0 552 730 rC 0 0 1 1 rF 372 344 :M psb /wp$x1 352 def /wp$x2 356 def /wp$y 348 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 364 344 :M 0 G f0_12 sf S 385 346 :M f0_9 sf (2)S 390 344 :M f0_12 sf 2.668 .267(, in )J f3_12 sf (Y)S 428 346 :M f0_9 sf (2)S 433 344 :M f0_12 sf 2.891 .289( \(and )J f3_12 sf (Y)S 480 346 :M f0_9 sf (1)S 485 344 :M f0_12 sf (\).)S 1 G 0 0 1 1 rF 491 344 :M psb /wp$x1 357 def /wp$x2 363 def /wp$y 346 def [1 2] 0 setdash .5 setlinewidth 0 setgray wp$x1 wp$y moveto wp$x2 wp$y lineto stroke pse 60 362 :M 0 G f2_12 sf (Supsepset)S 111 362 :M f0_12 sf (\(X)S 124 364 :M f0_9 sf (0)S 129 362 :M f0_12 sf (,V,X)S f0_9 sf 0 2 rm (2)S 0 -2 rm 157 362 :M f0_12 sf 4.892 .489(\) is the smallest set containing )J 350 362 :M f2_12 sf (Sepset)S 383 362 :M f0_12 sf (\(X)S 396 364 :M f0_9 sf (0)S 401 362 :M f0_12 sf (,X)S 413 364 :M f0_9 sf (2)S 418 362 :M f0_12 sf <29>S 422 362 :M f1_12 sf 1.12A f0_12 sf 3.545 .355({V} which)J 60 380 :M .08 .008(d-separates X)J 125 382 :M f0_9 sf (0)S 130 380 :M f0_12 sf .119 .012( and X)J f0_9 sf 0 2 rm (2)S 0 -2 rm 167 380 :M f0_12 sf .126 .013(, in the sense that no subset of )J 316 380 :M f2_12 sf (Supsepset)S 367 380 :M f0_12 sf (\(X)S 380 382 :M f0_9 sf (0)S 385 380 :M f0_12 sf (,V,X)S f0_9 sf 0 2 rm (2)S 0 -2 rm 413 380 :M f0_12 sf .101 .01(\) which contains)J 60 398 :M f2_12 sf (Sepset)S 93 398 :M f0_12 sf (\(X)S 106 400 :M f0_9 sf (0)S 111 398 :M f0_12 sf (,X)S 123 400 :M f0_9 sf (2)S 128 398 :M f0_12 sf .509 .051(\) )J f1_12 sf .805A f0_12 sf 2.156 .216({V} d-separates X)J 242 400 :M f0_9 sf 2.781 .278(0 )J 252 398 :M f0_12 sf 2.185 .219(and X)J 285 400 :M f0_9 sf (2)S 290 398 :M f0_12 sf 2.147 .215(. It follows from )J 384 398 :M 1.912 .191(Lemma 7)J 433 398 :M 2.477 .248( \(with )J f2_12 sf 2.263 .226(R )J 485 398 :M f0_12 sf (=)S 60 416 :M f2_12 sf (Sepset)S 93 416 :M f0_12 sf (\(X)S 106 418 :M f0_9 sf (0)S 111 416 :M f0_12 sf (,X)S 123 418 :M f0_9 sf (2)S 128 416 :M f0_12 sf .125 .013(\) )J f1_12 sf .198A f0_12 sf .353 .035({V}\) that every vertex in )J 270 416 :M f2_12 sf (Supsepset)S 321 416 :M f0_12 sf (\(X)S 334 418 :M f0_9 sf (0)S 339 416 :M f0_12 sf (,V,X)S f0_9 sf 0 2 rm (2)S 0 -2 rm 367 416 :M f0_12 sf .336 .034(\) is an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 471 416 :M f0_12 sf .417 .042(, X)J 487 418 :M f0_9 sf (2)S 60 434 :M f0_12 sf (or )S f2_12 sf .01(Sepset)A 106 434 :M f0_12 sf (\(X)S 119 436 :M f0_9 sf (0)S 124 434 :M f0_12 sf (,X)S 136 436 :M f0_9 sf (2)S 141 434 :M f0_12 sf (\) )S f1_12 sf S f0_12 sf .05 .005({V}. Since every vertex in )J 289 434 :M f2_12 sf (Sepset)S 322 434 :M f0_12 sf (\(X)S 335 436 :M f0_9 sf (0)S 340 434 :M f0_12 sf (,X)S 352 436 :M f0_9 sf (2)S 357 434 :M f0_12 sf .048 .005(\) is an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 459 434 :M f0_12 sf .06 .006( or X)J 484 436 :M f0_9 sf (2)S 489 434 :M f0_12 sf (,)S 60 452 :M .412 .041(it follows that every vertex in )J 209 452 :M f2_12 sf (Supsepset)S 260 452 :M f0_12 sf (\(X)S 273 454 :M f0_9 sf (0)S 278 452 :M f0_12 sf (,V,X)S f0_9 sf 0 2 rm (2)S 0 -2 rm 306 452 :M f0_12 sf .421 .042(\) is an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 411 452 :M f0_12 sf .368 .037(, X)J f0_9 sf 0 2 rm (2)S 0 -2 rm 431 452 :M f0_12 sf .435 .043(, or V. X)J f0_9 sf 0 2 rm (1)S 0 -2 rm 480 452 :M f0_12 sf .521 .052( is)J 60 470 :M 1.618 .162(not an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 172 470 :M f0_12 sf 2.115 .212(, X)J 190 472 :M f0_9 sf (2)S 195 470 :M f0_12 sf 1.69 .169(, or V in )J f4_12 sf (G)S 257 472 :M f0_9 sf (2)S 262 470 :M f0_12 sf 1.731 .173(. Hence in step \246D of the algorithm given a)J 60 488 :M .452 .045(d-separation oracle for )J f4_12 sf (G)S 182 490 :M f0_9 sf (2)S 187 488 :M f0_12 sf .658 .066( as input X)J 242 490 :M f0_9 sf (1)S 247 488 :M f1_12 sf S 256 488 :M f0_12 sf .877 .088( )J 260 488 :M f2_12 sf (Supsepset)S 311 488 :M f0_12 sf (\(X)S 324 490 :M f0_9 sf (0)S 329 488 :M f0_12 sf (,V,X)S f0_9 sf 0 2 rm (2)S 0 -2 rm 357 488 :M f0_12 sf .619 .062(\). Thus step \246E of the CCD)J 60 506 :M .631 .063(algorithm will orient W)J 177 506 :M f1_12 sf .255(*)A f0_12 sf .255A f1_12 sf .255(*)A f0_12 sf .408 .041(V in )J f3_12 sf (Y)S 231 508 :M f0_9 sf (2)S 236 506 :M f0_12 sf .526 .053( as W<\320)J f1_12 sf .212(*)A f0_12 sf .713 .071(V \(unless they have already been oriented)J 60 524 :M (this way in a previous stage of the algorithm\). Thus )S f3_12 sf (Y)S 320 526 :M f0_9 sf (1)S 325 524 :M f0_12 sf ( and )S f3_12 sf (Y)S 358 526 :M f0_9 sf (2)S 363 524 :M f0_12 sf ( are not the same.)S 60 548 :M f2_12 sf (Subcase 2:)S f0_12 sf ( n > 1, or W is not equal to X)S f0_9 sf 0 2 rm (1)S 0 -2 rm 259 548 :M f0_12 sf (.)S 60 572 :M f2_12 sf (Claim:)S f0_12 sf ( X)S 107 574 :M f0_9 sf (0)S 112 572 :M f0_12 sf ( and X)S 144 574 :M f0_10 sf (n)S f0_9 sf (+1)S 159 572 :M f0_12 sf ( are d-connected given )S 271 572 :M f2_12 sf (Supsepset)S 322 572 :M f0_12 sf (\(X)S 335 574 :M f0_9 sf (0)S 340 572 :M f0_12 sf (,V,X)S f0_10 sf 0 2 rm (n)S 0 -2 rm f0_9 sf 0 2 rm (+1)S 0 -2 rm 378 572 :M f0_12 sf <29>S 382 572 :M f1_12 sf S f0_12 sf ({W} in )S f4_12 sf (G)S 438 574 :M f0_9 sf (2)S 443 572 :M f0_12 sf (.)S 60 596 :M f2_12 sf (Proof.)S 92 596 :M f0_12 sf .121 .012( We have already shown that W is a descendant of X)J 348 598 :M f0_9 sf (1)S 353 596 :M f0_12 sf .145 .014(, and so also of X)J f0_10 sf 0 2 rm .059(n)A 0 -2 rm f0_12 sf .078 .008( in )J 458 596 :M f4_12 sf (G)S 467 598 :M f0_9 sf (1)S 472 596 :M f0_12 sf .134 .013( and)J 60 614 :M f4_12 sf (G)S 69 616 :M f0_9 sf (2)S 74 614 :M f0_12 sf .627 .063(. Since in both )J f4_12 sf (G)S 159 616 :M f0_9 sf (1)S 164 614 :M f0_12 sf .574 .057( and )J f4_12 sf (G)S 198 616 :M f0_9 sf .894 .089(2 )J 206 614 :M f0_12 sf (X)S 215 616 :M f0_9 sf (0)S 220 614 :M f0_12 sf .819 .082( is p)J 242 614 :M .596 .06(-adjacent to X)J 311 616 :M f0_9 sf (1)S 316 614 :M f0_12 sf .756 .076(, but X)J 351 616 :M f0_9 sf (1)S 356 614 :M f0_12 sf .675 .068( is not an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 478 614 :M f0_12 sf .756 .076(, it)J 60 632 :M .223 .022(follows that X)J 130 634 :M f0_9 sf (0)S 135 632 :M f0_12 sf .267 .027( is an ancestor of X)J 230 634 :M f0_9 sf (1)S 235 632 :M f0_12 sf .236 .024(. Hence in both )J f4_12 sf (G)S 321 634 :M f0_9 sf (1)S 326 632 :M f0_12 sf .215 .021( and )J f4_12 sf (G)S 359 634 :M f0_9 sf (2)S 364 632 :M f0_12 sf .245 .024( there is a directed path P)J f0_9 sf 0 2 rm (0)S 0 -2 rm 60 650 :M f0_12 sf .593 .059(from X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 101 650 :M f0_12 sf .722 .072( to X)J f0_9 sf 0 2 rm (1)S 0 -2 rm 132 650 :M f0_12 sf .696 .07( on which every vertex except for X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 317 650 :M f0_12 sf .766 .077( is a descendant of X)J 423 652 :M f0_9 sf (1)S 428 650 :M f0_12 sf .748 .075(. \(In the case)J 60 668 :M (X)S 69 670 :M f0_9 sf (0)S 74 668 :M f1_12 sf S 86 668 :M f0_12 sf (X)S 95 670 :M f0_9 sf (1)S 100 668 :M f0_12 sf .578 .058(, the last assertion is trivial. In the case where X)J 339 670 :M f0_9 sf (0)S 344 668 :M f0_12 sf .698 .07( and X)J 378 670 :M f0_9 sf (1)S 383 668 :M f0_12 sf .592 .059( have a common child)J 60 686 :M -.004(that is an ancestor of X)A 171 688 :M f0_9 sf (0)S 176 686 :M f0_12 sf ( or X)S 201 688 :M f0_9 sf (1)S 206 686 :M f0_12 sf (, and X)S 241 688 :M f0_9 sf (1)S 246 686 :M f0_12 sf -.005( is not an ancestor of X)A f0_9 sf 0 2 rm (0)S 0 -2 rm 362 686 :M f0_12 sf -.003(, it merely states a property)A endp %%Page: 57 57 %%BeginPageSetup initializepage (peter; page: 57 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (57)S gR gS 0 0 552 730 rC 60 56 :M f0_12 sf 1.161 .116(of the path X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 133 56 :M f1_12 sf S 145 56 :M f0_12 sf (C)S f1_12 sf S 165 56 :M f0_12 sf S 186 58 :M f0_9 sf (1)S 191 56 :M f0_12 sf 1.239 .124(, where C is a common child of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 372 56 :M f0_12 sf 1.426 .143( and X)J 408 58 :M f0_9 sf (1)S 413 56 :M f0_12 sf .295(.)A f0_9 sf 0 2 rm .221(.)A 0 -2 rm f0_12 sf 1.248 .125(\) Since W is a)J 60 74 :M .182 .018(descendant of X)J 139 76 :M f0_9 sf (1)S 144 74 :M f0_12 sf .212 .021(, it follows that there is a directed path )J 333 74 :M f4_12 sf (P)S f0_9 sf 0 2 rm (1)S 0 -2 rm 345 74 :M f0_12 sf .239 .024( from X)J 384 76 :M f0_9 sf (1)S 389 74 :M f0_12 sf .19 .019( to W. Concatenating)J 60 92 :M f4_12 sf (P)S f0_9 sf 0 2 rm (0)S 0 -2 rm 72 92 :M f0_12 sf ( and )S f4_12 sf (P)S f0_9 sf 0 2 rm (1)S 0 -2 rm 107 92 :M f0_12 sf ( we construct a directed path )S 248 92 :M f4_12 sf (P)S f0_12 sf (* from X)S 299 94 :M f0_9 sf (0)S 304 92 :M f0_12 sf ( to W on which every vertex except X)S 487 94 :M f0_9 sf (0)S 60 110 :M f0_12 sf .782 .078(is a descendant of X)J f0_9 sf 0 2 rm (1)S 0 -2 rm 167 110 :M f0_12 sf .768 .077(. Since X)J f0_9 sf 0 2 rm (1)S 0 -2 rm 218 110 :M f0_12 sf .842 .084( is not an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 341 110 :M f0_12 sf 1.022 .102(, X)J 358 112 :M f0_10 sf (n)S f0_9 sf (+1)S 373 110 :M f0_12 sf .876 .088( or V, it follows that no)J 60 128 :M .698 .07(vertex on )J f4_12 sf .32(P)A f0_10 sf 0 -3 rm .218(*)A 0 3 rm f0_12 sf .737 .074(, except X)J 174 130 :M f0_9 sf (0)S 179 128 :M f0_12 sf .802 .08(, is an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 285 128 :M f0_12 sf .993 .099(, X)J 302 130 :M f0_10 sf (n)S f0_9 sf (+1)S 317 128 :M f0_12 sf .802 .08( or V. Similarly we can construct a)J 60 146 :M (path from )S 110 146 :M f4_12 sf (Q)S 119 143 :M f0_10 sf (*)S f0_12 sf 0 3 rm ( from X)S 0 -3 rm 162 148 :M f0_10 sf (n)S f0_9 sf (+1)S 177 146 :M f0_12 sf ( to W on which no vertex, except X)S 348 148 :M f0_10 sf (n)S f0_9 sf (+1)S 363 146 :M f0_12 sf (, is an ancestor of X)S 459 148 :M f0_9 sf (0)S 464 146 :M f0_12 sf (, X)S 479 148 :M f0_10 sf (n)S f0_9 sf (+1)S 60 164 :M f0_12 sf (or V.)S 60 188 :M 4.47 .447(Since every vertex in )J 192 188 :M f2_12 sf (Supsepset)S 243 188 :M f0_12 sf (\(X)S 256 190 :M f0_9 sf (0)S 261 188 :M f0_12 sf (,V,X)S f0_10 sf 0 2 rm (n)S 0 -2 rm f0_9 sf 0 2 rm (+1)S 0 -2 rm 299 188 :M f0_12 sf 4.588 .459(\) is an ancestor of X)J f0_9 sf 0 2 rm (0)S 0 -2 rm 435 188 :M f0_12 sf 5.681 .568(, X)J 457 190 :M f0_10 sf (n)S f0_9 sf (+1)S 472 188 :M f0_12 sf 5.681 .568( or)J 60 206 :M f2_12 sf (Sepset)S 93 206 :M f0_12 sf (\(X)S 106 208 :M f0_9 sf (0)S 111 206 :M f0_12 sf (,X)S 123 208 :M f0_10 sf (n)S f0_9 sf (+1)S 138 206 :M f0_12 sf <29>S 142 206 :M f1_12 sf .06A f0_12 sf .103 .01({V}, it follows as before that every vertex in )J f2_12 sf .041(Supsepset)A 421 206 :M f0_12 sf (\(X)S 434 208 :M f0_9 sf (0)S 439 206 :M f0_12 sf (,V,X)S f0_10 sf 0 2 rm (n)S 0 -2 rm f0_9 sf 0 2 rm (+1)S 0 -2 rm 477 206 :M f0_12 sf .153 .015(\) is)J 60 224 :M .375 .038(an ancestor of X)J 141 226 :M f0_9 sf (0)S 146 224 :M f0_12 sf .217 .022(, X)J f0_10 sf 0 2 rm .096(n)A 0 -2 rm f0_9 sf 0 2 rm .184(+1)A 0 -2 rm 176 224 :M f0_12 sf .419 .042( or V. Thus no vertex in )J 298 224 :M f2_12 sf (Supsepset)S 349 224 :M f0_12 sf (\(X)S 362 226 :M f0_9 sf (0)S 367 224 :M f0_12 sf (,V,X)S f0_10 sf 0 2 rm (n)S 0 -2 rm f0_9 sf 0 2 rm (+1)S 0 -2 rm 405 224 :M f0_12 sf .433 .043(\) lies on )J 449 224 :M f4_12 sf .223(P)A f0_10 sf 0 -3 rm .152(*)A 0 3 rm f0_12 sf .203 .02( or )J f4_12 sf (Q)S 487 221 :M f0_10 sf (*)S 60 242 :M f0_12 sf (\(X)S 73 244 :M f0_9 sf (0)S 78 242 :M f0_12 sf .704 .07(, X)J 94 244 :M f0_10 sf (n)S f0_9 sf (+1)S 109 242 :M f0_12 sf .845 .084( )J 113 242 :M f1_12 sf S 122 242 :M f0_12 sf .845 .084( )J 126 242 :M f2_12 sf (Supsepset)S 177 242 :M f0_12 sf (\(X)S 190 244 :M f0_9 sf (0)S 195 242 :M f0_12 sf (,V,X)S f0_10 sf 0 2 rm (n)S 0 -2 rm f0_9 sf 0 2 rm (+1)S 0 -2 rm 233 242 :M f0_12 sf .554 .055(\) by definition\). It now follows by Lemma 1)J 452 242 :M .626 .063( that we)J 60 260 :M 2.109 .211(can concatenate )J 147 260 :M f4_12 sf 1.221(P)A f0_10 sf 0 -3 rm .833(*)A 0 3 rm f0_12 sf 1.554 .155( and )J f4_12 sf (Q)S 198 257 :M f0_10 sf .74(*)A f0_12 sf 0 3 rm 2.436 .244( to form a path which d-connects X)J 0 -3 rm 399 262 :M f0_9 sf (0)S 404 260 :M f0_12 sf 1.95 .195( and X)J f0_10 sf 0 2 rm .792(n)A 0 -2 rm f0_9 sf 0 2 rm 1.518(+1)A 0 -2 rm 458 260 :M f0_12 sf 2.46 .246( given)J 60 278 :M f2_12 sf (Supsepset)S 111 278 :M f0_12 sf (\(X)S 124 280 :M f0_9 sf (0)S 129 278 :M f0_12 sf (,V,X)S f0_10 sf 0 2 rm (n)S 0 -2 rm f0_9 sf 0 2 rm (+1)S 0 -2 rm 167 278 :M f0_12 sf <29>S 171 278 :M f1_12 sf S f0_12 sf ({W}.)S 60 302 :M .437 .044(It follows directly from this claim that step \246F of the CCD algorithm will orient V*\320*W)J 60 320 :M .41 .041(as V)J 83 320 :M f2_12 sf .233A f0_12 sf .249 .025(>W in )J f3_12 sf (Y)S 139 322 :M f1_9 sf .484 .048(2 )J 147 320 :M f0_12 sf .336 .034(\(unless they have already been oriented this way in a previous stage of)J 60 338 :M (the algorithm\). Hence )S 168 338 :M f3_12 sf (Y)S 178 340 :M f0_9 sf (1)S 183 338 :M f0_12 sf ( and )S f3_12 sf (Y)S 216 340 :M f0_9 sf (2)S 221 338 :M f0_12 sf ( are different.)S 60 362 :M (Since Cases 1-5 exhaust the possible ways in which )S 312 362 :M f4_12 sf (G)S 321 364 :M f0_9 sf (1 )S 328 362 :M f0_12 sf (and )S f4_12 sf (G)S 357 364 :M f0_9 sf (2)S 362 362 :M f0_12 sf ( may fail to satisfy CET\(I\)-)S 60 380 :M .084 .008(\(V\), this completes the proof that the CCD algorithm locates the d)J 379 380 :M .042 .004(-separation equivalence)J 60 398 :M (class. )S f1_12 sf <5C>S endp %%Page: 58 58 %%BeginPageSetup initializepage (peter; page: 58 of 60)setjob %%EndPageSetup gS 0 0 552 730 rC 517 5 29 24 rC 534 26 :M f0_12 sf (58)S gR gS 0 0 552 730 rC 60 65 :M f2_14 sf (8)S 67 65 :M (.)S 70 65 :M ( )S 96 65 :M (Refe)S 122 65 :M (rences)S 60 92 :M f0_12 sf .135 .013(Aho, A.V., Hopcroft, J.E., & Ullman, J.D., \(1974\) The Design and Analysis of Computer)J 60 104 :M (Algorithms. 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