%!PS-Adobe-3.0 %%Title: (Microsoft Word - cyclesai) %%Creator: (Microsoft Word: LaserWriter 8 8.3.4) %%CreationDate: (12:24 AM Thursday, March 27, 1997) %%For: (peter) %%Pages: 37 %%DocumentFonts: Times-Roman Symbol Times-Bold Times-Italic %%DocumentNeededFonts: Times-Roman Symbol Times-Bold Times-Italic %%DocumentSuppliedFonts: %%DocumentData: Clean7Bit %%PageOrder: Ascend %%Orientation: Portrait %%DocumentMedia: Default 612 792 0 () () %ADO_ImageableArea: 31 31 583 761 %%EndComments userdict begin/dscInfo 5 dict dup begin /Title(Microsoft Word - cyclesai)def /Creator(Microsoft Word: LaserWriter 8 8.3.4)def /CreationDate(12:24 AM Thursday, March 27, 1997)def /For(peter)def /Pages 37 def end def end save /version23-manualfeedpatch where { pop false } { true }ifelse % we don't do an explicit 'get' since product and version MAY % be in systemdict or statusdict - this technique gets the lookup % without failure statusdict begin product (LaserWriter) eq % true if LaserWriter version cvr 23.0 eq % true if version 23 end and % only install this patch if both are true and % true only if patch is not installed and is for this printer % save object and boolean on stack dup { exch restore }if % either true OR saveobject false dup { /version23-manualfeedpatch true def /oldversion23-showpage /showpage load def /showpage % this showpage will wait extra time if manualfeed is true {% statusdict /manualfeed known {% manualfeed known in statusdict statusdict /manualfeed get {% if true then we loop for 5 seconds usertime 5000 add % target usertime { % loop dup usertime sub 0 lt { exit }if }loop pop % pop the usertime off the stac }if }if oldversion23-showpage }bind def }if not{ restore }if /md 222 dict def md begin/currentpacking where {pop /sc_oldpacking currentpacking def true setpacking}if %%BeginFile: adobe_psp_basic %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /bd{bind def}bind def /xdf{exch def}bd /xs{exch store}bd /ld{load def}bd /Z{0 def}bd /T/true /F/false /:L/lineto /lw/setlinewidth /:M/moveto /rl/rlineto /rm/rmoveto /:C/curveto /:T/translate /:K/closepath /:mf/makefont /gS/gsave /gR/grestore /np/newpath 14{ld}repeat /$m matrix def /av 83 def /por true def /normland false def /psb-nosave{}bd /pse-nosave{}bd /us Z /psb{/us save store}bd /pse{us restore}bd /level2 /languagelevel where { pop languagelevel 2 ge }{ false }ifelse def /featurecleanup { stopped cleartomark countdictstack exch sub dup 0 gt { {end}repeat }{ pop }ifelse }bd /noload Z /startnoload { {/noload save store}if }bd /endnoload { {noload restore}if }bd level2 startnoload /setjob { statusdict/jobname 3 -1 roll put }bd /setcopies { userdict/#copies 3 -1 roll put }bd level2 endnoload level2 not startnoload /setjob { 1 dict begin/JobName xdf currentdict end setuserparams }bd /setcopies { 1 dict begin/NumCopies xdf currentdict end setpagedevice }bd level2 not endnoload /pm Z /mT Z /sD Z /realshowpage Z /initializepage { /pm save store mT concat }bd /endp { pm restore showpage }def /$c/DeviceRGB def /rectclip where { pop/rC/rectclip ld }{ /rC { np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K clip np }bd }ifelse /rectfill where { pop/rF/rectfill ld }{ /rF { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl fill gR }bd }ifelse /rectstroke where { pop/rS/rectstroke ld }{ /rS { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K stroke gR }bd }ifelse %%EndFile %%BeginFile: adobe_psp_colorspace_level1 %%Copyright: Copyright 1991-1993 Adobe Systems Incorporated. All Rights Reserved. /G/setgray ld /:F1/setgray ld /:F/setrgbcolor ld /:F4/setcmykcolor where { pop /setcmykcolor ld }{ { 3 { dup 3 -1 roll add dup 1 gt{pop 1}if 1 exch sub 4 1 roll }repeat pop setrgbcolor }bd }ifelse /:Fx { counttomark {0{G}0{:F}{:F4}} exch get exec pop }bd /:rg{/DeviceRGB :ss}bd /:sc{$cs :ss}bd /:dc{/$cs xdf}bd /:sgl{}def /:dr{}bd /:fCRD{pop}bd /:ckcs{}bd /:ss{/$c xdf}bd /$cs Z %%EndFile %%BeginFile: adobe_psp_uniform_graphics %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /@a { np :M 0 rl :L 0 exch rl 0 rl :L fill }bd /@b { np :M 0 rl 0 exch rl :L 0 rl 0 exch rl fill }bd /arct where { pop }{ /arct { arcto pop pop pop pop }bd }ifelse /x1 Z /x2 Z /y1 Z /y2 Z /rad Z /@q { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct fill }bd /@s { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct :K stroke }bd /@i { np 0 360 arc fill }bd /@j { gS np :T scale 0 0 .5 0 360 arc fill gR }bd /@e { np 0 360 arc :K stroke }bd /@f { np $m currentmatrix pop :T scale 0 0 .5 0 360 arc :K $m setmatrix stroke }bd /@k { gS np :T 0 0 :M 0 0 5 2 roll arc fill gR }bd /@l { gS np :T 0 0 :M scale 0 0 .5 5 -2 roll arc fill gR }bd /@m { np arc stroke }bd /@n { np $m currentmatrix pop :T scale 0 0 .5 5 -2 roll arc $m setmatrix stroke }bd %%EndFile %%BeginFile: adobe_psp_customps %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /$t Z /$p Z /$s Z /$o 1. def /2state? false def /ps Z level2 startnoload /pushcolor/currentrgbcolor ld /popcolor/setrgbcolor ld /setcmykcolor where { pop/currentcmykcolor where { pop/pushcolor/currentcmykcolor ld /popcolor/setcmykcolor ld }if }if level2 endnoload level2 not startnoload /pushcolor { currentcolorspace $c eq { currentcolor currentcolorspace true }{ currentcmykcolor false }ifelse }bd /popcolor { { setcolorspace setcolor }{ setcmykcolor }ifelse }bd level2 not endnoload /pushstatic { ps 2state? $o $t $p $s $cs }bd /popstatic { /$cs xs /$s xs /$p xs /$t xs /$o xs /2state? xs /ps xs }bd /pushgstate { save errordict/nocurrentpoint{pop 0 0}put currentpoint 3 -1 roll restore pushcolor currentlinewidth currentlinecap currentlinejoin currentdash exch aload length np clippath pathbbox $m currentmatrix aload pop }bd /popgstate { $m astore setmatrix 2 index sub exch 3 index sub exch rC array astore exch setdash setlinejoin setlinecap lw popcolor np :M }bd /bu { pushgstate gR pushgstate 2state? { gR pushgstate }if pushstatic pm restore mT concat }bd /bn { /pm save store popstatic popgstate gS popgstate 2state? { gS popgstate }if }bd /cpat{pop 64 div setgray 8{pop}repeat}bd %%EndFile %%BeginFile: adobe_psp_basic_text %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /S/show ld /A{ 0.0 exch ashow }bd /R{ 0.0 exch 32 exch widthshow }bd /W{ 0.0 3 1 roll widthshow }bd /J{ 0.0 32 4 2 roll 0.0 exch awidthshow }bd /V{ 0.0 4 1 roll 0.0 exch awidthshow }bd /fcflg true def /fc{ fcflg{ vmstatus exch sub 50000 lt{ (%%[ Warning: Running out of memory ]%%\r)print flush/fcflg false store }if pop }if }bd /$f[1 0 0 -1 0 0]def /:ff{$f :mf}bd /MacEncoding StandardEncoding 256 array copy def MacEncoding 39/quotesingle put MacEncoding 96/grave put /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 /dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash /infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation /product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash /questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft /guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge /ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl /daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave /Oacute/Ocircumflex/apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde /macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron MacEncoding 128 128 getinterval astore pop level2 startnoload /copyfontdict { findfont dup length dict begin { 1 index/FID ne{def}{pop pop}ifelse }forall }bd level2 endnoload level2 not startnoload /copyfontdict { findfont dup length dict copy begin }bd level2 not endnoload md/fontname known not{ /fontname/customfont def }if /Encoding Z /:mre { copyfontdict /Encoding MacEncoding def fontname currentdict end definefont :ff def }bd /:bsr { copyfontdict /Encoding Encoding 256 array copy def Encoding dup }bd /pd{put dup}bd /:esr { pop pop fontname currentdict end definefont :ff def }bd /scf { scalefont def }bd /scf-non { $m scale :mf setfont }bd /ps Z /fz{/ps xs}bd /sf/setfont ld /cF/currentfont ld /mbf { /makeblendedfont where { pop makeblendedfont /ABlend exch definefont }{ pop }ifelse def }def %%EndFile %%BeginFile: adobe_psp_derived_styles %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. 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page: 1 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 539 26 :M f0_12 sf (1)S gR gS 0 0 552 730 rC 108 158 :M f2_14 sf (Directed Cyclic Graphical Representations of Feedback)S 250 181 :M (Models)S 294 178 :M f2_8 sf (1)S 268 213 :M f0_12 sf (by)S 244 243 :M (Peter Spirtes)S 95 273 :M f2_12 sf (1. Introduction)S 95 303 :M f0_12 sf 1.892 .189(The introduction of statistical models represented by directed acyclic)J 95 321 :M .655 .065(graphs \(DAGs\) has proved fruitful in the construction of expert systems,)J 95 339 :M 3.792 .379(in allowing efficient updating algorithms that take advantage of)J 95 357 :M .797 .08(conditional independence relations \(Pearl, 1988, Lauritzen et. al., 1993\),)J 95 375 :M .669 .067(and in inferring causal structure from conditional independence relations)J 95 393 :M .644 .064(\(Spirtes and Glymour, 1991, Spirtes, Glymour and Scheines, 1993, Pearl)J 95 411 :M 1.176 .118(and Verma, 1991, Cooper, 1992\). As a framework for representing the)J 95 429 :M .517 .052(combination of causal and statistical hypotheses, DAG models have shed)J 95 447 :M -.003(light on a number of issues in statistics ranging from Simpson's Paradox to)A 95 465 :M .168 .017(experimental design \(Spirtes, Glymour and Scheines, 1993\). The relations)J 95 483 :M 4.706 .471(of DAGs with statistical constraints, and the equivalence and)J 95 501 :M 1.232 .123(distinguishability properties of DAG models, are now well understood,)J 95 519 :M 2.325 .232(and their characterization and computation involves three properties)J 95 537 :M 1.417 .142(connecting graphical structure and probability distributions: \(i\) a local)J 95 567 :M ( )S 95 564.48 -.48 .48 239.48 564 .48 95 564 @a 95 588 :M f0_8 sf (1)S 99 591 :M f0_10 sf 1.091 .109( Research for this paper was supported by the National Science Foundation through)J 95 602 :M .128 .013(grant 9102169 and the Navy Personnel Research and Development Center and the Office)J 95 613 :M .389 .039(of Naval Research through contract number N00014-93-1-0568. I am indebted to Clark)J endp %%Page: 2 2 %%BeginPageSetup initializepage (peter; page: 2 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 539 26 :M f0_12 sf (2)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .163 .016(directed Markov property, \(ii\) a global directed Markov property, \(iii\) and)J 95 122 :M 2.011 .201(factorizations of joint densities according to the structure of a graph)J 95 140 :M (\(Lauritizen, et al., 1990\).)S 95 170 :M 2.602 .26(Recursive structural equation models are one kind of DAG model.)J 95 188 :M .463 .046(However, non-recursive structural equation models are not DAG models,)J 95 206 :M .093 .009(and are instead naturally represented by directeed )J f4_12 sf .026(cyclic)A 365 206 :M f0_12 sf .126 .013( graphs in which a)J 95 224 :M 3.582 .358(finite series of edges representing influence leads from a vertex)J 95 242 :M .787 .079(representing a variable back to that same vertex. Such graphs have been)J 95 260 :M .83 .083(used to model feedback systems in electrical engineering \(Mason, 1953,)J 95 278 :M 4.651 .465(1956\), and to represent economic processes \(Haavelmo, 1943,)J 95 296 :M .415 .042(Goldberger, 1973\). In contrast to the acyclic case, almost nothing general)J 95 314 :M .578 .058(is known about how directed cyclic graphs \(DCGs\) represent conditional)J 95 332 :M 1.871 .187(independence constraints, or about their equivalence or identifiability)J 95 350 :M 1.985 .199(properties, or about characterizing classes of DCGs from conditional)J 95 368 :M 3.544 .354(independence relations or other statistical constraints. This paper)J 95 386 :M -.006(addresses the first of these problems, which is a prerequisite for the others.)A 95 404 :M 1.208 .121(The issues turn on how the relations among properties \(i\), \(ii\) and \(iii\))J 95 422 :M 3.408 .341(essential to the acyclic case generalize--or more typically fail to)J 95 440 :M 4.297 .43(generalize--to directed cyclic graphs and associated families of)J 95 458 :M .232 .023(distributions. It will be shown that when DCGs are interpreted by analogy)J 95 476 :M 1.012 .101(with DAGs as representing functional dependencies with independently)J 95 494 :M 1.29 .129(distributed noises or "error terms," the equivalence of the fundamental)J 95 512 :M 1.841 .184(global and local Markov conditions characteristc of DAGs no longer)J 95 530 :M 1.532 .153(holds, even in linear systems, and in non-linear systems both Markov)J 95 548 :M 2.569 .257(properties may fail. For linear systems associated with DCGs with)J 95 566 :M 4.762 .476(independent errors or noises, a characterisation of conditional)J 95 584 :M 2.137 .214(independence constraints is obtained, and it is shown that the result)J 95 614 :M ( )S 95 611.48 -.48 .48 455.48 611 .48 95 611 @a 95 626 :M f0_10 sf .222 .022(Glymour, Richard Scheines, Christopher Meek, Thomas Richardson, and Marek Druzdel)J endp %%Page: 3 3 %%BeginPageSetup initializepage (peter; page: 3 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 539 26 :M f0_12 sf (3)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf 1.272 .127(generalizes in a natural way to systems in which the error variables or)J 95 122 :M 4.613 .461(noises are statistically dependent. For non-linear systems with)J 95 140 :M .494 .049(independent errors a sufficient condition for conditional independence of)J 95 158 :M (variables in associated distributions is obtained.)S 95 188 :M 1.258 .126(The remainder of this paper is organized as follows: Section 2 defines)J 95 206 :M .195 .02(relevant mathematical ideas and gives some necessary technical results on)J 95 224 :M 2.351 .235(DAGs and DCGs. Section 3 obtains results for non-recursive linear)J 95 242 :M 1.607 .161(structural equations models. Section 4 treats non-linear models of the)J 95 260 :M (same kind.)S 95 290 :M f2_12 sf (2. Directed Graphs)S 95 320 :M f0_12 sf 1.455 .145(I place sets of variables and defined terms in boldface, and individual)J 95 338 :M .525 .053(variables in italics. A )J 204 338 :M f2_12 sf .36 .036(directed graph)J 281 338 :M f0_12 sf .586 .059( is an ordered pair of a finite set of)J 95 356 :M .406 .041(vertices )J f2_12 sf (V)S 145 356 :M f0_12 sf .599 .06(, and a set of directed edges )J f2_12 sf .353(E)A f0_12 sf .681 .068(. A directed edge from )J 411 356 :M f4_12 sf .503(A)A f0_12 sf .438 .044( to )J f4_12 sf .503(B)A f0_12 sf .629 .063( is)J 95 374 :M .064 .006(an ordered pair of distinct vertices <)J 270 374 :M f4_12 sf (A)S f0_12 sf (,)S f4_12 sf (B)S f0_12 sf .038 .004(> in )J f2_12 sf (V)S 318 374 :M f0_12 sf .079 .008( in which )J 366 374 :M f4_12 sf .058(A)A f0_12 sf .068 .007( is the )J 406 374 :M f2_12 sf (tail)S 423 374 :M f0_12 sf .079 .008( of the)J 95 392 :M .049 .005(edge and )J f4_12 sf (B)S f0_12 sf .026 .003( is the )J 180 392 :M f2_12 sf (head)S 205 392 :M f0_12 sf .039 .004(; the edge is )J 266 392 :M f2_12 sf .035 .003(out of)J 297 392 :M f0_12 sf ( )S f4_12 sf (A)S f0_12 sf .027 .003( and )J f2_12 sf .02(into)A 350 392 :M f0_12 sf ( )S f4_12 sf (B)S f0_12 sf .029 .003(, and )J f4_12 sf (A)S f0_12 sf ( is )S f2_12 sf .019(parent)A 441 392 :M f0_12 sf .043 .004( of)J 95 410 :M f4_12 sf .093(B)A f0_12 sf .118 .012( and )J f4_12 sf .093(B)A f0_12 sf .078 .008( is a )J f2_12 sf .064(child)A f0_12 sf .084 .008( of )J f4_12 sf .093(A)A f0_12 sf .212 .021(. A sequence of distinct edges <)J 358 410 :M f4_12 sf (E)S f0_9 sf 0 2 rm (1)S 0 -2 rm 370 410 :M f0_12 sf (,...,)S f4_12 sf (E)S f4_9 sf 0 2 rm (n)S 0 -2 rm 397 410 :M f0_12 sf .146 .015(> in )J f4_12 sf (G)S 428 410 :M f0_12 sf .204 .02( is an)J 95 429 :M f2_12 sf 3.473 .347(undirected path )J f0_12 sf 2.425 .243(if and only if there exists a sequence of vertices)J 95 447 :M (<)S 102 447 :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 152 447 :M f0_12 sf .533 .053(> such that for 1 )J cF f1_12 sf .053A sf .533 .053( )J 248 447 :M f4_12 sf .294(i)A f0_12 sf .505 .05( )J cF f1_12 sf .05A sf .505 .05( )J 265 447 :M f4_12 sf .15(n)A f0_12 sf .345 .035( either <)J f4_12 sf .183(V)A f4_10 sf 0 2 rm (i)S 0 -2 rm 322 447 :M f0_12 sf (,)S f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 335 449 :M f0_10 sf (+1)S 346 447 :M f0_12 sf .611 .061(> = )J 368 447 :M f4_12 sf (E)S f4_10 sf 0 2 rm (i)S 0 -2 rm 378 447 :M f0_12 sf .389 .039( or <)J f4_12 sf .313(V)A f4_10 sf 0 2 rm (i)S 0 -2 rm 412 449 :M f0_10 sf (+1)S 423 447 :M f0_12 sf (,)S f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 436 447 :M f0_12 sf .585 .059(> =)J 95 466 :M f4_12 sf (E)S f4_10 sf 0 2 rm (i)S 0 -2 rm 105 466 :M f0_12 sf (. A path )S f4_12 sf (U)S 155 466 :M f0_12 sf ( is )S f2_12 sf (acyclic)S 204 466 :M f0_12 sf .009 .001( if no vertex occurring on an edge in the path occurs)J 95 485 :M 2.311 .231(more than once. A sequence of distinct edges <)J 351 485 :M f4_12 sf (E)S f0_9 sf 0 2 rm (1)S 0 -2 rm 363 485 :M f0_12 sf (,...,)S f4_12 sf (E)S f4_9 sf 0 2 rm (n)S 0 -2 rm 390 485 :M f0_12 sf 2.075 .208(> in )J f4_12 sf (G)S 428 485 :M f0_12 sf 3.009 .301( is a)J 95 504 :M f2_12 sf 4.217 .422(directed path )J f0_12 sf 3.301 .33(if and only if there exists a sequence of vertices)J 95 522 :M (<)S 102 522 :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 152 522 :M f0_12 sf .324 .032(> such that for 1 )J cF f1_12 sf .032A sf .324 .032( )J f4_12 sf .095(i)A f0_12 sf .134 .013( )J cF f1_12 sf .013A sf .134 .013( )J 265 522 :M f4_12 sf .223(n)A f0_12 sf .215 .022( <)J 285 522 :M f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 295 522 :M f0_12 sf (,)S f4_12 sf (V)S f4_10 sf 0 2 rm (i)S 0 -2 rm 308 524 :M f0_10 sf (+1)S 319 522 :M f0_12 sf .191 .019(> = )J f4_12 sf .172(E)A f4_10 sf 0 2 rm (i)S 0 -2 rm 349 522 :M f0_12 sf .238 .024(. If there is an acyclic)J 95 541 :M .479 .048(directed path from )J 189 541 :M f4_12 sf .496(A)A f0_12 sf .451 .045( to )J 213 541 :M f4_12 sf .518(B)A f0_12 sf .47 .047( or )J f4_12 sf .518(B)A f0_12 sf .41 .041( = )J 259 541 :M f4_12 sf .305(A)A f0_12 sf .426 .043( then )J f4_12 sf .305(A)A f0_12 sf .318 .032( is an )J f2_12 sf .225(ancestor)A f0_12 sf .289 .029( of )J 392 541 :M f4_12 sf .373(B)A f0_12 sf .515 .051(, and )J f4_12 sf .373(B)A f0_12 sf .409 .041( is a)J 95 559 :M f2_12 sf .158(descendant)A f0_12 sf .183 .018( of )J f4_12 sf .202(A)A f0_12 sf .39 .039(. A directed graph is )J 281 559 :M f2_12 sf (acyclic)S 316 559 :M f0_12 sf .504 .05( if and only if it contains no)J 95 577 :M (directed cyclic paths.)S 197 574 :M f0_8 sf (2)S 95 610 :M f0_12 sf ( )S 95 607.48 -.48 .48 455.48 607 .48 95 607 @a 95 622 :M f0_10 sf (for helpful conversations.)S 95 630 :M f0_8 sf (2)S 99 633 :M f0_10 sf .449 .045(An undirected path is often defined as a sequence of vertices rather than a sequence of)J endp %%Page: 4 4 %%BeginPageSetup initializepage (peter; page: 4 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 539 26 :M f0_12 sf (4)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .431 .043(A )J 108 104 :M f2_12 sf .334 .033(directed acyclic graph)J f0_12 sf .19 .019( \(DAG\) )J 263 104 :M f4_12 sf (G)S 272 104 :M f0_12 sf .313 .031( with a set of vertices )J f2_12 sf (V)S 389 104 :M f0_12 sf .339 .034( can be given)J 95 122 :M .567 .057(two distinct interpretations. On the one hand, such graphs can be used to)J 95 140 :M .661 .066(represent causal relations between variables, where an edge from )J f4_12 sf .253(A)A f0_12 sf .22 .022( to )J f4_12 sf (B)S 95 158 :M f0_12 sf .134 .013(in )J f4_12 sf (G)S 116 158 :M f0_12 sf .216 .022( means that )J 175 158 :M f4_12 sf .113(A)A f0_12 sf .179 .018( is a direct cause of )J f4_12 sf .113(B)A f0_12 sf .195 .019( relative to )J f2_12 sf (V)S 350 158 :M f0_12 sf .078 .008(. A )J f2_12 sf .333 .033(causal graph)J 434 158 :M f0_12 sf .247 .025( is a)J 95 176 :M .104 .01(DAG given such an interpretation. On the other hand, a DAG with a set of)J 95 194 :M 1.615 .161(vertices )J f2_12 sf (V)S 148 194 :M f0_12 sf 2.326 .233( can also represent a set of probability measures over )J f2_12 sf (V)S 451 194 :M f0_12 sf (.)S 95 212 :M -.001(Following the terminology of Lauritzen et. al. \(1990\) say that a probability)A 95 230 :M 1.362 .136(measure over a set of variables )J f2_12 sf (V)S 268 230 :M f0_12 sf 1.506 .151( satisfies the )J 337 230 :M f2_12 sf 1.104 .11(local directed Markov)J 95 248 :M .072(property)A f0_12 sf .142 .014( for a DAG )J f4_12 sf (G)S 207 248 :M f0_12 sf .241 .024( with vertices )J 276 248 :M f2_12 sf (V)S 285 248 :M f0_12 sf .26 .026( if and only if for every )J 403 248 :M f4_12 sf (W)S 413 248 :M f0_12 sf .188 .019( in )J f2_12 sf (V)S 438 248 :M f0_12 sf .11 .011(, )J f4_12 sf (W)S 95 270 :M f0_12 sf 4.75 .475(is independent of )J 204 270 :M f2_12 sf (V)S 213 270 :M f0_12 sf <5C28>S f2_12 sf (Descendants)S 284 270 :M f0_12 sf <28>S 288 270 :M f4_12 sf (W)S 298 270 :M f0_12 sf (,)S f4_12 sf (G)S 310 270 :M f0_12 sf 4.515 .452(\) )J f1_12 sf 7.139A f0_12 sf 2.324 .232( )J 344 270 :M f2_12 sf (Parents)S f0_12 sf <28>S 387 270 :M f4_12 sf (W)S 397 270 :M f0_12 sf (,)S f4_12 sf (G)S 409 270 :M f0_12 sf 4.377 .438(\)\) given)J 95 292 :M f2_12 sf (Parents)S f0_12 sf <28>S 138 292 :M f4_12 sf (W)S 148 292 :M f0_12 sf (,)S f4_12 sf (G)S 160 292 :M f0_12 sf .407 .041(\), where )J f2_12 sf .157(Parents)A f0_12 sf <28>S 247 292 :M f4_12 sf (W)S 257 292 :M f0_12 sf (,)S f4_12 sf (G)S 269 292 :M f0_12 sf .87 .087(\) is the set of parents of )J 393 292 :M f4_12 sf (W)S 403 292 :M f0_12 sf 1.038 .104( in )J 421 292 :M f4_12 sf (G)S 430 292 :M f0_12 sf .853 .085(, and)J 95 310 :M f2_12 sf (Descendants)S 159 310 :M f0_12 sf <28>S 163 310 :M f4_12 sf (W)S 173 310 :M f0_12 sf (,)S f4_12 sf (G)S 185 310 :M f0_12 sf 2.155 .215(\) is the set of descendants of )J f4_12 sf (W)S 357 310 :M f0_12 sf 1.775 .177( in )J f4_12 sf (G)S 388 310 :M f0_12 sf 2.703 .27(. A DAG )J 445 310 :M f4_12 sf (G)S 95 328 :M f2_12 sf .038(represents)A f0_12 sf .132 .013( the set of probability measures which satisfy the local directed)J 95 346 :M .54 .054(Markov property for )J 200 346 :M f4_12 sf (G)S 209 346 :M f0_12 sf .579 .058(. The use of DAGs to simultaneously represent a)J 95 364 :M .514 .051(set of causal hypotheses and a family of probability distributions extends)J 95 382 :M .702 .07(back to the path diagrams introduced by Sewell Wright \(1934\). Variants)J 95 400 :M 2.405 .24(of DAG models were introduced in the 1980's in Wermuth \(1980\),)J 95 418 :M .198 .02(Wermuth and Lauritzen \(1983\), Kiiveri, Speed, and Carlin \(1984\), Kiiveri)J 95 436 :M (and Speed \(1982\), and Pearl \(1988\). )S 272 433 :M f0_8 sf (3)S 95 466 :M f0_12 sf .959 .096(Lauritzen et. al. also define a )J 246 466 :M f2_12 sf .819 .082(global directed Markov property )J 424 466 :M f0_12 sf .909 .091(that is)J 95 484 :M 2.206 .221(equivalent to the local directed Markov property for DAGs. Several)J 95 522 :M ( )S 95 519.48 -.48 .48 455.48 519 .48 95 519 @a 95 534 :M f0_10 sf .231 .023(edges. The two definitions are essentially equivalent for acyclic directed graphs, because)J 95 545 :M .714 .071(a pair of vertices can be identified with a unique edge in the graph. However, a cyclic)J 95 556 :M .763 .076(graph may contain more than one edge between a pair of vertices. In that case it is no)J 95 567 :M (longer possible to identify a pair of vertices with a unique edge.)S 95 575 :M f0_8 sf (3)S 99 578 :M f0_10 sf .77 .077(It is often the case that some further restrictions are placed on the set of distributions)J 95 589 :M .704 .07(represented by a DAG. For example, one could also require the Minimality Condition,)J 95 600 :M .981 .098(i.e. that for any distribution )J 215 600 :M f4_10 sf .413(P)A f0_10 sf .955 .096( represented by )J 290 600 :M f4_10 sf .928(G)A f0_10 sf .584 .058(, )J 304 600 :M f4_10 sf .405(P)A f0_10 sf .949 .095( does not satisfy the local directed)J 95 611 :M .112 .011(Markov Condition for any proper subgraph of )J f4_10 sf (G)S f0_10 sf .112 .011(. This condition, and others are discussed)J 95 622 :M 1.091 .109(in Pearl\(1988\) and Spirtes, Glymour, and Scheines\(1993\). We will not consider such)J 95 633 :M (further restrictions here.)S endp %%Page: 5 5 %%BeginPageSetup initializepage (peter; page: 5 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 539 26 :M f0_12 sf (5)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .676 .068(preliminary notions are required. Let )J 281 104 :M f2_12 sf (An)S f0_12 sf <28>S 300 104 :M f2_12 sf (X)S 309 104 :M f0_12 sf (,)S f4_12 sf (G)S 321 104 :M f0_12 sf .811 .081(\) be the set of ancestors of)J 95 122 :M .939 .094(members of )J f2_12 sf (X)S 167 122 :M f0_12 sf .887 .089( in )J f4_12 sf (G)S 194 122 :M f0_12 sf 1.332 .133(. Let )J 223 122 :M f4_12 sf (G)S 232 122 :M f0_12 sf <28>S 236 122 :M f2_12 sf (X)S 245 122 :M f0_12 sf 1.078 .108(\) be the subgraph of )J f4_12 sf (G)S 361 122 :M f0_12 sf 1.041 .104( that contains only)J 95 140 :M .121 .012(vertices in )J f2_12 sf (X)S 157 140 :M f0_12 sf .163 .016(, with an edge from )J 255 140 :M f4_12 sf .146(A)A f0_12 sf .132 .013( to )J 278 140 :M f4_12 sf .105(B)A f0_12 sf .092 .009( in )J f2_12 sf (X)S 309 140 :M f0_12 sf .162 .016( if and only if there is an edge)J 95 161 :M .226 .023(from )J f4_12 sf .095(A)A f0_12 sf .082 .008( to )J f4_12 sf .095(B)A f0_12 sf .086 .009( in )J 166 161 :M f4_12 sf (G)S 175 161 :M f0_12 sf .063 .006(. )J f4_12 sf (G)S 190 158 :M f4_10 sf .05(M)A f0_12 sf 0 3 rm ( )S 0 -3 rm f2_12 sf 0 3 rm .033(moralizes)A 0 -3 rm f0_12 sf 0 3 rm .087 .009( a directed graph )J 0 -3 rm f4_12 sf 0 3 rm (G)S 0 -3 rm 343 161 :M f0_12 sf .124 .012( if and only if )J f4_12 sf (G)S 421 158 :M f4_10 sf .091(M)A f0_12 sf 0 3 rm .111 .011( is an)J 0 -3 rm 95 179 :M .269 .027(undirected graph with the same vertices as )J f4_12 sf (G)S 314 179 :M f0_12 sf .323 .032(, and a pair of vertices )J 427 179 :M f4_12 sf .12(X)A f0_12 sf .255 .026( and)J 95 200 :M f4_12 sf (Y)S 102 200 :M f0_12 sf .815 .081( are adjacent in )J f4_12 sf (G)S 192 197 :M f4_10 sf .604(M)A f0_12 sf 0 3 rm .873 .087( if and only if either )J 0 -3 rm 307 200 :M f4_12 sf .619(X)A f0_12 sf .82 .082( and )J 340 200 :M f4_12 sf (Y)S 347 200 :M f0_12 sf .815 .081( are adjacent in )J f4_12 sf (G)S 437 200 :M f0_12 sf .995 .1(, or)J 95 218 :M .381 .038(they have a common child in )J f4_12 sf (G)S 249 218 :M f0_12 sf .368 .037(. In an undirected graph )J f4_12 sf (G)S 378 218 :M f0_12 sf .539 .054(, )J 385 218 :M f2_12 sf (X)S 394 218 :M f0_12 sf .37 .037( is separated)J 95 236 :M .139 .014(from )J f2_12 sf (Y)S 130 236 :M f0_12 sf .213 .021( given )J 164 236 :M f2_12 sf .085(Z)A f0_12 sf .178 .018( if and only if every undirected path between a member of)J 95 254 :M f2_12 sf (X)S 104 254 :M f0_12 sf .296 .03( and a member of )J 193 254 :M f2_12 sf (Y)S 202 254 :M f0_12 sf .275 .028( contains a member of )J 314 254 :M f2_12 sf .25(Z)A f0_12 sf .221 .022(. If )J 340 254 :M f2_12 sf (X)S 349 254 :M f0_12 sf .133 .013(, )J f2_12 sf (Y)S 364 254 :M f0_12 sf .166 .017( and )J f2_12 sf .142(Z)A f0_12 sf .315 .031( are disjoint)J 95 272 :M .277 .028(sets of variables, )J f2_12 sf (X)S 188 272 :M f0_12 sf .292 .029( and )J f2_12 sf (Y)S 221 272 :M f0_12 sf .417 .042( are )J 243 272 :M f2_12 sf (d-separated)S 304 272 :M f0_12 sf .385 .038( given )J 338 272 :M f2_12 sf .193(Z)A f0_12 sf .329 .033( in a directed graph )J 445 272 :M f4_12 sf (G)S 95 294 :M f0_12 sf .228 .023(just when )J f2_12 sf (X)S 154 294 :M f0_12 sf .235 .023( and )J f2_12 sf (Y)S 187 294 :M f0_12 sf .269 .027( are separated given )J 288 294 :M f2_12 sf .277(Z)A f0_12 sf .23 .023( in )J 311 294 :M f4_12 sf (G)S 320 291 :M f4_10 sf (M)S f0_12 sf 0 3 rm <28>S 0 -3 rm 332 294 :M f2_12 sf (An)S f0_12 sf <28>S 351 294 :M f2_12 sf (X)S 360 294 :M f1_12 sf .366 .037<20C8>J 373 294 :M f0_12 sf .094 .009( )J f2_12 sf (Y)S 385 294 :M f0_12 sf .403 .04( )J 389 294 :M f1_12 sf S f0_12 sf S f2_12 sf (Z)S f0_12 sf (,)S f4_12 sf (G)S 421 294 :M f0_12 sf .252 .025(\)\). The)J 95 316 :M .148 .015(relation defined here was stated in Lauritzen, et. al. \(1990\). "d-separation")J 95 334 :M .625 .063(is a graphical relation introduced by Pearl \(1986\). Since Lauritzen et. al.)J 95 352 :M 1.775 .178(\(1990\) proved that their graphical relation is equivalent to Pearl's for)J 95 370 :M .415 .041(acyclic graphs, and the proof is readily extended to the cyclic case, I will)J 95 388 :M 1.214 .121(also use "d-separation" to refer to the graphical relation just described.)J 95 406 :M 1.461 .146(Now the definition: A probability measure over )J f2_12 sf (V)S 353 406 :M f0_12 sf 1.727 .173( satisfies the )J 423 406 :M f2_12 sf (global)S 95 424 :M 1.38 .138(directed Markov property)J 236 424 :M f0_12 sf 2.343 .234( for DAG )J 294 424 :M f4_12 sf (G)S 303 424 :M f0_12 sf 2.119 .212( if and only if for any three)J 95 442 :M .767 .077(disjoint sets of variables )J f2_12 sf (X)S 228 442 :M f0_12 sf 1.253 .125(, )J 236 442 :M f2_12 sf (Y)S 245 442 :M f0_12 sf 1.102 .11(, and )J 273 442 :M f2_12 sf .489(Z)A f0_12 sf .868 .087( included in )J 345 442 :M f2_12 sf (V)S 354 442 :M f0_12 sf .72 .072(, if )J f2_12 sf (X)S 382 442 :M f0_12 sf .811 .081( is d-separated)J 95 460 :M 1.239 .124(from )J 122 460 :M f2_12 sf (Y)S 131 460 :M f0_12 sf 1.335 .133( given )J 168 460 :M f2_12 sf .6(Z)A f0_12 sf .823 .082(, then )J f2_12 sf (X)S 218 460 :M f0_12 sf 1.196 .12( is independent of )J 314 460 :M f2_12 sf (Y)S 323 460 :M f0_12 sf 1.335 .133( given )J 360 460 :M f2_12 sf .467(Z)A f0_12 sf .987 .099(. Lauritzen et. al.)J 95 478 :M 1.4 .14(\(1990\) shows that the global and local directed Markov properties are)J 95 496 :M .748 .075(equivalent in DAGs, even when the probability distributions represented)J 95 514 :M 1.089 .109(have no density function. In section 2, I show that the local and global)J 95 532 :M (directed Markov properties are not equivalent for cyclic directed graphs.)S 95 562 :M 2.034 .203(The following lemmas relate the global directed Markov property to)J 95 580 :M .629 .063(factorizations of a density function. Denote a density function over )J f2_12 sf (V)S 439 580 :M f0_12 sf .909 .091( by)J 95 598 :M f4_12 sf (f)S f0_12 sf <28>S 102 598 :M f2_12 sf (V)S 111 598 :M f0_12 sf .806 .081(\), where for any subset )J f2_12 sf (X)S 239 598 :M f0_12 sf 1.117 .112( of )J 258 598 :M f2_12 sf (V)S 267 598 :M f0_12 sf .482 .048(, )J f4_12 sf .321(f)A f0_12 sf <28>S 281 598 :M f2_12 sf (X)S 290 598 :M f0_12 sf .856 .086(\) denotes the marginal of )J 420 598 :M f4_12 sf (f)S f0_12 sf <28>S 427 598 :M f2_12 sf (V)S 436 598 :M f0_12 sf .917 .092(\). If)J 95 616 :M f4_12 sf (f)S f0_12 sf <28>S 102 616 :M f2_12 sf (V)S 111 616 :M f0_12 sf 2.457 .246(\) is the density function for a probability measure over a set of)J endp %%Page: 6 6 %%BeginPageSetup initializepage (peter; page: 6 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 539 26 :M f0_12 sf (6)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .925 .092(variables )J 142 104 :M f2_12 sf (V)S 151 104 :M f0_12 sf 1.318 .132(, say that )J 203 104 :M f4_12 sf (f)S f0_12 sf <28>S 210 104 :M f2_12 sf (V)S 219 104 :M f0_12 sf .273 .027(\) )J f2_12 sf 1.143 .114(factors according to directed graph)J 415 104 :M f0_12 sf 1.757 .176( )J 420 104 :M f4_12 sf (G)S 429 104 :M f0_12 sf 1.255 .126( with)J 95 122 :M (vertices )S f2_12 sf (V)S 144 122 :M f0_12 sf ( if and only if for every subset )S 292 122 :M f2_12 sf (X)S 301 122 :M f0_12 sf ( of )S 317 122 :M f2_12 sf (V)S 326 122 :M f0_12 sf (,)S 168 137 214 30 rC 382 167 :M psb currentpoint pse 168 137 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6848 div 960 3 -1 roll exch div scale currentpoint translate 64 40 translate 56 344 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-Italic f1 (f) show 229 344 moveto 384 /Times-Roman f1 (\() show 369 344 moveto 384 /Times-Bold f1 (An) show 860 344 moveto 384 /Times-Roman f1 (\() show 1000 344 moveto 384 /Times-Bold f1 (X) show 1274 344 moveto 384 /Times-Roman f1 (,) show 1404 344 moveto 384 /Times-Italic f1 (G) show 1694 344 moveto 384 /Times-Roman f1 (\)) show 1824 344 moveto (\)) show 2057 344 moveto 384 /Symbol f1 (=) show 3473 344 moveto 384 /Times-Italic f1 (g) show 3649 440 moveto 320 ns (V) show 3905 344 moveto 384 /Times-Roman f1 (\() show 4037 344 moveto 384 /Times-Italic f1 (V) show 4298 344 moveto 384 /Times-Roman f1 (,) show 4430 344 moveto 384 /Times-Bold f1 (Parents) show 5687 344 moveto 384 /Times-Roman f1 (\() show 5819 344 moveto 384 /Times-Italic f1 (V) show 6080 344 moveto 384 /Times-Roman f1 (,) show 6210 344 moveto 384 /Times-Italic f1 (G) show 6500 344 moveto 384 /Times-Roman f1 (\)) show 2377 810 moveto 320 /Times-Italic f1 (V) show 2605 810 moveto 320 /Symbol f1 (\316) show 2791 810 moveto 320 /Times-Bold f1 (An) show 3206 810 moveto 320 /Times-Roman f1 (\() show 3328 810 moveto 320 /Times-Bold f1 (X) show 3562 810 moveto 320 /Times-Roman f1 (,) show 3649 810 moveto 320 /Times-Italic f1 (G) show 3896 810 moveto 320 /Times-Roman f1 (\)) show 2950 432 moveto 576 /Symbol f1 (\325) show 6630 344 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 245 188 :M f2_12 sf ( Equation 1)S 95 218 :M f0_12 sf .771 .077(where )J f4_12 sf .229(g)A f4_10 sf 0 2 rm .233(V)A 0 -2 rm f0_12 sf .712 .071( is a non-negative function. The following result was proved in)J 95 237 :M (Lauritzen et. al. \(1990\).)S 95 267 :M f2_12 sf .336 .034(Lemma 1:)J 148 267 :M f0_12 sf .294 .029( If )J f2_12 sf (V)S 172 267 :M f0_12 sf .366 .037( is a set of random variables with a probability measure )J f4_12 sf (P)S 95 285 :M f0_12 sf .528 .053(that has a density function )J f4_12 sf .107(f)A f0_12 sf <28>S 235 285 :M f2_12 sf (V)S 244 285 :M f0_12 sf .645 .065(\), then )J 280 285 :M f4_12 sf (f)S f0_12 sf <28>S 287 285 :M f2_12 sf (V)S 296 285 :M f0_12 sf .531 .053(\) factors according to DAG )J f4_12 sf (G)S 443 285 :M f0_12 sf .726 .073( if)J 95 303 :M (and only if )S 150 303 :M f4_12 sf (P)S f0_12 sf ( satisfies the global directed Markov property for )S 396 303 :M f4_12 sf (G)S 405 303 :M f0_12 sf (.)S 95 333 :M .229 .023(As in the case of acyclic graphs, the existence of a factorization according)J 95 351 :M .338 .034(to a cyclic directed graph )J f4_12 sf (G)S 231 351 :M f0_12 sf .351 .035( does entail that a measure satisfies the global)J 95 369 :M 1.636 .164(directed Markov property for )J f4_12 sf (G)S 259 369 :M f0_12 sf 1.958 .196(. The proof given in Lauritzen et. al.)J 95 387 :M 1.705 .17(\(1990\) for the acyclic case carries over essentially unchanged for the)J 95 405 :M (cyclic case.)S 95 435 :M f2_12 sf .336 .034(Lemma 2:)J 148 435 :M f0_12 sf .294 .029( If )J f2_12 sf (V)S 172 435 :M f0_12 sf .366 .037( is a set of random variables with a probability measure )J f4_12 sf (P)S 95 453 :M f0_12 sf 1.484 .148(that has a density function )J f4_12 sf .299(f)A f0_12 sf <28>S 243 453 :M f2_12 sf (V)S 252 453 :M f0_12 sf 1.484 .148(\) and )J f4_12 sf .471(f)A f0_12 sf <28>S 291 453 :M f2_12 sf (V)S 300 453 :M f0_12 sf 1.398 .14(\) factors according to directed)J 95 471 :M .766 .077(\(cyclic or acyclic\) graph )J 220 471 :M f4_12 sf (G)S 229 471 :M f0_12 sf .957 .096(, then )J 262 471 :M f4_12 sf .295(P)A f0_12 sf .739 .074( satisfies the global directed Markov)J 95 489 :M (property for )S 156 489 :M f4_12 sf (G)S 165 489 :M f0_12 sf (.)S 95 519 :M .43 .043(However, unlike the case of acyclic graphs, if a probability measure over)J 95 537 :M .218 .022(a set of variable )J f2_12 sf (V)S 184 537 :M f0_12 sf .206 .021( satisfies the global directed Markov property for cyclic)J 95 555 :M 1.308 .131(graph )J f4_12 sf (G)S 137 555 :M f0_12 sf 1.738 .174( and has a density function )J f4_12 sf .388(f)A f0_12 sf <28>S 291 555 :M f2_12 sf (V)S 300 555 :M f0_12 sf 1.748 .175(\), it does not follow that )J f4_12 sf .429(f)A f0_12 sf <28>S 441 555 :M f2_12 sf (V)S 450 555 :M f0_12 sf <29>S 95 573 :M (factors according to )S f4_12 sf (G)S 202 573 :M f0_12 sf (.)S endp %%Page: 7 7 %%BeginPageSetup initializepage (peter; page: 7 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 539 26 :M f0_12 sf (7)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf 1.49 .149(The following weaker result relating factorization of densities and the)J 95 122 :M 1.361 .136(global directed Markov property does hold for both cyclic and acyclic)J 95 140 :M (directed graphs.)S 95 170 :M f2_12 sf .336 .034(Lemma 3:)J 148 170 :M f0_12 sf .294 .029( If )J f2_12 sf (V)S 172 170 :M f0_12 sf .366 .037( is a set of random variables with a probability measure )J f4_12 sf (P)S 95 188 :M f0_12 sf .101 .01(that has a positive density function )J 266 188 :M f4_12 sf (f)S f0_12 sf <28>S 273 188 :M f2_12 sf (V)S 282 188 :M f0_12 sf .063 .006(\), and )J f4_12 sf (P)S f0_12 sf .11 .011( satisfies the global directed)J 95 206 :M .034 .003(Markov property for directed \(cyclic or acyclic\) graph )J 359 206 :M f4_12 sf (G)S 368 206 :M f0_12 sf .044 .004(, then )J 398 206 :M f4_12 sf (f)S f0_12 sf <28>S 405 206 :M f2_12 sf (V)S 414 206 :M f0_12 sf .032 .003(\) factors)J 95 224 :M (according to )S 158 224 :M f4_12 sf (G)S 167 224 :M f0_12 sf (.)S 95 254 :M f2_12 sf (3. Non-recursive Linear Structural Equation Models)S 95 284 :M f0_12 sf .357 .036(The problem considered in this section is to investigate the generalization)J 95 302 :M 2.152 .215(of the Markov properties to linear, non-recursive structural equation)J 95 320 :M .338 .034(models. First we must relate the social scientific terminology to graphical)J 95 338 :M (representations, and clarify the questions.)S 95 368 :M 1.066 .107(Linear structural equation models \(which, following the terminology of)J 95 386 :M .336 .034(Bollen \(1989\), will be referred to as linear SEMs\) can also be represented)J 95 404 :M 2.067 .207(as directed graph models. In a linear SEM the random variables are)J 95 422 :M 1.393 .139(divided into two disjoint sets, the error terms and the non-error terms.)J 95 440 :M .332 .033(Corresponding to each non-error random variable )J f4_12 sf .109(V)A f0_12 sf .22 .022( is a unique error term)J 95 462 :M f5_12 sf .281(e)A f4_10 sf 0 2 rm .326(V)A 0 -2 rm f0_12 sf .846 .085(. A linear SEM contains a set of linear equations in which each non-)J 95 484 :M .566 .057(error random variable )J 206 484 :M f4_12 sf .266(V)A f0_12 sf .6 .06( is written as a linear function of other non-error)J 95 506 :M .524 .052(random variables and )J 204 506 :M f5_12 sf .151(e)A f4_10 sf 0 2 rm .176(V)A 0 -2 rm f0_12 sf .516 .052(. A linear SEM also specifies a joint distribution)J 95 528 :M 1.642 .164(over the error terms. So, for example, the following is a linear SEM,)J 95 550 :M 1.058 .106(where )J 129 550 :M f4_12 sf .46(a)A f0_12 sf .715 .071( and )J f4_12 sf .46(b)A f0_12 sf 1.159 .116( are real constants, )J f5_12 sf .404(e)A f4_10 sf 0 2 rm .468(X)A 0 -2 rm f0_12 sf .418 .042( ,)J 284 550 :M f5_12 sf (e)S f4_10 sf 0 2 rm (Y)S 0 -2 rm 295 550 :M f0_12 sf .865 .086(, and )J f5_12 sf .45(e)A f4_10 sf 0 2 rm (Z)S 0 -2 rm 335 550 :M f0_12 sf .898 .09( are jointly independent)J 95 572 :M ("error terms", and )S f4_12 sf (X)S f0_12 sf (, )S f4_12 sf (Y)S 204 572 :M f0_12 sf (, )S f4_12 sf (Z)S 217 572 :M f0_12 sf (, are random variables:)S 244 594 :M f4_12 sf (X)S f0_12 sf ( = )S 264 594 :M f4_12 sf (a)S f0_12 sf ( )S f4_12 sf (Y)S 280 594 :M f0_12 sf ( + )S 293 594 :M f5_12 sf (e)S f4_10 sf 0 2 rm (X)S 0 -2 rm 243 611 :M f4_12 sf (Y)S 250 611 :M f0_12 sf ( = )S 263 611 :M f4_12 sf (b)S f0_12 sf ( )S f4_12 sf (Z)S 282 611 :M f0_12 sf ( + )S 295 611 :M f5_12 sf (e)S f4_10 sf 0 2 rm (Y)S 0 -2 rm 259 628 :M f4_12 sf (Z)S 266 628 :M f0_12 sf ( = )S 279 628 :M f5_12 sf (e)S f4_10 sf 0 2 rm (Z)S 0 -2 rm endp %%Page: 8 8 %%BeginPageSetup initializepage (peter; page: 8 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 539 26 :M f0_12 sf (8)S gR gS 0 0 552 730 rC 130 108 :M f5_12 sf (e)S f4_10 sf 0 2 rm (X)S 0 -2 rm f0_12 sf ( , )S f5_12 sf (e)S f4_10 sf 0 2 rm (Y)S 0 -2 rm 161 108 :M f0_12 sf ( , )S f5_12 sf (e)S f4_10 sf 0 2 rm (Z)S 0 -2 rm 181 108 :M f0_12 sf ( are jointly independent and normally distributed)S 247 127 :M f2_12 sf (Equation 2)S 95 157 :M f0_12 sf 1.294 .129(The directed graph of a linear SEM with uncorrelated errors is written)J 95 175 :M 2.069 .207(with the convention that an edge does not appear if and only if the)J 95 193 :M .835 .083(corresponding entry in the coefficient matrix is zero; the graph does not)J 95 211 :M 1.438 .144(contain the error terms. )J 220 211 :M 1.508 .151(Figure 1 is the DAG that represents the SEM)J 95 229 :M .504 .05(shown above. A linear SEM is )J 250 229 :M f2_12 sf .137(recursive)A f0_12 sf .383 .038( if and only if its directed graph)J 95 247 :M (is acyclic.)S 213 268 122 18 rC 217 280 :M f4_12 sf .104 .01(Z Y X)J 10 156 204 269 275 @k 232 276 -1 1 261 275 1 232 275 @a 10 156 204 319 275 @k 282 276 -1 1 311 275 1 282 275 @a gR gS 0 0 552 730 rC 179 307 :M f2_12 sf (Figure )S 216 307 :M (1: Example of Recursive SEM)S 95 337 :M f0_12 sf 1.379 .138(Initially I will consider only linear SEMs in which the error terms are)J 95 355 :M .271 .027(jointly independent, but we will see that in the linear case in an important)J 95 373 :M 1.634 .163(sense nothing is lost by this restriction: a linear SEM with dependent)J 95 391 :M 1.225 .122(errors generates the same restrictions on the covariance matrix as does)J 95 409 :M 1.341 .134(some linear SEM with extra variables and independent errors. Further,)J 95 427 :M 1.38 .138(such an SEM with extra variables can always be found with the same)J 95 445 :M (graphical structure on the original variables as obtain in the original graph.)S 95 475 :M .837 .084(A linear SEM containing disjoint sets of variables )J 349 475 :M f2_12 sf (X)S 358 475 :M f0_12 sf .464 .046(, )J f2_12 sf (Y)S 374 475 :M f0_12 sf .539 .054(, and )J f2_12 sf .426(Z)A f0_12 sf .145 .015( )J f2_12 sf .299(linearly)A 95 493 :M .04(entails)A f0_12 sf .081 .008( that )J 153 493 :M f2_12 sf (X)S 162 493 :M f0_12 sf .168 .017( is independent of )J 252 493 :M f2_12 sf (Y)S 261 493 :M f0_12 sf .189 .019( given )J f2_12 sf .125(Z)A f0_12 sf .164 .016( if and only if )J 372 493 :M f2_12 sf (X)S 381 493 :M f0_12 sf .143 .014( is independent)J 95 511 :M 1.828 .183(of )J 111 511 :M f2_12 sf (Y)S 120 511 :M f0_12 sf 1.037 .104( given )J f2_12 sf .686(Z)A f0_12 sf 1.445 .145( for all values of the non-zero linear coefficients and all)J 95 529 :M 2.117 .212(distributions of the exogenous variables in which they have positive)J 95 551 :M (variances. Let )S f5_12 sf (r)S 172 553 :M f4_10 sf (XY)S 184 553 :M f0_10 sf (.)S 187 553 :M f2_10 sf (Z)S 194 551 :M f0_12 sf ( be the partial correlation of )S 331 551 :M f4_12 sf (X)S f0_12 sf ( and )S f4_12 sf (Y)S 368 551 :M f0_12 sf ( given )S 401 551 :M f2_12 sf (Z)S f0_12 sf (. A linear)S 95 577 :M 1.36 .136(SEM containing )J 181 577 :M f4_12 sf .847(X)A f0_12 sf .578 .058(, )J f4_12 sf (Y)S 203 577 :M f0_12 sf 1.85 .185(, and )J 234 577 :M f2_12 sf .878(Z)A f0_12 sf 1.501 .15(, where )J f4_12 sf .804(X)A f0_12 sf .299 .03( )J f1_12 sf S 303 577 :M f0_12 sf S f4_12 sf (Y)S 313 577 :M f0_12 sf 1.927 .193( and )J 341 577 :M f4_12 sf 1.106(X)A f0_12 sf 1.466 .147( and )J 376 577 :M f4_12 sf 2.102 .21(Y )J 388 577 :M f0_12 sf 1.378 .138(are not in )J f2_12 sf .936(Z)A f0_12 sf (,)S 95 603 :M f2_12 sf .226 .023(linearly entails)J 172 603 :M f0_12 sf .341 .034( that )J f5_12 sf (r)S 204 605 :M f4_10 sf (XY)S 216 605 :M f0_10 sf (.)S 219 605 :M f2_10 sf (Z)S 226 603 :M f0_12 sf .406 .041( = 0 if and only )J f5_12 sf (r)S 313 605 :M f4_10 sf (XY)S 325 605 :M f0_10 sf (.)S 328 605 :M f2_10 sf (Z)S 335 603 :M f0_12 sf .417 .042( = 0 for all values of the)J 95 625 :M 3.027 .303(non-zero linear coefficients and all distributions of the exogenous)J endp %%Page: 9 9 %%BeginPageSetup initializepage (peter; page: 9 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 539 26 :M f0_12 sf (9)S gR gS 0 0 552 730 rC 95 108 :M f0_12 sf 1.364 .136(variables in which they have positive variances and in which )J 412 108 :M f5_12 sf (r)S 419 110 :M f4_10 sf (XY)S 431 110 :M f0_10 sf (.)S 434 110 :M f2_10 sf (Z)S 441 108 :M f0_12 sf 1.818 .182( is)J 95 130 :M .79 .079(defined. It follows from Kiiveri and Speed \(1982\) that if the error terms)J 95 148 :M .087 .009(are jointly independent, then any distribution that forms a linear, recursive)J 95 166 :M .358 .036(SEM with a directed graph )J 230 166 :M f4_12 sf (G)S 239 166 :M f0_12 sf .31 .031( satisfies the local directed Markov property)J 95 184 :M 2.298 .23(for )J 115 184 :M f4_12 sf (G)S 124 184 :M f0_12 sf 1.922 .192(. One can therefore apply d-separation to the DAG in a linear,)J 95 202 :M .033 .003(recursive SEM to compute the conditional independencies and zero partial)J 95 220 :M 2.585 .259(correlations it linearly entails. The d-separation relation provides a)J 95 238 :M 2.444 .244(polynomial \(in the number of vertices\) time algorithm for deciding)J 95 256 :M 1.698 .17(whether a given vanishing partial correlation is linearly entailed by a)J 95 274 :M (DAG.)S 95 304 :M 2.973 .297(Linear non-recursive structural equation models \(linear SEMs\) are)J 95 322 :M 2.357 .236(commonly used in the econometrics literature to represent feedback)J 95 340 :M .572 .057(processes that have reached equilibrium.)J 294 337 :M f0_8 sf (4)S 298 340 :M f0_12 sf .753 .075( Corresponding to a set of non-)J 95 358 :M .045 .005(recursive linear equations is a cyclic graph, as the following example from)J 95 376 :M (Whittaker \(1990\) illustrates.)S 253 398 :M f4_12 sf (X)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf ( = )S 278 398 :M f5_12 sf (e)S f0_10 sf 0 2 rm (X1)S 0 -2 rm 253 415 :M f4_12 sf (X)S f0_10 sf 0 2 rm (2)S 0 -2 rm f0_12 sf ( = )S 278 415 :M f5_12 sf (e)S f0_10 sf 0 2 rm (X2)S 0 -2 rm 211 432 :M f4_12 sf (X)S f0_10 sf 0 2 rm (3)S 0 -2 rm f0_12 sf ( = )S 236 432 :M f5_12 sf (b)S 243 434 :M f0_10 sf (31)S f4_12 sf 0 -2 rm (X)S 0 2 rm f0_10 sf (1)S f0_12 sf 0 -2 rm ( + )S 0 2 rm 278 432 :M f5_12 sf (b)S 285 434 :M f0_10 sf (34)S f4_12 sf 0 -2 rm (X)S 0 2 rm f0_10 sf (4)S f0_12 sf 0 -2 rm ( + )S 0 2 rm 320 432 :M f5_12 sf (e)S f0_10 sf 0 2 rm (X3)S 0 -2 rm 211 449 :M f4_12 sf (X)S f0_10 sf 0 2 rm (4)S 0 -2 rm f0_12 sf ( = )S 236 449 :M f5_12 sf (b)S 243 451 :M f0_10 sf (42)S f4_12 sf 0 -2 rm (X)S 0 2 rm f0_10 sf (2)S f0_12 sf 0 -2 rm ( + )S 0 2 rm 278 449 :M f5_12 sf (b)S 285 451 :M f0_10 sf (43)S f4_12 sf 0 -2 rm (X)S 0 2 rm f0_10 sf (3)S f0_12 sf 0 -2 rm ( + )S 0 2 rm 320 449 :M f5_12 sf (e)S f0_10 sf 0 2 rm (X4)S 0 -2 rm 114 466 :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 247 485 :M f2_12 sf (Equation 3)S 95 577 :M f0_12 sf ( )S 95 574.48 -.48 .48 239.48 574 .48 95 574 @a 95 586 :M f0_8 sf (4)S 99 589 :M f0_10 sf 4.156 .416(Cox and Wermuth \(1993\), Wermuth and Lauritzen\(1990\) and \(indirectly\))J 95 600 :M 1.928 .193(Frydenberg\(1990\) consider a class of non-recursive linear models they call )J f4_10 sf .605(block)A 95 611 :M .213(recursive)A f0_10 sf .767 .077(. The block recursive models overlap the class of SEMs, but they are neither)J 95 622 :M 1.514 .151(properly included in that class, nor properly include it. Frydenberg \(1990\) presents)J 95 633 :M (necessary and sufficient conditions for the equivalence of two block recursive models.)S endp %%Page: 10 10 %%BeginPageSetup initializepage (peter; page: 10 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (10)S gR gS 212 95 125 80 rC 216 107 :M f4_12 sf (X)S 227 111 :M f0_10 sf (1)S 219 165 :M f4_12 sf (X)S 230 169 :M f0_10 sf (2)S 308 166 :M f4_12 sf (X)S 320 169 :M f0_10 sf (4)S 307 107 :M f4_12 sf (X)S 319 110 :M f0_10 sf (3)S 10 156 204 303 104 @k 236 105 -1 1 295 104 1 236 104 @a 10 156 204 302 163 @k 240 164 -1 1 294 163 1 240 163 @a 10 66 114 326 112 @k -1 -1 326 156 1 1 325 120 @b 10 -114 -66 310 155 @k -1 -1 310 148 1 1 309 115 @b gR gS 0 0 552 730 rC 168 190 :M f2_12 sf (Figure )S 205 190 :M (2: Example of Non-recursive SEM)S 95 220 :M f0_12 sf 1.036 .104(In DAGs the global directed Markov property entails the local directed)J 95 238 :M 2.232 .223(Markov property, because a variable )J 293 238 :M f4_12 sf .977(V)A f0_12 sf 2.307 .231( is d-separated from its non-)J 95 256 :M .484 .048(parental non-descendants given its parents. This is not always the case in)J 95 274 :M .11 .011(cyclic graphs. For example, in figure 4, )J f4_12 sf (X)S f0_10 sf 0 2 rm (4)S 0 -2 rm f0_12 sf .106 .011( is not d-separated from its non-)J 95 293 :M 1.465 .146(parental non-descendant )J 222 293 :M f4_12 sf 1.044(X)A f0_10 sf 0 2 rm .712(1)A 0 -2 rm f0_12 sf 2.044 .204( given its parents )J f4_12 sf 1.044(X)A f0_10 sf 0 2 rm .712(2)A 0 -2 rm f0_12 sf 1.385 .138( and )J 373 293 :M f4_12 sf .962(X)A f0_10 sf 0 2 rm .656(3)A 0 -2 rm .547 .055(, )J f0_12 sf 2.314 .231(so the local)J 95 312 :M (directed Markov property does not hold.)S 292 309 :M f0_8 sf (5)S 95 342 :M f2_12 sf .524 .052(Theorem 1:)J 156 342 :M f0_12 sf .744 .074( The probability measure )J f4_12 sf .328(P)A f0_12 sf .55 .055( of a linear SEM )J f4_12 sf (L)S 388 342 :M f0_12 sf .604 .06( \(recursive or)J 95 360 :M 1.293 .129(non-recursive\) with jointly independent error terms satisfies the global)J 95 378 :M .097 .01(directed Markov property for the directed \(cyclic or acyclic\) graph )J f4_12 sf (G)S 428 378 :M f0_12 sf .098 .01( of )J f4_12 sf (L)S 451 378 :M f0_12 sf (,)S 95 396 :M .385 .038(i.e. if )J f2_12 sf (X)S 133 396 :M f0_12 sf .641 .064(, )J 140 396 :M f2_12 sf (Y)S 149 396 :M f0_12 sf .329 .033(, and )J f2_12 sf .26(Z)A f0_12 sf .472 .047( are disjoint sets of variables in )J f4_12 sf (G)S 350 396 :M f0_12 sf .588 .059( and )J 375 396 :M f2_12 sf (X)S 384 396 :M f0_12 sf .415 .042( is d-separated)J 95 414 :M (from )S f2_12 sf (Y)S 130 414 :M f0_12 sf ( given )S 163 414 :M f2_12 sf (Z)S f0_12 sf ( in )S f4_12 sf (G)S 195 414 :M f0_12 sf (, then )S 225 414 :M f2_12 sf (X)S 234 414 :M f0_12 sf ( and )S f2_12 sf (Y)S 266 414 :M f0_12 sf ( are independent given )S 378 414 :M f2_12 sf (Z)S f0_12 sf ( in )S f4_12 sf (P)S f0_12 sf (.)S 411 411 :M f0_8 sf (6)S 95 474 :M f0_12 sf ( )S 95 471.48 -.48 .48 239.48 471 .48 95 471 @a 95 483 :M f0_8 sf (5)S 99 486 :M f0_10 sf .488 .049( Note that this use of cyclic directed graphs to represent feedback processes represents)J 95 497 :M 2.112 .211(an extension of the causal interpretation of directed graphs. The causal structure)J 95 508 :M .685 .068(corresponding to )J 168 508 :M .725 .072(Figure 2)J 203 508 :M .734 .073( is described by an infinite acyclic directed graph containing)J 95 519 :M 2.683 .268(each variable indexed by time. The cyclic graph can be viewed as a compact)J 95 530 :M .37 .037(representation of such a causal graph. I am indebted to C. Glymour for pointing out that)J 95 541 :M .471 .047(the local Markov condition fails in Whittaker's model. Indeed, there is )J f4_10 sf .151(no)A f0_10 sf .515 .051( acyclic graph)J 95 552 :M 3.187 .319(\(even with additional variables\) that linearly entails all and only conditional)J 95 563 :M .707 .071(independence relations linearly entailed by Figure 2)J 311 563 :M .737 .074(, although Thomas Richardson has)J 95 574 :M .041 .004(pointed out that the directed cyclic graph of Figure 2)J 306 574 :M .047 .005( is equivalent to one in which in the)J 95 585 :M .172 .017(edges from )J f4_10 sf .069(X)A f0_10 sf 0 2 rm .056(1)A 0 -2 rm .06 .006( to )J f4_10 sf .069(X)A f0_10 sf 0 2 rm .056(3)A 0 -2 rm .091 .009( and )J 197 585 :M f4_10 sf .056(X)A f0_10 sf 0 2 rm (2)S 0 -2 rm .049 .005( to )J f4_10 sf .056(X)A f0_10 sf 0 2 rm (4)S 0 -2 rm .138 .014( are replaced, respectively, by edges from )J f4_10 sf .056(X)A f0_10 sf 0 2 rm (1)S 0 -2 rm .051 .005( to )J 427 585 :M f4_10 sf (X)S f0_10 sf 0 2 rm (4)S 0 -2 rm .089 .009( and)J 95 598 :M (from )S 117 598 :M f4_10 sf (X)S f0_10 sf 0 2 rm (2)S 0 -2 rm ( to )S 141 598 :M f4_10 sf (X)S f0_10 sf 0 2 rm (3)S 0 -2 rm 95 608 :M f0_8 sf (6)S 99 611 :M f0_10 sf .44 .044( This theorem has been independently proved by Jan Koster of the Erasmus University)J 95 622 :M 1.341 .134(Rotterdam, in a paper which has not yet been published but has been submitted to)J 95 633 :M (Statistical Science.)S endp %%Page: 11 11 %%BeginPageSetup initializepage (peter; page: 11 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (11)S gR gS 0 0 552 730 rC 95 104 :M f2_12 sf .553 .055(Theorem 2:)J 156 104 :M f0_12 sf .795 .08( In a linear SEM )J 244 104 :M f4_12 sf (L)S 251 104 :M f0_12 sf .63 .063( with jointly independent error terms and)J 95 122 :M (directed \(cyclic or acyclic\) graph )S 257 122 :M f4_12 sf (G)S 266 122 :M f0_12 sf ( containing disjoint sets of variables )S 442 122 :M f2_12 sf (X)S 451 122 :M f0_12 sf (,)S 95 140 :M f2_12 sf (Y)S 104 140 :M f0_12 sf .272 .027( and )J f2_12 sf .233(Z)A f0_12 sf .19 .019(, if )J f2_12 sf (X)S 162 140 :M f0_12 sf .277 .028( is not d-separated from )J 281 140 :M f2_12 sf (Y)S 290 140 :M f0_12 sf .315 .032( given )J 324 140 :M f2_12 sf .159(Z)A f0_12 sf .204 .02( then )J f4_12 sf (L)S 366 140 :M f0_12 sf .283 .028( does not linearly)J 95 158 :M (entail that )S 146 158 :M f2_12 sf (X)S 155 158 :M f0_12 sf ( is independent of )S 244 158 :M f2_12 sf (Y)S 253 158 :M f0_12 sf ( given )S 286 158 :M f2_12 sf (Z)S f0_12 sf (.)S 95 188 :M .991 .099(Applying Theorems 1 and 2 to the directed graph in )J 362 188 :M .949 .095(Figure 2, only two)J 95 206 :M .775 .077(conditional independence relations are entailed: )J f4_12 sf .242(X)A f0_10 sf 0 2 rm .165(1)A 0 -2 rm f0_12 sf .505 .05( is independent of )J 439 206 :M f4_12 sf (X)S f0_10 sf 0 2 rm (2)S 0 -2 rm f0_12 sf (,)S 95 225 :M (and )S f4_12 sf (X)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf ( is independent of )S 216 225 :M f4_12 sf (X)S f0_10 sf 0 2 rm (2)S 0 -2 rm f0_12 sf ( given )S 261 225 :M f4_12 sf (X)S f0_10 sf 0 2 rm (3)S 0 -2 rm f0_12 sf ( and )S f4_12 sf (X)S f0_10 sf 0 2 rm (4)S 0 -2 rm f0_12 sf (. .)S 95 256 :M f2_12 sf .553 .055(Theorem 3:)J 156 256 :M f0_12 sf .795 .08( In a linear SEM )J 244 256 :M f4_12 sf (L)S 251 256 :M f0_12 sf .63 .063( with jointly independent error terms and)J 95 278 :M .687 .069(\(cyclic or acyclic\) directed graph )J 263 278 :M f4_12 sf (G)S 272 278 :M f0_12 sf .744 .074( containing )J 331 278 :M f4_12 sf .423(X)A f0_12 sf .288 .029(, )J f4_12 sf (Y)S 352 278 :M f0_12 sf .961 .096( and )J 378 278 :M f2_12 sf .438(Z)A f0_12 sf .749 .075(, where )J f4_12 sf .401(X)A f0_12 sf .149 .015( )J f1_12 sf S 444 278 :M f0_12 sf S f4_12 sf (Y)S 95 300 :M f0_12 sf .037 .004(and )J f2_12 sf (Z)S f0_12 sf .038 .004( does not contain )J 208 300 :M f4_12 sf (X)S f0_12 sf .026 .003( or )J f4_12 sf (Y)S 238 300 :M f0_12 sf (, )S f4_12 sf (X)S f0_12 sf .043 .004( is d-separated from )J 351 300 :M f4_12 sf (Y)S 358 300 :M f0_12 sf .045 .004( given )J 391 300 :M f2_12 sf (Z)S f0_12 sf .037 .004( if and only)J 95 322 :M (if )S f4_12 sf (L)S 112 322 :M f0_12 sf ( linearly entails that )S 210 322 :M f5_12 sf (r)S 217 324 :M f4_10 sf (XY)S 229 324 :M f0_10 sf (.)S 232 324 :M f2_10 sf (Z)S 239 322 :M f0_12 sf ( = 0.)S 95 356 :M .223 .022(As in the acyclic case, d-separation provides a polynomial time procedure)J 95 374 :M 1.206 .121(for deciding whether cyclic graphs entail a conditonal independence or)J 95 392 :M (vanishing partial correlation)S 95 422 :M .468 .047(Theorem 3 can be used to relax the restriction that the error terms in an a)J 95 444 :M .613 .061(linear SEM )J 155 444 :M f4_12 sf (L)S 162 444 :M f0_12 sf .608 .061( be jointly independent. If )J f5_12 sf .196(e)A f4_10 sf 0 2 rm .228(X)A 0 -2 rm f0_12 sf .348 .035( and )J f5_12 sf .196(e)A f4_10 sf 0 2 rm (Y)S 0 -2 rm 340 444 :M f0_12 sf .593 .059( are not independent in)J 95 466 :M .36 .036(linear SEM )J f4_12 sf (L)S 161 466 :M f0_12 sf .419 .042(, there is a linear SEM )J f4_12 sf (L)S 282 466 :M f0_12 sf .37 .037(' with independent error terms such)J 95 484 :M .848 .085(that the marginal distribution of )J f4_12 sf (L)S 265 484 :M f0_12 sf 1.017 .102(' over the variables in )J 379 484 :M f4_12 sf (L)S 386 484 :M f0_12 sf 1.061 .106( has the same)J 95 502 :M .357 .036(covariance matrix as )J f4_12 sf (L)S 206 502 :M f0_12 sf .41 .041(. Form the graph )J f4_12 sf (G)S 300 502 :M f0_12 sf .378 .038(' of )J f4_12 sf (L)S 326 502 :M f0_12 sf .408 .041(' from the graph )J f4_12 sf (G)S 417 502 :M f0_12 sf .38 .038( of )J f4_12 sf (L)S 441 502 :M f0_12 sf .538 .054( in)J 95 520 :M .396 .04(the following way. Add a latent variable )J 296 520 :M f4_12 sf (T)S 303 520 :M f0_12 sf .331 .033( to )J f4_12 sf (G)S 328 520 :M f0_12 sf .412 .041(, and add edges from )J f4_12 sf (T)S 441 520 :M f0_12 sf .518 .052( to)J 95 538 :M f4_12 sf (X)S f0_12 sf .03 .003( and )J f4_12 sf (Y)S 132 538 :M f0_12 sf .035 .004(. In )J f4_12 sf (L)S 158 538 :M f0_12 sf .039 .004(', modify the equation for )J 284 538 :M f4_12 sf (X)S f0_12 sf .038 .004( by making it a linear functions of)J 95 560 :M 1.021 .102(the parents of )J f4_12 sf .493(X)A f0_12 sf .913 .091( \(including )J f4_12 sf (T)S 245 560 :M f0_12 sf .871 .087(\) in )J f4_12 sf (G)S 276 560 :M f0_12 sf 1.075 .107(', and replace )J 347 560 :M f5_12 sf .327(e)A f4_10 sf 0 2 rm .379(X)A 0 -2 rm f0_12 sf .466 .047( by )J f5_12 sf .327(e)A f0_12 sf .134(')A f4_10 sf 0 2 rm .379(X)A 0 -2 rm f0_12 sf 1.175 .117(; modify the)J 95 582 :M .201 .02(equation for )J 157 582 :M f4_12 sf (Y)S 164 582 :M f0_12 sf .194 .019( in an analogous way. There always exist linear coefficients)J 95 600 :M .81 .081(and distributions over )J f4_12 sf (T)S 214 600 :M f0_12 sf 1.005 .101( and the new error terms such that the marginal)J 95 622 :M .129 .013(covariance matrix for )J f4_12 sf (L)S 208 622 :M f0_12 sf .154 .015(' is equal to the covariance matrix of )J f4_12 sf (L)S 394 622 :M f0_12 sf .19 .019(, and )J 421 622 :M f5_12 sf (e)S f0_12 sf (')S f4_10 sf 0 2 rm (X)S 0 -2 rm f0_12 sf .11 .011( and)J endp %%Page: 12 12 %%BeginPageSetup initializepage (peter; page: 12 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (12)S gR gS 0 0 552 730 rC 95 108 :M f5_12 sf (e)S f0_12 sf (')S f4_10 sf 0 2 rm (Y)S 0 -2 rm 108 108 :M f0_12 sf .181 .018( are independent. The process can be repeated for each pair of variables)J 95 130 :M .347 .035(with correlated errors in )J 216 130 :M f4_12 sf (L)S 223 130 :M f0_12 sf .335 .033(. Hence the zero partial correlations entailed by)J 95 148 :M f4_12 sf (L)S 102 148 :M f0_12 sf 1.585 .159( can be derived by applying Theorem 3 to the graph of )J 394 148 :M f4_12 sf (L)S 401 148 :M f0_12 sf 1.905 .19('. )J 412 148 :M 1.345 .134(Figure 3)J 95 166 :M .737 .074(illustrates this process. The set of variables )J f2_12 sf (V)S 322 166 :M f0_12 sf .926 .093( in the graph on the left is)J 95 184 :M ({)S 101 184 :M f4_12 sf .154(X)A f0_10 sf 0 2 rm .105(1)A 0 -2 rm f0_12 sf .063(,)A f4_12 sf .154(X)A f0_10 sf 0 2 rm .105(2)A 0 -2 rm f0_12 sf .063(,)A f4_12 sf .154(X)A f0_10 sf 0 2 rm .105(3)A 0 -2 rm f0_12 sf .063(,)A f4_12 sf .154(X)A f0_10 sf 0 2 rm .105(4)A 0 -2 rm f0_12 sf .349 .035(}. The graph on the left correlates the errors between )J f4_12 sf .154(X)A f0_10 sf 0 2 rm .105(1)A 0 -2 rm f0_12 sf .329 .033( and)J 95 203 :M f4_12 sf .39(X)A f0_10 sf 0 2 rm .266(2)A 0 -2 rm f0_12 sf .996 .1( \(indicated by the undirected edges between them.\) The graph on the)J 95 222 :M .853 .085(right has no correlated errors, but does have a latent variable )J f4_12 sf (T)S 410 222 :M f0_12 sf 1.061 .106( that is a)J 95 240 :M .666 .067(parent of )J 143 240 :M f4_12 sf .259(X)A f0_10 sf 0 2 rm .177(1)A 0 -2 rm f0_12 sf .33 .033( and )J f4_12 sf .259(X)A f0_10 sf 0 2 rm .177(2)A 0 -2 rm f0_12 sf .624 .062(. The two graphs linearly entail the same zero partial)J 95 259 :M .229 .023(correlations involving only variables in )J 289 259 :M f2_12 sf (V)S 298 259 :M f0_12 sf .281 .028( \(in this case they both entail no)J 95 277 :M (non-trivial zero partial correlations\).)S 136 317 :M f4_12 sf (X)S f0_10 sf 0 2 rm (3)S 0 -2 rm f0_12 sf ( = )S 161 317 :M f4_12 sf (a)S f0_12 sf ( )S f1_12 sf S 183 317 :M f4_12 sf (X)S f0_10 sf 0 2 rm (2)S 0 -2 rm f0_12 sf ( + )S 208 317 :M f4_12 sf (b)S f0_12 sf ( )S f1_12 sf S 233 317 :M f4_12 sf (X)S f0_10 sf 0 2 rm (4)S 0 -2 rm f0_12 sf ( + )S 258 317 :M f5_12 sf (e)S f0_10 sf 0 2 rm (3)S 0 -2 rm 280 317 :M f4_12 sf (X)S f0_10 sf 0 2 rm (3)S 0 -2 rm f0_12 sf ( = )S 305 317 :M f4_12 sf (a)S f0_12 sf ( )S f1_12 sf S 327 317 :M f4_12 sf (X)S f0_10 sf 0 2 rm (2)S 0 -2 rm f0_12 sf ( + )S 352 317 :M f4_12 sf (b)S f0_12 sf ( )S f1_12 sf S 377 317 :M f4_12 sf (X)S f0_10 sf 0 2 rm (4)S 0 -2 rm f0_12 sf ( +)S 399 317 :M f5_12 sf ( e)S f0_10 sf 0 2 rm (3)S 0 -2 rm 138 334 :M f4_12 sf (X)S f0_10 sf 0 2 rm (4)S 0 -2 rm f0_12 sf ( = )S 163 334 :M f4_12 sf (c)S f0_12 sf ( )S f1_12 sf S 184 334 :M f4_12 sf (X)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf ( + )S 209 334 :M f4_12 sf (d)S f0_12 sf ( )S f1_12 sf S 231 334 :M f4_12 sf (X)S f0_10 sf 0 2 rm (3)S 0 -2 rm f0_12 sf ( + )S 256 334 :M f5_12 sf (e)S f0_10 sf 0 2 rm (4)S 0 -2 rm f0_12 sf ( )S 282 334 :M f4_12 sf (X)S f0_10 sf 0 2 rm (4)S 0 -2 rm f0_12 sf ( = )S 307 334 :M f4_12 sf (c)S f0_12 sf ( )S f1_12 sf S 328 334 :M f4_12 sf (X)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf ( + )S 353 334 :M f4_12 sf (d)S f0_12 sf ( )S f1_12 sf S 375 334 :M f4_12 sf (X)S f0_10 sf 0 2 rm (3)S 0 -2 rm f0_12 sf ( + )S 400 334 :M f5_12 sf (e)S f0_10 sf 0 2 rm (4)S 0 -2 rm 163 351 :M f4_12 sf (X)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf ( = )S 188 351 :M f5_12 sf (e)S f0_10 sf 0 2 rm (1)S 0 -2 rm 307 351 :M f4_12 sf (X)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf ( = )S 332 351 :M f4_12 sf (e)S f0_12 sf ( )S f1_12 sf S 353 351 :M f5_12 sf (T)S f1_12 sf ( +)S 370 351 :M f0_12 sf ( )S f5_12 sf (e)S f0_12 sf (')S f0_10 sf 0 2 rm (1)S 0 -2 rm 164 368 :M f4_12 sf (X)S f0_10 sf 0 2 rm (2)S 0 -2 rm f0_12 sf ( = )S 189 368 :M f5_12 sf (e)S f0_10 sf 0 2 rm (2)S 0 -2 rm 308 368 :M f4_12 sf (X)S f0_10 sf 0 2 rm (2)S 0 -2 rm f0_12 sf ( = )S 333 368 :M f4_12 sf (f)S f0_12 sf ( )S f1_12 sf S 352 368 :M f5_12 sf (T)S f1_12 sf ( +)S 369 368 :M f0_12 sf ( )S f5_12 sf (e)S f0_12 sf (')S f0_10 sf 0 2 rm (2)S 0 -2 rm 130 385 :M f5_12 sf (e)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf ( and )S f5_12 sf (e)S f0_10 sf 0 2 rm (2)S 0 -2 rm f0_12 sf ( correlated)S 274 385 :M f5_12 sf (e)S f0_12 sf (')S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf ( and )S f5_12 sf (e)S f0_12 sf (')S f0_10 sf 0 2 rm (2)S 0 -2 rm f0_12 sf ( uncorrelated)S 247 404 :M f2_12 sf (Equation 4)S 107 431 335 81 rC 139 443 :M f4_12 sf (X)S 150 447 :M f0_10 sf (1)S 142 501 :M f4_12 sf (X)S 153 505 :M f0_10 sf (2)S 231 502 :M f4_12 sf (X)S 243 505 :M f0_10 sf (4)S 230 443 :M f4_12 sf (X)S 242 446 :M f0_10 sf (3)S 180 270 40 60 130.5 470.5 @n 90 180 44 58 132.5 469.5 @n 326 444 :M f4_12 sf (X)S 337 448 :M f0_10 sf (1)S 329 502 :M f4_12 sf (X)S 340 506 :M f0_10 sf (2)S 418 503 :M f4_12 sf (X)S 430 506 :M f0_10 sf (4)S 417 444 :M f4_12 sf (X)S 429 447 :M f0_10 sf (3)S 292 479 :M f4_12 sf (T)S 10 193 241 230 498 @k 160 446 -1 1 224 493 1 160 445 @a 10 118 166 228 447 @k -1 -1 163 500 1 1 221 452 @b 10 193 241 415 496 @k 345 444 -1 1 409 491 1 345 443 @a 10 116 164 413 444 @k -1 -1 350 499 1 1 406 449 @b 10 -114 -66 235 489 @k -1 -1 235 482 1 1 234 450 @b 10 66 114 245 450 @k -1 -1 245 488 1 1 244 458 @b 10 -114 -66 419 490 @k -1 -1 419 483 1 1 418 451 @b 10 66 114 429 447 @k -1 -1 429 489 1 1 428 455 @b 10 111 159 324 444 @k -1 -1 303 466 1 1 317 450 @b 10 187 235 329 496 @k 303 482 -1 1 322 492 1 303 481 @a gR gS 0 0 552 730 rC 95 521 :M f2_12 sf (Graph with Correlated Error Graph without Correlated Error)S 95 533 :M ( and Same Partial )S 275 545 :M ( Correlations Over V)S 203 563 :M (Figure )S 240 563 :M (3: Correlated Errors)S 95 611 :M (3. Non-linear Structural Equation Models)S endp %%Page: 13 13 %%BeginPageSetup initializepage (peter; page: 13 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (13)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .234 .023(A linear SEM is a special case of a SEM in which the equations relating a)J 95 122 :M 1.57 .157(given variable to other variables and a unique error term need not be)J 95 140 :M .741 .074(linear. In a SEM the random variables are divided into two disjoint sets,)J 95 158 :M .613 .061(the error terms and the non-error terms. Corresponding to each non-error)J 95 180 :M 1.681 .168(random variable )J f4_12 sf .546(V)A f0_12 sf .969 .097( is a unique error term )J f5_12 sf .392(e)A f4_10 sf 0 2 rm .455(V)A 0 -2 rm f0_12 sf .987 .099(. A SEM contains a set of)J 95 202 :M 2.248 .225(equations in which each non-error random variable )J f4_12 sf .797(V)A f0_12 sf 1.369 .137( is written as a)J 95 224 :M 1.185 .118(measureable function of other non-error random variables and )J 413 224 :M f5_12 sf .28(e)A f4_10 sf 0 2 rm .324(V)A 0 -2 rm f0_12 sf .979 .098(.. The)J 95 246 :M .362 .036(convention is that in the directed graph of a SEM there is an edge from )J 447 246 :M f4_12 sf (A)S 95 264 :M f0_12 sf .492 .049(to )J 108 264 :M f4_12 sf .226(B)A f0_12 sf .318 .032( if and only if )J f4_12 sf .226(B)A f0_12 sf .446 .045( is an argument in the function for )J 365 264 :M f4_12 sf .199(A)A f0_12 sf .38 .038(. As in the linear)J 95 282 :M .198 .02(case, I will still assume that density functions exist for both the probabilty)J 95 300 :M .446 .045(measure over the error terms and the non-error terms, that each non-error)J 95 318 :M .388 .039(term )J f4_12 sf .171(V)A f0_12 sf .355 .036( can also be written as a function of the error terms of its ancestors)J 95 340 :M .074 .007(in )J f4_12 sf (G)S 116 340 :M f0_12 sf .121 .012(, that each )J 169 340 :M f5_12 sf (e)S f4_10 sf 0 2 rm .052(V)A 0 -2 rm f0_12 sf .098 .01( is a function of )J f4_12 sf .063(V)A f0_12 sf .107 .011( and its parents in )J f4_12 sf (G)S 363 340 :M f0_12 sf .114 .011( \(which will be the)J 95 362 :M .64 .064(case if the errors are additive or multiplicative\), and that the Jacobean of)J 95 380 :M 1.597 .16(the transformation between the error terms and the non-error terms is)J 95 398 :M 1.959 .196(well-defined. Call such a set of equations and its associated graph a)J 95 416 :M f2_12 sf 5.078 .508(pseudo-indeterministic )J f0_12 sf 3.085 .309(SEM \(because the equations are actually)J 95 434 :M 2.091 .209(deterministic if the unmeasured error terms are included, but appear)J 95 452 :M .535 .054(indeterministic when the error terms are not measured.\) A directed graph)J 95 470 :M f4_12 sf (G)S 104 470 :M f0_12 sf ( )S f2_12 sf .137 .014(pseudo-indeterministically entails)J 280 470 :M f0_12 sf .17 .017( that )J f2_12 sf (X)S 313 470 :M f0_12 sf .207 .021( is independent of )J 403 470 :M f2_12 sf (Y)S 412 470 :M f0_12 sf .231 .023( given )J 446 470 :M f2_12 sf (Z)S 95 488 :M f0_12 sf 1.05 .105(if and only if in every pseudo-indeterministic SEM with graph )J f4_12 sf (G)S 424 488 :M f0_12 sf 1.591 .159(, )J 432 488 :M f2_12 sf (X)S 441 488 :M f0_12 sf 1.458 .146( is)J 95 506 :M (independent of )S 170 506 :M f2_12 sf (Y)S 179 506 :M f0_12 sf ( given )S 212 506 :M f2_12 sf (Z)S f0_12 sf (.)S 95 536 :M .687 .069(This section establishes that d-separation again provides a fast algorithm)J 95 554 :M 3.627 .363(for deciding whether a DAG pseudo-indeterministically entails a)J 95 572 :M 2.433 .243(conditional independence relation, but in a cyclic directed graph d-)J 95 590 :M 2.928 .293(separation may not pseudo-indeterministically entails a conditional)J 95 608 :M 4.094 .409(independence relation. Instead, a different condition, yielding a)J endp %%Page: 14 14 %%BeginPageSetup initializepage (peter; page: 14 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (14)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .155 .016(polynomial time algorithm, is found to suffice for a cyclic direected graph)J 95 122 :M (to pseudo-indeterministically entail a conditional independence relation.)S 95 152 :M 1.5 .15(By Theorem 2, d-separation is a necessary condition for a conditional)J 95 170 :M 1.407 .141(independence claim to be entailed by an SEM. The following remarks)J 95 188 :M .816 .082(show d-separation is also sufficient for acyclic SEMs, but not for cyclic)J 95 206 :M (SEMS.)S 95 236 :M f2_12 sf .647 .065(Theorem 4:)J f0_12 sf .126 .013( If )J f4_12 sf (G)S 179 236 :M f0_12 sf .391 .039( is a DAG containing disjoint sets of variables )J 409 236 :M f2_12 sf (X)S 418 236 :M f0_12 sf .203 .02(, )J f2_12 sf (Y)S 433 236 :M f0_12 sf .457 .046( and)J 95 254 :M f2_12 sf 3.857(Z)A f0_12 sf 2.629 .263(, )J 116 254 :M f2_12 sf (X)S 125 254 :M f0_12 sf 4.499 .45( is d-separated from )J 251 254 :M f2_12 sf (Y)S 260 254 :M f0_12 sf 4.965 .497( given )J f2_12 sf 3.285(Z)A f0_12 sf 4.577 .458( if and only )J 399 254 :M f4_12 sf (L)S 406 254 :M f0_12 sf 3.97 .397( pseudo-)J 95 272 :M (indeterministically entails that )S f2_12 sf (X)S 252 272 :M f0_12 sf ( is independent of )S 341 272 :M f2_12 sf (Y)S 350 272 :M f0_12 sf ( given )S 383 272 :M f2_12 sf (Z)S f0_12 sf (.)S 95 302 :M .72 .072(The following example gives a concrete illustration that there is a cyclic)J 95 320 :M .671 .067(graph )J f4_12 sf (G)S 135 320 :M f0_12 sf 1.062 .106( in which )J 187 320 :M f4_12 sf .382(X)A f0_12 sf .857 .086( is d-separated from )J 299 320 :M f4_12 sf (Y)S 306 320 :M f0_12 sf 1.023 .102( given {)J 346 320 :M f4_12 sf (Z)S 353 320 :M f0_12 sf (,)S f4_12 sf (W)S 366 320 :M f0_12 sf 1.062 .106(}, but )J 399 320 :M f4_12 sf (G)S 408 320 :M f0_12 sf .986 .099( does not)J 95 338 :M (pseudo-indeterminstically entail that )S f4_12 sf (X)S f0_12 sf ( is independent of )S 369 338 :M f4_12 sf (Y)S 376 338 :M f0_12 sf ( given {)S f4_12 sf (Z)S 421 338 :M f0_12 sf (,)S f4_12 sf (W)S 434 338 :M f0_12 sf (}.)S 217 359 115 77 rC 221 371 :M f4_12 sf (X)S 224 429 :M (Y)S 313 430 :M (Z)S 312 371 :M (W)S 10 156 204 308 368 @k 241 369 -1 1 300 368 1 241 368 @a 10 156 204 307 427 @k 245 428 -1 1 299 427 1 245 427 @a 10 66 114 322 373 @k -1 -1 322 417 1 1 321 381 @b 10 -114 -66 315 419 @k -1 -1 315 412 1 1 314 379 @b gR gS 0 0 552 730 rC 227 457 :M f2_12 sf (Figure )S 264 457 :M (4: Graph G)S 239 479 :M ( )S f4_12 sf (X)S f0_12 sf ( = )S 280 479 :M f5_12 sf (e)S f4_10 sf 0 2 rm (X)S 0 -2 rm 259 496 :M f4_12 sf (Y)S 266 496 :M f0_12 sf ( = )S 279 496 :M f5_12 sf (e)S f4_10 sf 0 2 rm (Y)S 0 -2 rm 237 513 :M f4_12 sf (Z)S 244 513 :M f0_12 sf ( = )S 257 513 :M f4_12 sf (W)S 267 513 :M f0_12 sf ( )S f1_12 sf S 277 513 :M f0_12 sf ( )S f4_12 sf (Y)S 287 513 :M f0_12 sf ( + )S 300 513 :M f5_12 sf (e)S f4_10 sf 0 2 rm (Z)S 0 -2 rm 237 530 :M f4_12 sf (W)S 247 530 :M f0_12 sf ( = )S 260 530 :M f4_12 sf (Z)S 267 530 :M f0_12 sf ( )S f1_12 sf S 277 530 :M f0_12 sf ( )S f4_12 sf (X)S f0_12 sf ( + )S 300 530 :M f5_12 sf (e)S f4_10 sf 0 2 rm (Y)S 0 -2 rm 128 547 :M f5_12 sf (e)S f4_10 sf 0 2 rm (X, )S 0 -2 rm f5_12 sf (e)S f4_10 sf 0 2 rm (X, )S 0 -2 rm f5_12 sf (e)S f4_10 sf 0 2 rm (X, )S 0 -2 rm f5_12 sf (e)S f4_10 sf 0 2 rm (X )S 0 -2 rm 190 547 :M f0_12 sf ( with independent standard normal distributions)S 247 566 :M f2_12 sf (Equation 5)S 95 600 :M f0_12 sf .411 .041(The transformation from )J 220 600 :M f5_12 sf .136(e)A f4_10 sf 0 2 rm .157(X)A 0 -2 rm f0_12 sf .129 .013(, )J f5_12 sf .136(e)A f4_10 sf 0 2 rm (Y)S 0 -2 rm 248 600 :M f0_12 sf .726 .073( )J 252 600 :M f5_12 sf (e)S f4_10 sf 0 2 rm (Z)S 0 -2 rm 263 600 :M f0_12 sf .66 .066(, )J 270 600 :M f5_12 sf .338(e)A f4_10 sf 0 2 rm .534(W)A 0 -2 rm f0_12 sf .336 .034( to )J f4_12 sf .47(X)A f0_12 sf .35 .035(, )J 317 600 :M f4_12 sf (Y)S 324 600 :M f0_12 sf .286 .029(, )J f4_12 sf (Z)S 337 600 :M f0_12 sf .66 .066(, )J 344 600 :M f4_12 sf (W)S 354 600 :M f0_12 sf .492 .049( is 1-1 except where)J 95 626 :M f5_12 sf (e)S f4_10 sf 0 2 rm (X)S 0 -2 rm f0_12 sf ( )S f1_12 sf S 116 626 :M f0_12 sf ( )S f5_12 sf (e)S f4_10 sf 0 2 rm (Y)S 0 -2 rm 130 626 :M f0_12 sf ( = 1 because)S endp %%Page: 15 15 %%BeginPageSetup initializepage (peter; page: 15 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (15)S gR gS 0 0 552 730 rC 259 108 :M f4_12 sf (X)S f0_12 sf ( = )S 279 108 :M f5_12 sf (e)S f4_10 sf 0 2 rm (X)S 0 -2 rm 259 125 :M f4_12 sf (Y)S 266 125 :M f0_12 sf ( = )S 279 125 :M f5_12 sf (e)S f4_10 sf 0 2 rm (Y)S 0 -2 rm 226 129 97 67 rC 323 196 :M psb currentpoint pse 226 129 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3104 div 2144 3 -1 roll exch div scale currentpoint translate 64 62 translate 7 510 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-Italic f1 (Z) show 354 510 moveto 384 /Symbol f1 (=) show 712 261 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) show 901 357 moveto 320 /Times-Italic f1 (W) show 1303 261 moveto 384 /Symbol f1 (\264) show 1595 261 moveto 384 /Symbol f2 (e) show 1781 357 moveto 320 /Times-Italic f1 (Y) show 2098 261 moveto 384 /Symbol f1 (+) show 2392 261 moveto 384 /Symbol f2 (e) show 2598 357 moveto 320 /Times-Italic f1 (Z) show 728 811 moveto 384 /Times-Roman f1 (1) show 971 811 moveto 384 /Symbol f1 (-) show 1254 811 moveto 384 /Times-Roman f1 (\() show 1394 811 moveto 384 /Symbol f2 (e) show 1608 907 moveto 320 /Times-Italic f1 (X) show 1932 811 moveto 384 /Symbol f1 (\264) show 2224 811 moveto 384 /Symbol f2 (e) show 2410 907 moveto 320 /Times-Italic f1 (Y) show 2655 811 moveto 384 /Times-Roman f1 (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 688 411 moveto 2159 0 rlineto stroke -13 1649 moveto 384 /Times-Italic f1 (W) show 447 1649 moveto 384 /Symbol f1 (=) show 805 1400 moveto 384 /Symbol f2 (e) show 1011 1496 moveto 320 /Times-Italic f1 (Z) show 1318 1400 moveto 384 /Symbol f1 (\264) show 1610 1400 moveto 384 /Symbol f2 (e) show 1824 1496 moveto 320 /Times-Italic f1 (X) show 2151 1400 moveto 384 /Symbol f1 (+) show 2445 1400 moveto 384 /Symbol f2 (e) show 2634 1496 moveto 320 /Times-Italic f1 (W) show 840 1950 moveto 384 /Times-Roman f1 (1) show 1083 1950 moveto 384 /Symbol f1 (-) show 1366 1950 moveto 384 /Times-Roman f1 (\() show 1506 1950 moveto 384 /Symbol f2 (e) show 1720 2046 moveto 320 /Times-Italic f1 (X) show 2044 1950 moveto 384 /Symbol f1 (\264) show 2336 1950 moveto 384 /Symbol f2 (e) show 2522 2046 moveto 320 /Times-Italic f1 (Y) show 2767 1950 moveto 384 /Times-Roman f1 (\)) show 781 1550 moveto 2197 0 rlineto stroke end pse gR gS 0 0 552 730 rC 247 211 :M f2_12 sf (Equation 6)S 95 245 :M f0_12 sf .559 .056(The Jacobean of the transformation from the )J 319 245 :M f5_12 sf .347(e)A f0_12 sf .658 .066('s is 1/\(1 + )J f4_12 sf .483(X)A f0_12 sf .197 .02( )J 391 245 :M f1_12 sf S 398 245 :M f0_12 sf .902 .09( )J 402 245 :M f4_12 sf (Y)S 409 245 :M f0_12 sf .501 .05(\). Hence,)J 95 271 :M (transforming the joint normal density of the )S 308 271 :M f5_12 sf (e)S f0_12 sf ('s yields)S 95 290 360 95 rC 455 385 :M psb currentpoint pse 95 290 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11520 div 3040 3 -1 roll exch div scale currentpoint translate 64 37 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 1144 moveto 693 0 rlineto stroke 1629 1144 moveto 684 0 rlineto stroke 3521 1144 moveto 672 0 rlineto stroke 5401 1144 moveto 2098 0 rlineto stroke 8707 1144 moveto 2110 0 rlineto stroke 4866 2442 moveto 1686 0 rlineto stroke 4813 1943 moveto 0 998 rlineto stroke 6589 1943 moveto 0 998 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) 4619 262 sh (X) 4951 262 sh (Y) 5324 262 sh (Z) 5707 262 sh (W) 6063 262 sh (x) 1902 996 sh (y) 3793 996 sh (z) 5790 996 sh (w) 6323 996 sh (y) 6972 996 sh (w) 9091 996 sh (z) 9733 996 sh (x) 10279 996 sh (X) 5546 2842 sh (Y) 6163 2842 sh 384 /Times-Roman f1 (\() 4789 262 sh (,) 5199 262 sh (,) 5559 262 sh (,) 5935 262 sh (\)) 6417 262 sh (exp\() 930 1243 sh (\)) 2331 1243 sh (exp\() 2822 1243 sh (\)) 4211 1243 sh (exp\() 4702 1243 sh (\() 5645 996 sh (\)) 7151 996 sh (\)) 7517 1243 sh (exp\() 8008 1243 sh (\() 8951 996 sh (\)) 10469 996 sh (\)) 10835 1243 sh (\() 5384 2842 sh (\)) 6411 2842 sh 384 /Symbol f1 (=) 6650 262 sh (-) 1657 996 sh (\264) 2530 1243 sh (-) 3549 996 sh (\264) 4410 1243 sh (-) 5429 996 sh (-) 6027 996 sh (\264) 6662 996 sh (\264) 7716 1243 sh (-) 8735 996 sh (-) 9432 996 sh (\264) 9968 996 sh (\346) 742 932 sh (\350) 742 1599 sh (\347) 742 1380 sh (\366) 10953 932 sh (\370) 10953 1599 sh (\367) 10953 1380 sh (\264) 11215 1243 sh (+) 5100 2842 sh (\264) 5879 2842 sh 384 /Times-Roman f1 (1) 250 996 sh (4) 28 1579 sh (2) 1875 1544 sh (2) 3761 1544 sh (2) 6354 1544 sh (2) 9666 1544 sh (1) 5613 2294 sh (1) 4856 2842 sh 320 ns (2) 477 1409 sh (2) 2097 826 sh (2) 3977 826 sh (2) 7283 826 sh (2) 10601 826 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (p) 214 1579 sh end MTsave restore pse gR gS 0 0 552 730 rC 247 394 :M f2_12 sf (Equation 7)S 95 424 :M f4_12 sf .063(X)A f0_12 sf .127 .013( is not independent of )J 210 424 :M f4_12 sf (Y)S 217 424 :M f0_12 sf .145 .014( given {)J 256 424 :M f4_12 sf (Z)S 263 424 :M f0_12 sf (,)S f4_12 sf (W)S 276 424 :M f0_12 sf .125 .013(} in this distribution because it is not)J 95 442 :M .119 .012(possible to factor it into a product of terms, no one of which contains both)J 95 460 :M f4_12 sf (X)S f0_12 sf ( and )S f4_12 sf (Y)S 132 460 :M f0_12 sf (.)S 95 490 :M 1.897 .19(However, it is possible to modify the graphical representation of the)J 95 508 :M 1.289 .129(functional relations in such a way that d-separation applied to the new)J 95 526 :M .207 .021(graph does entail conditional independence. In a directed graph )J f4_12 sf (G)S 414 526 :M f0_12 sf .109 .011(, a )J f2_12 sf .116(cycle)A 95 544 :M f0_12 sf 1.694 .169(is a cyclic directed path, in which each vertex occurs on exactly two)J 95 562 :M .489 .049(edges. A set of cycles )J 206 562 :M f2_12 sf (C)S 215 562 :M f0_12 sf .167 .017( is a )J f2_12 sf .168(cyclegroup)A 295 562 :M f0_12 sf .506 .051( if and only if it is a smallest set)J 95 580 :M 2.002 .2(of cycles such that for each cycle )J f4_12 sf 1.037(C)A f0_10 sf 0 2 rm .648(1)A 0 -2 rm f0_12 sf .828 .083( in )J f2_12 sf (C)S 320 580 :M f0_12 sf .978 .098(, )J f2_12 sf (C)S 337 580 :M f0_12 sf 1.624 .162( contains the transitive)J 95 599 :M .283 .028(closure of all of the cycles intersecting )J f4_12 sf .135(C)A f0_10 sf 0 2 rm .084(1)A 0 -2 rm f0_12 sf .244 .024(, i.e. it contains all of the cycles)J 95 618 :M .882 .088(that intersect )J 163 618 :M f4_12 sf .462(C)A f0_10 sf 0 2 rm .289(1)A 0 -2 rm f0_12 sf .913 .091(, all of the cycles that intersect cycles that intersect )J 438 618 :M f4_12 sf (C)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf (,)S endp %%Page: 16 16 %%BeginPageSetup initializepage (peter; page: 16 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (16)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf -.006(etc. For example, in figure 7, there are two distinct cyclegroups: the first is)A 95 122 :M ({)S 101 122 :M f4_12 sf (C)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf (,)S f4_12 sf (C)S f0_10 sf 0 2 rm (2)S 0 -2 rm f0_12 sf (,)S f4_12 sf (C)S f0_10 sf 0 2 rm (3)S 0 -2 rm f0_12 sf (}, and the second is {)S 249 122 :M f4_12 sf (C)S f0_10 sf 0 2 rm (4)S 0 -2 rm f0_12 sf (, )S f4_12 sf (C)S f0_10 sf 0 2 rm (5)S 0 -2 rm f0_12 sf (}.)S 171 144 207 194 rC 10 156 204 234 200 @k 188 201 -1 1 226 200 1 188 200 @a 10 156 204 294 201 @k 248 202 -1 1 286 201 1 248 201 @a 10 156 204 360 203 @k 314 204 -1 1 352 203 1 314 203 @a 175 204 :M f4_12 sf (A)S 232 195 :M (B)S 293 194 :M (D)S 364 207 :M (E)S 228 253 :M (F)S 228 332 :M (G)S 296 273 :M (H)S 295 161 :M (J)S 188 176 :M (C)S 196 181 :M f0_10 sf (1)S 188 232 :M f4_12 sf (C)S 198 237 :M f0_10 sf (2)S 184 288 :M f4_12 sf (C)S 194 293 :M f0_10 sf (3)S 334 171 :M f4_12 sf (C)S 342 176 :M f0_10 sf (4)S 336 238 :M f4_12 sf (C)S 346 243 :M f0_10 sf (5)S 234 160 :M f4_12 sf (I)S 90 180 48 62 236.5 169.5 @n 180 270 32 44 228.5 169.5 @n 10 156 204 246 147 @k 228 148 -1 1 238 147 1 228 147 @a -90 0 36 52 245.5 174.5 @n 0 90 38 60 244.5 171.5 @n 10 -24 24 236 200 @k 243 201 -1 1 247 200 1 243 200 @a 90 180 48 62 232.5 228.5 @n 180 270 32 44 224.5 228.5 @n 10 156 204 242 206 @k 224 207 -1 1 234 206 1 224 206 @a -90 0 36 52 241.5 233.5 @n 0 90 38 60 240.5 230.5 @n 10 -24 24 232 259 @k 239 260 -1 1 243 259 1 239 259 @a 90 180 48 62 228.5 286.5 @n 180 270 32 44 220.5 286.5 @n 10 156 204 238 264 @k 220 265 -1 1 230 264 1 220 264 @a -90 0 36 52 237.5 291.5 @n 0 90 38 60 236.5 288.5 @n 10 -24 24 228 317 @k 235 318 -1 1 239 317 1 235 317 @a 90 180 48 62 296.5 169.5 @n 180 270 32 44 288.5 169.5 @n 10 156 204 306 147 @k 288 148 -1 1 298 147 1 288 147 @a -90 0 36 52 305.5 174.5 @n 0 90 38 60 304.5 171.5 @n 10 -24 24 296 200 @k 303 201 -1 1 307 200 1 303 200 @a 90 180 48 62 293.5 228.5 @n 180 270 32 44 285.5 228.5 @n 10 156 204 303 206 @k 285 207 -1 1 295 206 1 285 206 @a -90 0 36 52 302.5 233.5 @n 0 90 38 60 301.5 230.5 @n 10 -24 24 293 259 @k 300 260 -1 1 304 259 1 300 259 @a gR gS 0 0 552 730 rC 218 359 :M f2_12 sf (Figure )S 255 359 :M (5: Cyclegroups)S 95 389 :M f0_12 sf .069 .007(Let the set of all cycles in )J f4_12 sf (G)S 231 389 :M f0_12 sf .027 .003( be )J f2_12 sf .021(Cycles)A f0_12 sf <28>S 285 389 :M f4_12 sf (G)S 294 389 :M f0_12 sf .077 .008(\). If a vertex )J 357 389 :M f4_12 sf (V)S f0_12 sf .054 .005( or an edge <)J f4_12 sf (V)S f0_12 sf (,)S f4_12 sf (W)S 447 389 :M f0_12 sf (>)S 95 411 :M .774 .077(occurs in some set )J f2_12 sf (C)S 200 411 :M f0_12 sf .902 .09( of cycles, for brevity write )J f4_12 sf .446(V)A f0_12 sf .182 .018( )J 353 411 :M f1_12 sf S 362 411 :M f0_12 sf .292 .029( )J f2_12 sf (C)S 375 411 :M f0_12 sf .528 .053( or <)J f4_12 sf .425(V)A f0_12 sf .174(,)A f4_12 sf (W)S 420 411 :M f0_12 sf 1.136 .114(> )J 432 411 :M f1_12 sf S 441 411 :M f0_12 sf .292 .029( )J f2_12 sf (C)S 95 433 :M f0_12 sf .812 .081(respectively, although strictly speaking neither a vertex nor an edge is a)J 95 451 :M .633 .063(member of a set of cycles. Form the )J 279 451 :M f2_12 sf .379 .038(collapsed graph)J 361 451 :M f0_12 sf .213 .021( )J f4_12 sf (G)S 373 451 :M f0_12 sf .529 .053(' from )J f4_12 sf (G)S 415 451 :M f0_12 sf .738 .074( by the)J 95 469 :M (following operations on each cyclegroup:)S 95 517 :M (1. remove all of the edges between members of the cyclegroup;)S 95 535 :M (2. arbitrarily number the vertices in the cyclegroup;)S 95 553 :M 1.361 .136(3. add an edge from each lower number vertex to each higher number)J 95 571 :M (vertex;)S 95 589 :M .298 .03(4. for each parent )J 184 589 :M f4_12 sf .143(A)A f0_12 sf .287 .029( of a member of the cyclegroup that is not itself in the)J 95 607 :M (cyclegroup, add an edge from )S 241 607 :M f4_12 sf (A)S f0_12 sf ( to each member of the cyclegroup.)S endp %%Page: 17 17 %%BeginPageSetup initializepage (peter; page: 17 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (17)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf 2.049 .205(\(The procedure does not define a unique collapsed graph due to the)J 95 122 :M .445 .044(arbitrariness of the numbering, but since all of the collapsed graphs share)J 95 140 :M .328 .033(the same d-separation relations, it does not matter.\) Note that even if )J f4_12 sf (G)S 442 140 :M f0_12 sf .449 .045( is)J 95 158 :M .354 .035(a cyclic graph, the collapsed graph is acyclic. The collapsed graph can be)J 95 176 :M (generated in polynomial time.)S 95 206 :M f2_12 sf .928 .093(Theorem 5:)J 157 206 :M f0_12 sf 1.157 .116( In an SEM with directed graph )J f4_12 sf (G)S 332 206 :M f0_12 sf 1.191 .119( \(cyclic or acyclic\) and)J 95 224 :M .081 .008(collapsed graph )J 174 224 :M f4_12 sf (G)S 183 224 :M f0_12 sf .087 .009(' containing disjoint sets of variables )J 362 224 :M f2_12 sf (X)S 371 224 :M f0_12 sf .049 .005(, )J f2_12 sf (Y)S 386 224 :M f0_12 sf .119 .012( and )J 410 224 :M f2_12 sf .069(Z)A f0_12 sf .056 .006(, if )J f2_12 sf (X)S 443 224 :M f0_12 sf .119 .012( is)J 95 242 :M 2.179 .218(d-separated from )J 188 242 :M f2_12 sf (Y)S 197 242 :M f0_12 sf 3.018 .302( given )J 238 242 :M f2_12 sf 1.963(Z)A f0_12 sf 1.566 .157( in )J f4_12 sf (G)S 278 242 :M f0_12 sf 2.706 .271(' then the SEM entails that )J 430 242 :M f2_12 sf (X)S 439 242 :M f0_12 sf 3.269 .327( is)J 95 260 :M (independent of )S 170 260 :M f2_12 sf (Y)S 179 260 :M f0_12 sf ( given )S 212 260 :M f2_12 sf (Z)S f0_12 sf (.)S 95 290 :M (A collapsed graph for the graph in figure 7 is shown in figure 8a, and a)S 95 308 :M (collapsed graph for the graph in figure 4 is shown in figure 8b.)S 95 338 :M (I do not know whether the follow conjecture holds:)S 95 368 :M f2_12 sf (Conjecture)S f0_12 sf (:Let )S f4_12 sf (G)S 183 368 :M f0_12 sf ( \(cyclic or acyclic\) have collapsed graph )S 383 368 :M f4_12 sf (G)S 392 368 :M f0_12 sf (' containing)S 95 386 :M (disjoint sets of variables )S f2_12 sf (X)S 223 386 :M f0_12 sf (, )S f2_12 sf (Y)S 238 386 :M f0_12 sf ( and )S f2_12 sf (Z)S f0_12 sf (. If )S 286 386 :M f4_12 sf (G)S 295 386 :M f0_12 sf ( pseudo-indeterministically)S 95 404 :M (entails that )S f2_12 sf (X)S 159 404 :M f0_12 sf ( is independent of )S 248 404 :M f2_12 sf (Y)S 257 404 :M f0_12 sf ( given )S 290 404 :M f2_12 sf (Z)S f0_12 sf (, then in )S 340 404 :M f4_12 sf (G')S f0_12 sf ( )S f2_12 sf (X)S 363 404 :M f0_12 sf ( is d-separated)S 95 422 :M (from )S f2_12 sf (Y)S 130 422 :M f0_12 sf ( given )S 163 422 :M f2_12 sf (Z)S f0_12 sf (.)S endp %%Page: 18 18 %%BeginPageSetup initializepage (peter; page: 18 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (18)S gR gS 95 95 407 207 rC 10 156 204 216 148 @k 170 149 -1 1 208 148 1 170 148 @a 10 156 204 276 149 @k 230 150 -1 1 268 149 1 230 149 @a 10 156 204 342 151 @k 296 152 -1 1 334 151 1 296 151 @a 157 152 :M f4_12 sf (A)S 210 163 :M (B)S 279 164 :M (D)S 346 155 :M (E)S 212 219 :M (F)S 163 257 :M (G)S 282 225 :M (H)S 215 107 :M (I)S 277 109 :M (J)S 10 -114 -66 219 146 @k -1 -1 219 139 1 1 218 110 @b 10 -114 -66 220 205 @k -1 -1 220 198 1 1 219 168 @b 10 -89 -41 174 245 @k -1 -1 177 239 1 1 209 167 @b 10 -58 -10 179 248 @k -1 -1 186 245 1 1 214 223 @b 10 122 170 278 113 @k -1 -1 226 149 1 1 270 117 @b 10 208 256 282 218 @k 228 150 -1 1 277 212 1 228 149 @a 180 270 198 132 213.5 170.5 @n 10 156 204 163 253 @k 151 254 -1 1 155 253 1 151 253 @a 90 180 74 168 151.5 169.5 @n 10 243 291 284 151 @k 281 116 -1 1 284 143 1 281 115 @a 10 245 293 287 210 @k 285 169 -1 1 287 202 1 285 168 @a 10 -24 24 295 219 @k 302 220 -1 1 316 219 1 302 219 @a 10 -24 24 223 215 @k 230 216 -1 1 244 215 1 230 215 @a -90 0 52 80 239.5 255.5 @n 0 90 94 90 217.5 254.5 @n 144 300 -1 1 218 299 1 144 299 @a 386 146 :M (X)S 397 150 :M f0_10 sf (1)S 389 204 :M f4_12 sf (X)S 400 208 :M f0_10 sf (2)S 478 205 :M f4_12 sf (X)S 490 208 :M f0_10 sf (4)S 477 146 :M f4_12 sf (X)S 489 149 :M f0_10 sf (3)S 10 156 204 473 143 @k 406 144 -1 1 465 143 1 406 143 @a 10 156 204 472 202 @k 410 203 -1 1 464 202 1 410 202 @a 10 -114 -66 480 194 @k -1 -1 480 187 1 1 479 154 @b 10 185 233 467 191 @k 403 156 -1 1 460 187 1 403 155 @a 10 129 177 474 151 @k -1 -1 402 190 1 1 466 155 @b -90 0 152 78 289.5 150.5 @n 0 90 100 142 315.5 148.5 @n 90 180 92 246 144.5 176.5 @n 180 270 226 158 211.5 178.5 @n 10 123 171 213 111 @k -1 -1 168 142 1 1 205 115 @b 10 206 254 212 211 @k 166 158 -1 1 207 205 1 166 157 @a 10 240 288 170 246 @k 160 157 -1 1 169 238 1 160 156 @a gR gS 0 0 552 730 rC 95 317 :M f0_12 sf ( \(a\) \(b\))S 204 335 :M f2_12 sf (Figure )S 241 335 :M (6: Collapsed Graphs)S 95 365 :M (5. Conclusion)S 95 395 :M f0_12 sf .443 .044(These results raise a number of interesting questions whose answers may)J 95 413 :M .532 .053(be of practical importance. Under what conditions, for example, are their)J 95 431 :M .679 .068(results about conditional independence comparable to the equivalence of)J 95 449 :M 1.069 .107(vanishing partial correlations in models with depndent errors and latent)J 95 467 :M 4.143 .414(variable models with independent errors? There are polynomial)J 95 485 :M 1.17 .117(algorithms \(Verma and Pearl, 1990, Frydenberg, 1990\) for determining)J 95 503 :M 3.059 .306(when two arbitrary directed acyclic graphs entail the same set of)J 95 521 :M 1.102 .11(conditional independence relations. Is there a polynomial algorithm for)J 95 539 :M .07 .007(determining when two arbitrary directed graphs \(cyclic or acyclic\) linearly)J 95 557 :M 2.07 .207(entail the same set of conditional independence relations? There are)J 95 575 :M .9 .09(polynomial algorithms \(Spirtes and Verma, 1992\) for determining when)J 95 593 :M 1.297 .13(two arbitrary directed acyclic graphs entail the same set of conditional)J 95 611 :M 1.129 .113(independence relations over a common subset of variable )J 389 611 :M f2_12 sf .9(O)A f0_12 sf 1.175 .117(. Is there a)J 95 629 :M .413 .041(polynomial algorithm for determining when two arbitrary directed graphs)J endp %%Page: 19 19 %%BeginPageSetup initializepage (peter; page: 19 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (19)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .199 .02(\(cyclic or acyclic\) linearly entail the same set of conditional independence)J 95 122 :M 2.614 .261(relations over a common subset of variables )J 339 122 :M f2_12 sf .734(O)A f0_12 sf 2.14 .214(? Assuming Markov)J 95 140 :M .461 .046(properties hold and completely characterize the condiitonal independence)J 95 158 :M .644 .064(facts in distributions considered, there are correct polynomial algorithms)J 95 176 :M .045 .004(for inferring features of \(sparse\) directed acyclic graphs from a probability)J 95 194 :M 1.609 .161(distribution when there are no latent common causes \(see Spirtes and)J 95 212 :M 1.635 .163(Glymour, 1991, Cooper and Herskovitz, 1992\). Are there comparable)J 95 230 :M 1.181 .118(correct, polynomial algorithms for inferring features of directed graphs)J 95 248 :M .351 .035(\(cyclic or acyclic\) from a probability distribution when there are no latent)J 95 266 :M 3.207 .321(common causes? There are similarly correct, but not polynomial,)J 95 284 :M 2.852 .285(algorithms for inferring features of directed acyclic graphs from a)J 95 302 :M 1.182 .118(probability distribution even when there may be latent common causes)J 95 320 :M .92 .092(\(see Spirtes, 1992 and Spirtes, Glymour and Scheines, 1993\). Are there)J 95 338 :M .519 .052(comparable algorithms for inferring features of directed graphs \(cyclic or)J 95 356 :M 1.353 .135(acyclic\) from a probability distribution even when there may be latent)J 95 374 :M (common causes?)S 246 404 :M f2_12 sf (References)S 95 428 :M f0_12 sf -.002(Bollen, K., 1989, Structural Equations with Latent Variables. \(Wiley, New)A 95 440 :M (York\).)S 95 464 :M 1.408 .141(Frydenberg, M., 1990, The chain graph Markov property,)J 389 464 :M f4_12 sf (Scandinvaian)S 95 476 :M (Journal of Statistics,)S 194 476 :M f0_12 sf ( )S f2_12 sf (17)S f0_12 sf (, 333-353.)S 95 500 :M 1.452 .145(Geiger, D., and Pearl, J., 1988, Logical and Algorithmic properties of)J 95 512 :M 1.566 .157(Conditional Independence. Technical Report R-97, Cognitive Systems)J 95 524 :M (Laboratory, University of California, Los Angeles.)S 95 548 :M .957 .096(Goldberger, A., Duncan, O. \(eds.\), 1973, Structural Equation Models in)J 95 560 :M (the Social Sciences \(Seminar Press, New York\).)S 95 584 :M 3.722 .372(Haavelmo, T., 1943, The statistical implications of a system of)J 95 596 :M (simultaneous equations, )S f4_12 sf (Econometrica)S f0_12 sf (, )S f2_12 sf (11)S f0_12 sf (, 1-12.)S endp %%Page: 20 20 %%BeginPageSetup initializepage (peter; page: 20 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (20)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .204 .02(Kiiveri, H. and Speed, T.,1982, Structural analysis of multivariate data: A)J 95 116 :M .975 .098(review, )J 136 116 :M f4_12 sf .502 .05(Sociological Methodology)J 264 116 :M f0_12 sf .942 .094(, Leinhardt, S. \(ed.\). Jossey-Bass, San)J 95 128 :M (Francisco.)S 95 152 :M 1.651 .165(Kiiveri, H., Speed, T., and Carlin, J., 1984, Recursive causal models,)J 95 164 :M f4_12 sf (Journal of the Australian Mathematical Society)S f0_12 sf (, )S f2_12 sf (36)S f0_12 sf (, 30-52.)S 95 188 :M 1.909 .191(Lauritzen, S., Dawid, A., Larsen, B., Leimer, H.,1990, Independence)J 95 200 :M (properties of directed Markov fields, )S f4_12 sf (Networks)S f0_12 sf (, )S f2_12 sf (20)S f0_12 sf (, 491-505.)S 95 224 :M .43 .043(Mason, S., 1953, Feedback theory-some properties of signal flow graphs,)J 95 236 :M f4_12 sf (Proceedings of the IRE)S 207 236 :M f0_12 sf (, )S f2_12 sf (41)S f0_12 sf (.)S 95 260 :M 2.635 .264(Mason, S., 1956, Feedback theory-further properties of signal flow)J 95 272 :M (graphs, )S 133 272 :M f4_12 sf (Proceedings of the IRE)S 245 272 :M f0_12 sf (, )S f2_12 sf (44)S f0_12 sf (.)S 95 296 :M 1.407 .141(Pearl, J.,1986, Fusion, propagation, and structuring in belief networks,)J 95 308 :M f4_12 sf (Artificial Intelligence)S 198 308 :M f0_12 sf ( )S f2_12 sf (29)S f0_12 sf (, 241-88.)S 95 332 :M 2.272 .227(Pearl, J., 1988,. )J f2_12 sf 3.893 .389(Probabilistic Reasoning in Intelligent Systems)J 451 332 :M f0_12 sf (,)S 95 344 :M (\(Morgan Kaufman: San Mateo, CA\).)S 95 368 :M 2.659 .266(Pearl, J. and Verma, T. \(1991\). A theory of inferred causation, in)J 95 380 :M f2_12 sf .504 .05(Principles of Knowledge Representation and Reasoning: Proceedings)J 95 392 :M 2.758 .276(of the Second International Conference)J f0_12 sf 2.57 .257( \(Morgan Kaufmann, San)J 95 404 :M (Mateo, CA\).)S 95 428 :M 1.328 .133(Spirtes, P. and Glymour, C., 1990, Causal Structure Among Measured)J 95 440 :M .255 .026(Variables Preserved with Unmeasured Variables. Technical Report Â鶹´å-)J 95 452 :M .917 .092(LCL-90-5, Laboratory for Computational Linguistics, Carnegie Mellon)J 95 464 :M (University.)S 95 488 :M 1.586 .159(Spirtes, P., and Glymour, C., 1991, An algorithm for fast recovery of)J 95 500 :M (sparse causal graphs, )S 199 500 :M f4_12 sf (Social Science Computer Review, )S f2_12 sf (9)S f0_12 sf (, 62-72.)S 95 524 :M .431 .043(Spirtes, P., Glymour, C., and Scheines, R., 1993, )J f2_12 sf .871 .087(Causation, Prediction,)J 95 536 :M (and Search,)S 156 536 :M f0_12 sf ( \(Springer-Verlag Lecture Notes in Statistics 81, New York\).)S 95 560 :M .876 .088(Wermuth, N.,1980, Linear recursive equations, covariance selection and)J 95 572 :M .925 .092(path analysis, )J f4_12 sf 1.12 .112(Journal of the American Statistical Association)J 404 572 :M f0_12 sf 1.817 .182(, )J 412 572 :M f2_12 sf .3(75)A f0_12 sf .933 .093(, 963-)J 95 584 :M (972.)S 95 608 :M .578 .058(Wermuth, N. and Lauritzen, S.,1983, Graphical and recursive models for)J 95 620 :M (contingency tables, )S 191 620 :M f4_12 sf (Biometrika)S f0_12 sf (, )S f2_12 sf (72)S f0_12 sf (, 537-552.)S endp %%Page: 21 21 %%BeginPageSetup initializepage (peter; page: 21 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (21)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf 4.04 .404(Wermuth, N. and Lauritzen, S., 1990,. On substantive research)J 95 116 :M .276 .028(hypotheses, conditional independence graphs and graphical chain models,)J 95 128 :M f4_12 sf (Journal of the Royal Statistical Society, Series B)S 331 128 :M f0_12 sf (, )S f2_12 sf (52)S f0_12 sf (, 21-50.)S 95 152 :M 3.986 .399(Whittaker, J.,1990, )J 206 152 :M f2_12 sf 3.935 .394(Graphical Models in Applied Multivariate)J 95 164 :M (Statistics)S 141 164 :M f0_12 sf ( \(Wiley, New York\).)S 95 188 :M 4.252 .425(Wright, S. \(1934\). The method of path coefficients, )J 401 188 :M f4_12 sf 3.868 .387(Annals of)J 95 200 :M (Mathematical Statistics )S 211 200 :M f2_12 sf (5)S f0_12 sf (, 161-215.)S endp %%Page: 22 22 %%BeginPageSetup initializepage (peter; page: 22 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (22)S gR gS 0 0 552 730 rC 225 104 :M f2_12 sf (Proofs for Referees)S 95 134 :M .336 .034(Lemma 3:)J 148 134 :M f0_12 sf .294 .029( If )J f2_12 sf (V)S 172 134 :M f0_12 sf .366 .037( is a set of random variables with a probability measure )J f4_12 sf (P)S 95 152 :M f0_12 sf .101 .01(that has a positive density function )J 266 152 :M f4_12 sf (f)S f0_12 sf <28>S 273 152 :M f2_12 sf (V)S 282 152 :M f0_12 sf .063 .006(\), and )J f4_12 sf (P)S f0_12 sf .11 .011( satisfies the global directed)J 95 170 :M .034 .003(Markov property for directed \(cyclic or acyclic\) graph )J 359 170 :M f4_12 sf (G)S 368 170 :M f0_12 sf .044 .004(, then )J 398 170 :M f4_12 sf (f)S f0_12 sf <28>S 405 170 :M f2_12 sf (V)S 414 170 :M f0_12 sf .032 .003(\) factors)J 95 188 :M (according to )S 158 188 :M f4_12 sf (G)S 167 188 :M f0_12 sf (.)S 95 218 :M 2.485 .248(Proof. Assume that probability measure over )J f2_12 sf (V)S 349 218 :M f0_12 sf 2.687 .269( satisfies the global)J 95 236 :M .677 .068(directed Markov property for directed \(cyclic or acyclic\) graph )J 411 236 :M f4_12 sf (G)S 420 236 :M f0_12 sf .852 .085(. I will)J 95 254 :M .729 .073(now show that for any disjoint sets of variables )J 335 254 :M f2_12 sf (R)S 344 254 :M f0_12 sf 1.016 .102(, )J 352 254 :M f2_12 sf (S)S 359 254 :M f0_12 sf .498 .05(, and )J f2_12 sf .394(T)A f0_12 sf .898 .09( included in)J 95 276 :M f2_12 sf (An)S f0_12 sf <28>S 114 276 :M f2_12 sf (X)S 123 276 :M f1_12 sf 1.023 .102<20C8>J f0_12 sf .274 .027( )J f2_12 sf (Y)S 149 276 :M f0_12 sf 1.2 .12( )J 154 276 :M f1_12 sf S f0_12 sf S f2_12 sf (Z)S f0_12 sf (,)S f4_12 sf (G)S 186 276 :M f0_12 sf .647 .065(\), if )J f2_12 sf (R)S 217 276 :M f0_12 sf .746 .075( and )J f2_12 sf (S)S 249 276 :M f0_12 sf .8 .08( are separated given )J 353 276 :M f2_12 sf .823(T)A f0_12 sf .686 .069( in )J 379 276 :M f4_12 sf (G)S 388 273 :M f4_10 sf (M)S f0_12 sf 0 3 rm <28>S 0 -3 rm 400 276 :M f2_12 sf (An)S f0_12 sf <28>S 419 276 :M f2_12 sf (X)S 428 276 :M f1_12 sf 1.023 .102<20C8>J f0_12 sf .274 .027( )J f2_12 sf (Y)S 95 302 :M f1_12 sf S f0_12 sf S f2_12 sf (Z)S f0_12 sf (,)S f4_12 sf (G)S 127 302 :M f0_12 sf 1.972 .197(\)\), then )J 171 302 :M f2_12 sf (R)S 180 302 :M f0_12 sf 1.717 .172( and )J f2_12 sf (S)S 215 302 :M f0_12 sf 1.781 .178( are independent given )J 338 302 :M f2_12 sf 1.768(T)A f0_12 sf 1.564 .156(. If )J 369 302 :M f2_12 sf (R)S 378 302 :M f0_12 sf 2.509 .251(, )J 387 302 :M f2_12 sf (S)S 394 302 :M f0_12 sf 1.613 .161(, and )J f2_12 sf 1.275(T)A f0_12 sf 2.163 .216( are)J 95 328 :M 1.174 .117(included in )J 156 328 :M f2_12 sf (An)S f0_12 sf <28>S 175 328 :M f2_12 sf (X)S 184 328 :M f1_12 sf 2.434 .243<20C8>J f0_12 sf .717 .072( )J 203 328 :M f2_12 sf (Y)S 212 328 :M f0_12 sf .142 .014( )J f1_12 sf .481A f0_12 sf .156A f2_12 sf .417(Z)A f0_12 sf .156(,)A f4_12 sf (G)S 248 328 :M f0_12 sf 1.347 .135(\), then )J 286 328 :M f2_12 sf (An)S f0_12 sf <28>S 305 328 :M f2_12 sf (R)S 314 328 :M f1_12 sf 2.434 .243<20C8>J f0_12 sf .717 .072( )J 333 328 :M f2_12 sf (S)S 340 328 :M f0_12 sf 1.819 .182( )J 345 328 :M f1_12 sf S f0_12 sf S f2_12 sf (T)S f0_12 sf (,)S f4_12 sf (G)S 377 328 :M f0_12 sf 1.213 .121(\) is included in)J 95 354 :M f2_12 sf (An)S f0_12 sf <28>S 114 354 :M f2_12 sf (X)S 123 354 :M f1_12 sf .669 .067<20C8>J f0_12 sf .197 .02( )J 139 354 :M f2_12 sf (Y)S 148 354 :M f0_12 sf ( )S f1_12 sf .132A f0_12 sf S f2_12 sf .115(Z)A f0_12 sf (,)S f4_12 sf (G)S 183 354 :M f0_12 sf .343 .034(\). Any pair of vertices )J 295 354 :M f4_12 sf .175(A)A f0_12 sf .222 .022( and )J f4_12 sf .175(B)A f0_12 sf .323 .032( adjacent in )J f4_12 sf (G)S 401 351 :M f4_10 sf (M)S f0_12 sf 0 3 rm <28>S 0 -3 rm 413 354 :M f2_12 sf (An)S f0_12 sf <28>S 432 354 :M f2_12 sf (R)S 441 354 :M f1_12 sf .455 .045<20C8>J 95 380 :M f2_12 sf (S)S 102 380 :M f0_12 sf ( )S f1_12 sf .088A f0_12 sf S f2_12 sf .076(T)A f0_12 sf (,)S f4_12 sf (G)S 137 380 :M f0_12 sf .231 .023(\)\) is also adjacent in )J 239 380 :M f4_12 sf (G)S 248 377 :M f4_10 sf (M)S f0_12 sf 0 3 rm <28>S 0 -3 rm 260 380 :M f2_12 sf (An)S f0_12 sf <28>S 279 380 :M f2_12 sf (X)S 288 380 :M f1_12 sf .446 .045<20C8>J f0_12 sf .131 .013( )J 304 380 :M f2_12 sf (Y)S 313 380 :M f0_12 sf ( )S f1_12 sf .088A f0_12 sf S f2_12 sf .076(Z)A f0_12 sf (,)S f4_12 sf (G)S 348 380 :M f0_12 sf .222 .022(\)\) because )J 401 380 :M f4_12 sf (G)S 410 380 :M f0_12 sf <28>S 414 380 :M f2_12 sf (An)S f0_12 sf <28>S 433 380 :M f2_12 sf (R)S 442 380 :M f1_12 sf .303 .03<20C8>J 95 406 :M f2_12 sf (S)S 102 406 :M f0_12 sf .615 .062( )J 106 406 :M f1_12 sf S f0_12 sf S f2_12 sf (T)S f0_12 sf (,)S f4_12 sf (G)S 138 406 :M f0_12 sf .401 .04(\)\) is a subgraph of )J f4_12 sf (G)S 240 406 :M f0_12 sf <28>S 244 406 :M f2_12 sf (An)S f0_12 sf <28>S 263 406 :M f2_12 sf (X)S 272 406 :M f1_12 sf .823 .082<20C8>J f0_12 sf .243 .024( )J 288 406 :M f2_12 sf (Y)S 297 406 :M f0_12 sf .615 .062( )J 301 406 :M f1_12 sf S f0_12 sf S f2_12 sf (Z)S f0_12 sf (,)S f4_12 sf (G)S 333 406 :M f0_12 sf .424 .042(\)\). Hence )J 382 406 :M f4_12 sf (G)S 391 403 :M f4_10 sf (M)S f0_12 sf 0 3 rm <28>S 0 -3 rm 403 406 :M f2_12 sf (An)S f0_12 sf <28>S 422 406 :M f2_12 sf (R)S 431 406 :M f1_12 sf .823 .082<20C8>J f0_12 sf .243 .024( )J 447 406 :M f2_12 sf (S)S 95 432 :M f1_12 sf S f0_12 sf S f2_12 sf (T)S f0_12 sf (,)S f4_12 sf (G)S 127 432 :M f0_12 sf .248 .025(\)\) is a subgraph of )J f4_12 sf (G)S 228 429 :M f4_10 sf (M)S f0_12 sf 0 3 rm <28>S 0 -3 rm 240 432 :M f2_12 sf (An)S f0_12 sf <28>S 259 432 :M f2_12 sf (X)S 268 432 :M f1_12 sf .324 .032<20C8>J f0_12 sf .087 .009( )J f2_12 sf (Y)S 292 432 :M f0_12 sf .38 .038( )J 296 432 :M f1_12 sf S f0_12 sf S f2_12 sf (Z)S f0_12 sf (,)S f4_12 sf (G)S 328 432 :M f0_12 sf .264 .026(\)\). It follows that if )J 425 432 :M f2_12 sf (R)S 434 432 :M f0_12 sf .293 .029( and)J 95 458 :M f2_12 sf (S)S 102 458 :M f0_12 sf .098 .01( are separated given )J 202 458 :M f2_12 sf .074(T)A f0_12 sf .059 .006( in )J f4_12 sf (G)S 234 455 :M f4_10 sf (M)S f0_12 sf 0 3 rm <28>S 0 -3 rm 246 458 :M f2_12 sf (An)S f0_12 sf <28>S 265 458 :M f2_12 sf (X)S 274 458 :M f1_12 sf .126 .013<20C8>J f0_12 sf ( )S f2_12 sf (Y)S 298 458 :M f0_12 sf ( )S f1_12 sf S f0_12 sf S f2_12 sf (Z)S f0_12 sf (,)S f4_12 sf (G)S 333 458 :M f0_12 sf .091 .009(\)\) they are also separated)J 95 484 :M .248 .025(in )J 108 484 :M f4_12 sf (G)S 117 481 :M f4_10 sf (M)S f0_12 sf 0 3 rm <28>S 0 -3 rm 129 484 :M f2_12 sf (An)S f0_12 sf <28>S 148 484 :M f2_12 sf (R)S 157 484 :M f1_12 sf .277 .028<20C8>J f0_12 sf .074 .007( )J f2_12 sf (S)S 179 484 :M f0_12 sf .298 .03( )J 183 484 :M f1_12 sf S f0_12 sf S f2_12 sf (T)S f0_12 sf (,)S f4_12 sf (G)S 215 484 :M f0_12 sf .185 .018(\)\). But by the global directed Markov property, if)J 95 510 :M f2_12 sf (R)S 104 510 :M f0_12 sf ( and )S f2_12 sf (S)S 134 510 :M f0_12 sf -.004( are separated given )A 233 510 :M f2_12 sf (T)S f0_12 sf ( in )S f4_12 sf (G)S 265 507 :M f4_10 sf (M)S f0_12 sf 0 3 rm <28>S 0 -3 rm 277 510 :M f2_12 sf (An)S f0_12 sf <28>S 296 510 :M f2_12 sf (R)S 305 510 :M f1_12 sf <20C8>S f0_12 sf ( )S 320 510 :M f2_12 sf (S)S 327 510 :M f0_12 sf ( )S 330 510 :M f1_12 sf S f0_12 sf S f2_12 sf (T)S f0_12 sf (,)S f4_12 sf (G)S 362 510 :M f0_12 sf (\)\) then )S 397 510 :M f2_12 sf (R)S 406 510 :M f0_12 sf ( and )S f2_12 sf (S)S 436 510 :M f0_12 sf ( are)S 95 532 :M 1.135 .114(independent given )J f2_12 sf .378(T)A f0_12 sf 1.078 .108(. It follows from the Hammersly-Clifford Theorem)J 95 554 :M (that the density function )S f4_12 sf (f)S f0_12 sf <28>S 221 554 :M f2_12 sf (An)S f0_12 sf <28>S 240 554 :M f2_12 sf (X)S 249 554 :M f1_12 sf <20C8>S f0_12 sf ( )S f2_12 sf (Y)S 273 554 :M f0_12 sf ( )S f1_12 sf S f0_12 sf S f2_12 sf (Z)S f0_12 sf (,)S f4_12 sf (G)S 308 554 :M f0_12 sf (\)\) can be factored as)S 138 579 273 30 rC 411 609 :M psb currentpoint pse 138 579 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8736 div 960 3 -1 roll exch div scale currentpoint translate 64 40 translate 56 344 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-Italic f1 (f) show 229 344 moveto 384 /Times-Roman f1 (\() show 369 344 moveto 384 /Times-Bold f1 (An) show 860 344 moveto 384 /Times-Roman f1 (\() show 1000 344 moveto 384 /Times-Bold f1 (X) show 1347 344 moveto 384 /Symbol f1 (\310) show 1717 344 moveto 384 /Times-Bold f1 (Y) show 2065 344 moveto 384 /Symbol f1 (\310) show 2428 344 moveto 384 /Times-Bold f1 (Z) show 2675 344 moveto 384 /Times-Roman f1 (,) show 2805 344 moveto 384 /Times-Italic f1 (G) show 3095 344 moveto 384 /Times-Roman f1 (\)) show 3225 344 moveto (\)) show 3458 344 moveto 384 /Symbol f1 (=) show 5363 344 moveto 384 /Times-Italic f1 (g) show 5539 440 moveto 320 ns (V) show 5795 344 moveto 384 /Times-Roman f1 (\() show 5927 344 moveto 384 /Times-Italic f1 (V) show 6188 344 moveto 384 /Times-Roman f1 (,) show 6320 344 moveto 384 /Times-Bold f1 (Parents) show 7577 344 moveto 384 /Times-Roman f1 (\() show 7709 344 moveto 384 /Times-Italic f1 (V) show 7970 344 moveto 384 /Times-Roman f1 (,) show 8100 344 moveto 384 /Times-Italic f1 (G) show 8390 344 moveto 384 /Times-Roman f1 (\)) show 3778 810 moveto 320 /Times-Italic f1 (V) show 4006 810 moveto 320 /Symbol f1 (\316) show 4192 810 moveto 320 /Times-Bold f1 (An) show 4607 810 moveto 320 /Times-Roman f1 (\() show 4729 810 moveto 320 /Times-Bold f1 (X) show 4971 810 moveto 320 /Symbol f1 (\310) show 5231 810 moveto 320 /Times-Bold f1 (Y) show 5474 810 moveto 320 /Symbol f1 (\310) show 5729 810 moveto 320 /Times-Bold f1 (Z) show 5940 810 moveto 320 /Times-Roman f1 (,) show 6027 810 moveto 320 /Times-Italic f1 (G) show 6274 810 moveto 320 /Times-Roman f1 (\)) show 4840 432 moveto 576 /Symbol f1 (\325) show 8520 344 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 247 630 :M f2_12 sf (Equation 8)S endp %%Page: 23 23 %%BeginPageSetup initializepage (peter; page: 23 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (23)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf 1.618 .162(where each )J f4_12 sf .525(g)A f4_10 sf 0 2 rm .535(V)A 0 -2 rm f0_12 sf 1.543 .154( is a positive function, i.e., the density function factors)J 95 123 :M (according to )S 158 123 :M f4_12 sf (G)S 167 123 :M f0_12 sf (. )S 173 114 9 9 rC gS 1.286 1 scale 134.556 123 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 95 153 :M f2_12 sf .524 .052(Theorem 1:)J 156 153 :M f0_12 sf .744 .074( The probability measure )J f4_12 sf .328(P)A f0_12 sf .55 .055( of a linear SEM )J f4_12 sf (L)S 388 153 :M f0_12 sf .604 .06( \(recursive or)J 95 171 :M 1.293 .129(non-recursive\) with jointly independent error terms satisfies the global)J 95 189 :M .097 .01(directed Markov property for the directed \(cyclic or acyclic\) graph )J f4_12 sf (G)S 428 189 :M f0_12 sf .098 .01( of )J f4_12 sf (L)S 451 189 :M f0_12 sf (,)S 95 207 :M .385 .038(i.e. if )J f2_12 sf (X)S 133 207 :M f0_12 sf .641 .064(, )J 140 207 :M f2_12 sf (Y)S 149 207 :M f0_12 sf .329 .033(, and )J f2_12 sf .26(Z)A f0_12 sf .472 .047( are disjoint sets of variables in )J f4_12 sf (G)S 350 207 :M f0_12 sf .588 .059( and )J 375 207 :M f2_12 sf (X)S 384 207 :M f0_12 sf .415 .042( is d-separated)J 95 225 :M (from )S f2_12 sf (Y)S 130 225 :M f0_12 sf ( given )S 163 225 :M f2_12 sf (Z)S f0_12 sf ( in )S f4_12 sf (G)S 195 225 :M f0_12 sf (, then )S 225 225 :M f2_12 sf (X)S 234 225 :M f0_12 sf ( and )S f2_12 sf (Y)S 266 225 :M f0_12 sf ( are independent given )S 378 225 :M f2_12 sf (Z)S f0_12 sf ( in )S f4_12 sf (P)S f0_12 sf (.)S 95 255 :M .327 .033(Proof. Let )J 148 255 :M f2_12 sf (Err)S 167 255 :M f0_12 sf <28>S 171 255 :M f2_12 sf (X)S 180 255 :M f0_12 sf .333 .033(\) be the set of error terms corresponding to a set of non-)J 95 273 :M .539 .054(error variables )J 170 273 :M f2_12 sf (X)S 179 273 :M f0_12 sf .58 .058(. In order to distinguish the density function for )J f2_12 sf (V)S 427 273 :M f0_12 sf .674 .067( from)J 95 291 :M 1.263 .126(the density function for the error terms we will use )J 361 291 :M f4_12 sf .219(f)A f2_10 sf 0 2 rm .475(V)A 0 -2 rm f0_12 sf 1.101 .11( to represent the)J 95 310 :M .991 .099(density function \(including marginal densities\) for the latter and )J f4_12 sf .167(f)A f2_10 sf 0 2 rm .39(Err)A 0 -2 rm 440 310 :M f0_12 sf 1.528 .153( to)J 95 329 :M .295 .03(represent the density function of the former. If )J 325 329 :M f2_12 sf (V)S 334 329 :M f0_12 sf .33 .033( is the set of variables in)J 95 347 :M f4_12 sf (G)S 104 347 :M f0_12 sf (, then by hypothesis,)S 198 368 152 30 rC 350 398 :M psb currentpoint pse 198 368 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4864 div 960 3 -1 roll exch div scale currentpoint translate 64 40 translate 56 344 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-Italic f1 (f) show 207 440 moveto 320 /Times-Bold f1 (Err) show 741 344 moveto 384 /Times-Roman f1 (\() show 881 344 moveto 384 /Times-Bold f1 (Err) show 1481 344 moveto 384 /Times-Roman f1 (\() show 1621 344 moveto 384 /Times-Bold f1 (V) show 1909 344 moveto 384 /Times-Roman f1 (\)) show 2039 344 moveto (\)) show 2272 344 moveto 384 /Symbol f1 (=) show 3608 344 moveto 384 /Times-Italic f1 (f) show 3759 440 moveto 320 /Times-Bold f1 (Err) show 4293 344 moveto 384 /Times-Roman f1 (\() show 4433 344 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) show 4642 344 moveto 384 /Times-Roman f1 (\)) show 2599 810 moveto 320 /Symbol f2 (e) show 2774 810 moveto 320 /Symbol f1 (\316) show 2961 810 moveto 320 /Times-Bold f1 (Err) show 3466 810 moveto 320 /Times-Roman f1 (\() show 3588 810 moveto 320 /Times-Bold f1 (V) show 3833 810 moveto 320 /Times-Roman f1 (\)) show 3026 432 moveto 576 /Symbol f1 (\325) show end pse gR gS 0 0 552 730 rC 247 419 :M f2_12 sf (Equation 9)S 95 449 :M f0_12 sf 1.359 .136(It is possible to integrate out the error terms not in )J 361 449 :M f2_12 sf (Err)S 380 449 :M f0_12 sf <28>S 384 449 :M f2_12 sf (An)S f0_12 sf <28>S 403 449 :M f2_12 sf (X)S 412 449 :M f0_12 sf (,)S f4_12 sf (G)S 424 449 :M f0_12 sf 1.351 .135(\)\) and)J 95 467 :M (obtain)S 172 488 204 30 rC 376 518 :M psb currentpoint pse 172 488 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6528 div 960 3 -1 roll exch div scale currentpoint translate 64 40 translate 56 344 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-Italic f1 (f) show 207 440 moveto 320 /Times-Bold f1 (Err) show 741 344 moveto 384 /Times-Roman f1 (\() show 881 344 moveto 384 /Times-Bold f1 (Err) show 1481 344 moveto 384 /Times-Roman f1 (\() show 1621 344 moveto 384 /Times-Bold f1 (An) show 2112 344 moveto 384 /Times-Roman f1 (\() show 2252 344 moveto 384 /Times-Bold f1 (X) show 2526 344 moveto 384 /Times-Roman f1 (,) show 2656 344 moveto 384 /Times-Italic f1 (G) show 2946 344 moveto 384 /Times-Roman f1 (\)) show 3076 344 moveto (\)) show 3206 344 moveto (\)) show 3439 344 moveto 384 /Symbol f1 (=) show 5262 344 moveto 384 /Times-Italic f1 (f) show 5413 440 moveto 320 /Times-Bold f1 (Err) show 5947 344 moveto 384 /Times-Roman f1 (\() show 6087 344 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) show 6296 344 moveto 384 /Times-Roman f1 (\)) show 3766 810 moveto 320 /Symbol f2 (e) show 3941 810 moveto 320 /Symbol f1 (\316) show 4128 810 moveto 320 /Times-Bold f1 (Err) show 4633 810 moveto 320 /Times-Roman f1 (\() show 4755 810 moveto 320 /Times-Bold f1 (An) show 5170 810 moveto 320 /Times-Roman f1 (\() show 5292 810 moveto 320 /Times-Bold f1 (X) show 5526 810 moveto 320 /Times-Roman f1 (,) show 5613 810 moveto 320 /Times-Italic f1 (G) show 5860 810 moveto 320 /Times-Roman f1 (\)) show 5973 810 moveto (\)) show 4680 432 moveto 576 /Symbol f1 (\325) show end pse gR gS 0 0 552 730 rC 244 539 :M f2_12 sf (Equation 10)S 95 569 :M f0_12 sf .273 .027(Because for each variable )J f4_12 sf .104(X)A f0_12 sf .091 .009( in )J f2_12 sf (V)S 255 569 :M f0_12 sf .341 .034(, )J 262 569 :M f4_12 sf .131(X)A f0_12 sf .238 .024( is a linear function of its parents in )J f4_12 sf (G)S 95 591 :M f0_12 sf .358 .036(plus a unique error term )J 216 591 :M f5_12 sf .128(e)A f4_10 sf 0 2 rm .149(X)A 0 -2 rm f0_12 sf .323 .032(, it follows that )J 305 591 :M f5_12 sf .162(e)A f4_10 sf 0 2 rm .187(X)A 0 -2 rm f0_12 sf .407 .041( is a linear function )J f4_12 sf .184(g)A f4_10 sf 0 2 rm .187(X)A 0 -2 rm f0_12 sf .171 .017( of )J 447 591 :M f4_12 sf (X)S 95 613 :M f0_12 sf .159 .016(and the parents of )J 185 613 :M f4_12 sf .108(X)A f0_12 sf .094 .009( in )J f4_12 sf (G)S 216 613 :M f0_12 sf .17 .017(. Hence )J 257 613 :M f2_12 sf (Err)S 276 613 :M f0_12 sf <28>S 280 613 :M f2_12 sf (An)S f0_12 sf <28>S 299 613 :M f2_12 sf (X)S 308 613 :M f0_12 sf (,)S f4_12 sf (G)S 320 613 :M f0_12 sf .134 .013(\)\) is a function of )J f2_12 sf .084(An)A f0_12 sf <28>S 426 613 :M f2_12 sf (X)S 435 613 :M f0_12 sf (,)S f4_12 sf (G)S 447 613 :M f0_12 sf (\).)S 95 631 :M .956 .096(Following Haavelmo \(1943\) it is possible to derive the density function)J endp %%Page: 24 24 %%BeginPageSetup initializepage (peter; page: 24 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (24)S gR gS 0 0 552 730 rC 95 108 :M f0_12 sf 1.97 .197(for the set of variables )J 220 108 :M f2_12 sf (An)S f0_12 sf <28>S 239 108 :M f2_12 sf (X)S 248 108 :M f0_12 sf (,)S f4_12 sf (G)S 260 108 :M f0_12 sf 1.968 .197(\) by replacing each )J f5_12 sf .657(e)A f4_10 sf 0 2 rm .762(X)A 0 -2 rm f0_12 sf .797 .08( in )J f4_12 sf .416(f)A f2_10 sf 0 2 rm .97(Err)A 0 -2 rm 417 108 :M f0_12 sf <28>S 421 108 :M f5_12 sf .506(e)A f4_10 sf 0 2 rm .587(X)A 0 -2 rm f0_12 sf 1.403 .14(\) by)J 95 130 :M f4_12 sf (g)S f4_10 sf 0 2 rm (X)S 0 -2 rm f0_12 sf <28>S 111 130 :M f4_12 sf (X)S f0_12 sf (,)S f2_12 sf (Parents)S f0_12 sf <28>S 164 130 :M f4_12 sf (X)S f0_12 sf (\)\) and multiplying by the absolute value of the Jacobean:)S 135 152 278 30 rC 413 182 :M psb currentpoint pse 135 152 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8896 div 960 3 -1 roll exch div scale currentpoint translate 64 40 translate 56 344 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-Italic f1 (f) show 206 440 moveto 320 /Times-Bold f1 (V) show 474 344 moveto 384 /Times-Roman f1 (\() show 614 344 moveto 384 /Times-Bold f1 (An) show 1105 344 moveto 384 /Times-Roman f1 (\() show 1245 344 moveto 384 /Times-Bold f1 (X) show 1519 344 moveto 384 /Times-Roman f1 (,) show 1649 344 moveto 384 /Times-Italic f1 (G) show 1939 344 moveto 384 /Times-Roman f1 (\)) show 2069 344 moveto (\)) show 2302 344 moveto 384 /Symbol f1 (=) show 3787 344 moveto 384 /Times-Italic f1 (f) show 3938 440 moveto 320 /Times-Bold f1 (Err) show 4472 344 moveto 384 /Times-Roman f1 (\() show 4617 344 moveto 384 /Times-Italic f1 (g) show 4821 440 moveto 320 ns (X) show 5069 344 moveto 384 /Times-Roman f1 (\() show 5234 344 moveto 384 /Times-Italic f1 (X) show 5485 344 moveto 384 /Times-Roman f1 (,) show 5617 344 moveto 384 /Times-Bold f1 (Parents) show 6874 344 moveto 384 /Times-Roman f1 (\() show 7039 344 moveto 384 /Times-Italic f1 (X) show 7290 344 moveto 384 /Times-Roman f1 (,) show 7420 344 moveto 384 /Times-Italic f1 (G) show 7710 344 moveto 384 /Times-Roman f1 (\)) show 7840 344 moveto (\)) show 7970 344 moveto (\)) show 2650 810 moveto 320 /Times-Italic f1 (X) show 2870 810 moveto 320 /Symbol f1 (\316) show 3056 810 moveto 320 /Times-Bold f1 (An) show 3471 810 moveto 320 /Times-Roman f1 (\() show 3593 810 moveto 320 /Times-Bold f1 (X) show 3827 810 moveto 320 /Times-Roman f1 (,) show 3914 810 moveto 320 /Times-Italic f1 (G) show 4161 810 moveto 320 /Times-Roman f1 (\)) show 3205 432 moveto 576 /Symbol f1 (\325) show 8169 344 moveto 384 ns (\264) show 8546 344 moveto 384 /Times-Italic f1 (J) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 8477 51 moveto 0 388 rlineto stroke 8769 51 moveto 0 388 rlineto stroke end pse gR gS 0 0 552 730 rC 244 203 :M f2_12 sf (Equation 11)S 95 233 :M f0_12 sf .412 .041(where )J f4_12 sf .109(J)A f0_12 sf .415 .041( is the Jacobian of the transformation. Because the transformation)J 95 251 :M .545 .054(is linear, the Jacobian is a constant. All of the terms in the multiplication)J 95 269 :M .833 .083(are non-negative because they are either a density function or a positive)J 95 287 :M .142 .014(constant. It follows from lemma 1 that if )J 295 287 :M f2_12 sf .195 .019(X )J 307 287 :M f0_12 sf .194 .019(and )J f2_12 sf .142 .014(Y )J 339 287 :M f0_12 sf .126 .013(are d-separated given )J 446 287 :M f2_12 sf (Z)S 95 305 :M f0_12 sf (then )S 119 305 :M f2_12 sf (X)S 128 305 :M f0_12 sf ( and )S f2_12 sf (Y)S 160 305 :M f0_12 sf ( are independent given )S 272 305 :M f2_12 sf (Z)S f0_12 sf (. )S 286 296 9 9 rC gS 1.286 1 scale 222.446 305 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 95 335 :M f2_12 sf 1.28 .128(Lemma 4:)J 150 335 :M f0_12 sf 1.554 .155( In a directed graph )J 256 335 :M f4_12 sf (G)S 265 335 :M f0_12 sf 1.484 .148( with vertices )J 340 335 :M f2_12 sf (V)S 349 335 :M f0_12 sf 1.137 .114(, if )J f2_12 sf (X)S 378 335 :M f0_12 sf .742 .074(, )J f2_12 sf (Y)S 395 335 :M f0_12 sf 1.741 .174(, and )J 426 335 :M f2_12 sf .679(Z)A f0_12 sf 1.152 .115( are)J 95 353 :M (disjoint subsets of )S 185 353 :M f2_12 sf (V)S 194 353 :M f0_12 sf (, and )S f2_12 sf (X)S 229 353 :M f0_12 sf ( is d-connected to )S 317 353 :M f2_12 sf (Y)S 326 353 :M f0_12 sf ( given )S 359 353 :M f2_12 sf (Z)S f0_12 sf ( in )S f4_12 sf (G)S 391 353 :M f0_12 sf (, then )S 421 353 :M f2_12 sf (X)S 430 353 :M f0_12 sf ( is d-)S 95 371 :M (connected to )S 159 371 :M f2_12 sf (Y)S 168 371 :M f0_12 sf ( given )S 201 371 :M f2_12 sf (Z)S f0_12 sf ( in an acyclic directed subgraph of )S f4_12 sf (G)S 386 371 :M f0_12 sf (.)S 95 401 :M .514 .051(Proof. I will use the sense of d-connection defined in Pearl \(1988\) which)J 95 419 :M .786 .079(Lauritzen et. al. \(1990\) proved equivalent to their sense of d-connection)J 95 437 :M .492 .049(for acyclic graphs. The proof of the equivalence given by Lauritzen et. al)J 95 455 :M 1.696 .17(can easily be extended to cyclic graphs. Vertex )J f4_12 sf .695(X)A f0_12 sf .588 .059( is a )J f2_12 sf .524(collider)A 420 455 :M f0_12 sf 2.051 .205( on an)J 95 473 :M .33 .033(acyclic undirected path )J f4_12 sf (U)S 220 473 :M f0_12 sf .44 .044( in directed graph )J 310 473 :M f4_12 sf (G)S 319 473 :M f0_12 sf .456 .046( if and only if there are two)J 95 491 :M .833 .083(adjacent edges on )J f4_12 sf (U)S 196 491 :M f0_12 sf .997 .1( directed into )J 267 491 :M f4_12 sf .297(X)A f0_12 sf .807 .081(. According to Pearl's definition, for)J 95 509 :M .395 .04(three disjoint sets )J 184 509 :M f2_12 sf (X)S 193 509 :M f0_12 sf .22 .022(, )J f2_12 sf (Y)S 208 509 :M f0_12 sf .516 .052(, and )J 236 509 :M f2_12 sf .228(Z)A f0_12 sf .142 .014(, )J f2_12 sf (X)S 259 509 :M f0_12 sf .376 .038( and )J f2_12 sf (Y)S 292 509 :M f0_12 sf .538 .054( are )J 315 509 :M f2_12 sf (d-separated)S 376 509 :M f0_12 sf .496 .05( given )J 410 509 :M f2_12 sf .323(Z)A f0_12 sf .258 .026( in )J f4_12 sf (G)S 443 509 :M f0_12 sf .538 .054( if)J 95 527 :M .191 .019(and only if there is no acyclic undirected path )J 320 527 :M f4_12 sf (U)S 329 527 :M f0_12 sf .196 .02( from a member of )J f2_12 sf (X)S 432 527 :M f0_12 sf .243 .024( to a)J 95 545 :M 1.23 .123(member of )J 154 545 :M f2_12 sf (Y)S 163 545 :M f0_12 sf 1.09 .109( such that every non-collider on )J f4_12 sf (U)S 337 545 :M f0_12 sf 1.427 .143( is not in )J 389 545 :M f2_12 sf .468(Z)A f0_12 sf 1.033 .103(, and every)J 95 563 :M (collider on )S 150 563 :M f4_12 sf (U)S 159 563 :M f0_12 sf ( has a descendant in )S 261 563 :M f2_12 sf (Z)S f0_12 sf (. For three disjoint sets )S 382 563 :M f2_12 sf (X)S 391 563 :M f0_12 sf (, )S f2_12 sf (Y)S 406 563 :M f0_12 sf (, and )S f2_12 sf (Z)S f0_12 sf (, )S f2_12 sf (X)S 95 581 :M f0_12 sf 1.025 .103(and )J 117 581 :M f2_12 sf (Y)S 126 581 :M f0_12 sf .266 .027( are )J f2_12 sf .2(d-connected)A 211 581 :M f0_12 sf 1.025 .103( given )J 247 581 :M f2_12 sf .667(Z)A f0_12 sf .532 .053( in )J f4_12 sf (G)S 282 581 :M f0_12 sf .896 .09( if and only if )J f2_12 sf (X)S 365 581 :M f0_12 sf 1.111 .111( and )J 391 581 :M f2_12 sf (Y)S 400 581 :M f0_12 sf 1.052 .105( are not )J 443 581 :M f2_12 sf (d-)S 95 599 :M (separated)S 145 599 :M f0_12 sf ( given )S 178 599 :M f2_12 sf (Z)S endp %%Page: 25 25 %%BeginPageSetup initializepage (peter; page: 25 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (25)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .104 .01(Suppose that )J 160 104 :M f4_12 sf (U)S 169 104 :M f0_12 sf .108 .011( is an undirected path that d-connects )J 352 104 :M f4_12 sf .079(X)A f0_12 sf .105 .011( and )J 383 104 :M f4_12 sf (Y)S 390 104 :M f0_12 sf .128 .013( given )J 423 104 :M f2_12 sf (Z)S f0_12 sf .088 .009(, and)J 95 122 :M f4_12 sf .462(C)A f0_12 sf .663 .066( is a collider on )J 185 122 :M f4_12 sf (U)S 194 122 :M f0_12 sf .785 .078(. Let )J 221 122 :M f4_12 sf (length)S 251 122 :M f0_12 sf <28>S 255 122 :M f4_12 sf .526(C)A f0_12 sf .197(,)A f2_12 sf .526(Z)A f0_12 sf .546 .055(\) be 0 if )J 319 122 :M f4_12 sf .458(C)A f0_12 sf .684 .068( is a member of )J 410 122 :M f2_12 sf .377(Z)A f0_12 sf .587 .059(, or the)J 95 140 :M .995 .099(length of a shortest directed path from )J f4_12 sf .47(C)A f0_12 sf .703 .07( to a member of )J f2_12 sf .47(Z)A f0_12 sf .587 .059(. Let )J 419 140 :M f4_12 sf (size)S 437 140 :M f0_12 sf <28>S 441 140 :M f4_12 sf (U)S 450 140 :M f0_12 sf <29>S 95 158 :M .453 .045(equal the number of collider on )J f4_12 sf (U)S 262 158 :M f0_12 sf .503 .05( plus the sum over all colliders )J 418 158 :M f4_12 sf .339(C)A f0_12 sf .317 .032( on )J f4_12 sf (U)S 95 176 :M f0_12 sf .793 .079(of )J 109 176 :M f4_12 sf (length)S 139 176 :M f0_12 sf <28>S 143 176 :M f4_12 sf .263(C)A f0_12 sf .098(,)A f2_12 sf .263(Z)A f0_12 sf .273 .027(\). )J 173 176 :M f4_12 sf (U)S 182 176 :M f0_12 sf .816 .082( is a )J 206 176 :M f2_12 sf .453 .045(minimal path)J 276 176 :M f0_12 sf .621 .062( that d-connects )J 358 176 :M f4_12 sf .374(X)A f0_12 sf .476 .048( and )J f4_12 sf (Y)S 397 176 :M f0_12 sf .732 .073( given )J 432 176 :M f2_12 sf .357(Z)A f0_12 sf .457 .046(, if)J 95 194 :M .049 .005(there is no other path )J f4_12 sf (U)S 208 194 :M f0_12 sf .058 .006(' that d-connects )J f4_12 sf (X)S f0_12 sf .033 .003( and )J 320 194 :M f4_12 sf (Y)S 327 194 :M f0_12 sf .059 .006( given )J 360 194 :M f2_12 sf (Z)S f0_12 sf .037 .004( such that )J f4_12 sf .018(size)A 435 194 :M f0_12 sf <28>S 439 194 :M f4_12 sf (U)S 448 194 :M f0_12 sf ('\))S 95 212 :M 1.286 .129(< )J 107 212 :M f4_12 sf (size)S 125 212 :M f0_12 sf <28>S 129 212 :M f4_12 sf (U)S 138 212 :M f0_12 sf .98 .098(\). If there is a path that d-connects )J f4_12 sf .508(X)A f0_12 sf .646 .065( and )J f4_12 sf (Y)S 356 212 :M f0_12 sf 1.088 .109( given )J 392 212 :M f2_12 sf .589(Z)A f0_12 sf .892 .089( there is at)J 95 230 :M (least one minimal path that d-connects )S 283 230 :M f4_12 sf (X)S f0_12 sf ( and )S f4_12 sf (Y)S 320 230 :M f0_12 sf ( given )S 353 230 :M f2_12 sf (Z.)S 95 260 :M f0_12 sf .034 .003(Suppose )J 139 260 :M f2_12 sf (X)S 148 260 :M f0_12 sf .036 .004( is d-connected to )J f2_12 sf (Y)S 245 260 :M f0_12 sf .045 .005( given )J 278 260 :M f2_12 sf (Z.)S f0_12 sf .038 .004( Then for some )J 366 260 :M f4_12 sf (X)S f0_12 sf .024 .002( in )J f2_12 sf (X)S 397 260 :M f0_12 sf .036 .004( and )J f4_12 sf (Y)S 427 260 :M f0_12 sf .031 .003( in )J f2_12 sf (Y)S 451 260 :M f0_12 sf (,)S 95 278 :M f4_12 sf .243(X)A f0_12 sf .504 .05( is d-connected to )J 194 278 :M f4_12 sf (Y)S 201 278 :M f0_12 sf .436 .044( given )J f2_12 sf .288(Z)A f0_12 sf .573 .057( by some minimal path )J 360 278 :M f4_12 sf (U)S 369 278 :M f0_12 sf .688 .069( in )J 386 278 :M f4_12 sf (G)S 395 278 :M f0_12 sf .552 .055(. First I will)J 95 296 :M .151 .015(show that no shortest acyclic directed path )J 304 296 :M f4_12 sf (D)S 313 298 :M f4_10 sf (i)S 316 296 :M f0_12 sf .15 .015( from a collider )J f4_12 sf .089(C)A f4_10 sf 0 2 rm (i)S 0 -2 rm 405 296 :M f4_12 sf .074 .007( )J f0_12 sf .339 .034(on )J 424 296 :M f4_12 sf (U)S 433 296 :M f0_12 sf .203 .02( to a)J 95 315 :M .573 .057(member of )J 152 315 :M f2_12 sf .224(Z)A f0_12 sf .464 .046( intersects )J 213 315 :M f4_12 sf (U)S 222 315 :M f0_12 sf .498 .05( except at )J f4_12 sf .334(C)A f4_10 sf 0 2 rm (i)S 0 -2 rm 284 315 :M f4_12 sf .107(.)A f0_12 sf .549 .055( Suppose this is false. I will show)J 95 334 :M .215 .022(that it follows that there is a path )J 259 334 :M f4_12 sf (U)S 268 334 :M f0_12 sf .199 .02(' that d-connects )J 349 334 :M f4_12 sf .149(X)A f0_12 sf .198 .02( and )J 380 334 :M f4_12 sf (Y)S 387 334 :M f0_12 sf .24 .024( given )J 421 334 :M f2_12 sf .076(Z)A f0_12 sf .169 .017( such)J 95 352 :M .279 .028(that )J f4_12 sf .12(size)A 134 352 :M f0_12 sf <28>S 138 352 :M f4_12 sf (U)S 147 352 :M f0_12 sf .647 .065('\) < )J 168 352 :M f4_12 sf (size)S 186 352 :M f0_12 sf <28>S 190 352 :M f4_12 sf (U)S 199 352 :M f0_12 sf .496 .05(\), contrary to the assumption that )J 366 352 :M f4_12 sf (U)S 375 352 :M f0_12 sf .518 .052( is minimal. See)J 95 370 :M (figure 9.)S 95 391 420 151 rC 99 434 :M f4_12 sf .092 .009(X W C W Y)J 15 156 204 135 432 @k 111 433 -2 2 122 431 2 111 431 @a 15 156 204 190 433 @k 163 434 -2 2 177 432 2 163 432 @a 15 -24 24 213 433 @k 224 434 -2 2 240 432 2 224 432 @a 15 156 204 284 432 @k 262 433 -2 2 271 431 2 262 431 @a 90 180 44 42 230.5 439.5 @n 10 66 114 249 439 @k -1 -1 249 452 1 1 248 447 @b 0 90 36 24 230.5 448.5 @n 10 -114 -66 147 422 @k -1 -1 147 415 1 1 146 408 @b 148 439 :M f4_10 sf (X)S 249 439 :M (Y)S 202 438 :M (i)S 10 -114 -66 151 475 @k -1 -1 151 468 1 1 150 444 @b 145 487 :M f4_12 sf (Z)S -90 0 110 56 197.5 422.5 @n 180 270 112 26 202.5 407.5 @n 107 506 -2 2 147 504 2 107 504 @a 109 528 -1 1 148 527 1 109 527 @a 157 509 :M (U)S 162 532 :M -.663(D )A 170 536 :M f4_10 sf (i)S 174 529 :M f0_12 sf (')S 317 435 :M f4_12 sf .092 .009(X W C W Y)J 15 156 204 353 433 @k 329 434 -2 2 340 432 2 329 432 @a 10 156 204 407 434 @k 384 435 -1 1 399 434 1 384 434 @a 10 -24 24 431 434 @k 438 435 -1 1 455 434 1 438 434 @a 15 156 204 498 433 @k 480 434 -2 2 485 432 2 480 432 @a 15 -114 -66 364 423 @k -2 -2 364 411 2 2 362 404 @b 366 441 :M f4_10 sf (X)S 467 440 :M (Y)S 420 439 :M (i)S 10 -114 -66 369 476 @k -1 -1 369 469 1 1 368 445 @b 363 488 :M f4_12 sf (Z)S 2 lw -90 0 98 56 419 423 @n 180 270 114 18 420 404 @n 325 507 -2 2 365 505 2 325 505 @a 375 510 :M (U)S 389 508 :M f0_12 sf (')S 1 lw 90 180 44 42 447.5 441.5 @n 10 66 114 466 441 @k -1 -1 466 454 1 1 465 449 @b 0 90 36 24 447.5 450.5 @n gR gS 0 0 552 730 rC 253 563 :M f2_12 sf (Figure )S 290 563 :M (7)S 95 593 :M f0_12 sf .187 .019(Form the path )J f4_12 sf (U)S 175 593 :M f0_12 sf .215 .021(' in the following way. If )J 299 593 :M f4_12 sf (D)S 308 595 :M f4_10 sf (i)S 311 593 :M f0_12 sf .199 .02( intersects )J 363 593 :M f4_12 sf (U)S 372 593 :M f0_12 sf .216 .022( at a vertex other)J 95 612 :M (than )S 119 612 :M f4_12 sf (C)S f4_10 sf 0 2 rm (i )S 0 -2 rm f0_12 sf (then let )S 171 612 :M f4_12 sf (W)S 181 614 :M f4_10 sf (X)S f0_12 sf 0 -2 rm .011 .001( be the vertex on )J 0 2 rm f4_12 sf 0 -2 rm (D)S 0 2 rm 279 614 :M f4_10 sf (i)S 282 612 :M f0_12 sf ( and )S f4_12 sf (U)S 314 612 :M f0_12 sf .012 .001( that is closest to )J f4_12 sf (X)S f0_12 sf ( on )S f4_12 sf (U)S 431 612 :M f0_12 sf (, and)S 95 631 :M f4_12 sf (W)S 105 633 :M f4_10 sf (Y)S 111 631 :M f0_12 sf .555 .055( be the vertex on )J f4_12 sf (D)S 207 633 :M f4_10 sf (i)S 210 631 :M f0_12 sf .694 .069( and )J 235 631 :M f4_12 sf (U)S 244 631 :M f0_12 sf .551 .055( that is closest to )J f4_12 sf (Y)S 338 631 :M f0_12 sf .469 .047( on )J f4_12 sf (U)S 366 631 :M f0_12 sf .463 .046(. Suppose without)J endp %%Page: 26 26 %%BeginPageSetup initializepage (peter; page: 26 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (26)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf -.003(loss of generality that )A 202 104 :M f4_12 sf (W)S 212 106 :M f4_10 sf (X)S f0_12 sf 0 -2 rm ( is after )S 0 2 rm 257 104 :M f4_12 sf (W)S 267 106 :M f4_10 sf (Y)S 273 104 :M f0_12 sf ( on )S 291 104 :M f4_12 sf (D)S 300 106 :M f4_10 sf (i)S 303 104 :M f0_12 sf (. Let )S 328 104 :M f4_12 sf (U)S 337 104 :M f0_12 sf -.003(' be the concatenation of)A 95 123 :M f4_12 sf (U)S 104 123 :M f0_12 sf <28>S 108 123 :M f4_12 sf (X)S f0_12 sf (,)S f4_12 sf (W)S 128 125 :M f4_10 sf .267(X)A f0_12 sf 0 -2 rm .364 .036(\), )J 0 2 rm 145 123 :M f4_12 sf (D)S 154 125 :M f4_10 sf (i)S 157 123 :M f0_12 sf <28>S 161 123 :M f4_12 sf (W)S 171 125 :M f4_10 sf (Y)S 177 123 :M f0_12 sf (,)S f4_12 sf (W)S 190 125 :M f4_10 sf .191(X)A f0_12 sf 0 -2 rm .351 .035(\), and )J 0 2 rm f4_12 sf 0 -2 rm (U)S 0 2 rm 236 123 :M f0_12 sf <28>S 240 123 :M f4_12 sf (W)S 250 125 :M f4_10 sf (Y)S 256 123 :M f0_12 sf (,)S f4_12 sf (Y)S 266 123 :M f0_12 sf .503 .05(\) \(where )J 311 123 :M f4_12 sf (U)S 320 123 :M f0_12 sf <28>S 324 123 :M f4_12 sf (X)S f0_12 sf (,)S f4_12 sf (W)S 344 125 :M f4_10 sf .12(X)A f0_12 sf 0 -2 rm .398 .04(\) denotes the subpath)J 0 2 rm 95 142 :M .145 .014(of )J f4_12 sf (U)S 117 142 :M f0_12 sf .224 .022( between )J 164 142 :M f4_12 sf .113(X)A f0_12 sf .144 .014( and )J f4_12 sf (W)S 204 144 :M f4_10 sf .115(X)A f0_12 sf 0 -2 rm .223 .022(.\) It is now easy to show that )J 0 2 rm 357 142 :M f4_12 sf (U)S 366 142 :M f0_12 sf .196 .02(' d-connects )J 427 142 :M f4_12 sf .083(X)A f0_12 sf .177 .018( and)J 95 161 :M f4_12 sf (Y)S 102 161 :M f0_12 sf .91 .091( given )J f2_12 sf .602(Z)A f0_12 sf .792 .079(, and )J 174 161 :M f4_12 sf (size)S 192 161 :M f0_12 sf <28>S 196 161 :M f4_12 sf (U)S 205 161 :M f0_12 sf .516 .052('\) < )J f4_12 sf .409(size)A 244 161 :M f0_12 sf <28>S 248 161 :M f4_12 sf (U)S 257 161 :M f0_12 sf .869 .087(\) because )J 308 161 :M f4_12 sf (U)S 317 161 :M f0_12 sf .752 .075(' contains no more colliders)J 95 179 :M .586 .059(than )J 120 179 :M f4_12 sf (U)S 129 179 :M f0_12 sf .516 .052( and a shortest directed path from )J f4_12 sf (W)S 308 181 :M f4_10 sf .25(X)A f0_12 sf 0 -2 rm .49 .049( to a member of )J 0 2 rm f2_12 sf 0 -2 rm .328(Z)A 0 2 rm f0_12 sf 0 -2 rm .646 .065( is shorter)J 0 2 rm 95 198 :M (than )S 119 198 :M f4_12 sf (D)S 128 200 :M f4_10 sf (i)S 131 198 :M f0_12 sf (. Hence )S 171 198 :M f4_12 sf (U)S 180 198 :M f0_12 sf ( is not minimal, contrary to the assumption.)S 95 229 :M .748 .075(Next, I will show that if )J 219 229 :M f4_12 sf (U)S 228 229 :M f0_12 sf .725 .073( is minimal, then it does not contain a pair of)J 95 247 :M .133 .013(colliders )J f4_12 sf (C)S f0_12 sf .056 .006( and )J f4_12 sf .064 .006(D )J 183 247 :M f0_12 sf .084 .008(such that a shortest directed path from )J 369 247 :M f4_12 sf (C)S f0_12 sf .088 .009( to a member of)J 95 265 :M f2_12 sf .586(Z)A f0_12 sf 1.194 .119( intersects a shortest path from )J 265 265 :M f4_12 sf 1.753 .175(D )J 279 265 :M f0_12 sf 1.151 .115(to a member of )J f2_12 sf .667(Z)A f0_12 sf 1.413 .141(. Suppose this is)J 95 283 :M (false. See figure 10.)S 95 304 457 175 rC 99 316 :M f4_12 sf .035 .004(X C M D Y)J 15 156 204 150 311 @k 115 312 -2 2 137 310 2 115 310 @a 15 -24 24 169 312 @k 180 313 -2 2 198 311 2 180 311 @a 15 156 204 258 312 @k 218 313 -2 2 245 311 2 218 311 @a 15 -24 24 278 312 @k 289 313 -2 2 313 311 2 289 311 @a 10 197 245 210 361 @k 165 324 -1 1 204 356 1 165 323 @a 10 -61 -13 217 361 @k -1 -1 223 357 1 1 267 322 @b 10 -114 -66 215 429 @k -1 -1 215 422 1 1 214 380 @b 209 375 :M (R)S 209 445 :M (Z)S 15 156 204 226 468 @k 197 469 -2 2 213 467 2 197 467 @a 178 473 :M (U)S 335 316 :M .035 .004(X C M D Y)J 15 156 204 386 311 @k 351 312 -2 2 373 310 2 351 310 @a 10 -24 24 405 312 @k 412 313 -1 1 434 312 1 412 312 @a 10 156 204 494 312 @k 455 313 -1 1 486 312 1 455 312 @a 15 -24 24 514 312 @k 525 313 -2 2 549 311 2 525 311 @a 15 197 245 446 361 @k 400 324 -2 2 436 351 2 400 322 @a 15 -61 -13 453 361 @k -2 -2 463 354 2 2 502 321 @b 10 -114 -66 451 429 @k -1 -1 451 422 1 1 450 380 @b 445 375 :M (R)S 445 445 :M (Z)S 15 156 204 462 468 @k 433 469 -2 2 449 467 2 433 467 @a 414 473 :M (U)S 425 470 :M f0_12 sf (')S gR gS 0 0 552 730 rC 253 500 :M f2_12 sf (Figure )S 290 500 :M (8)S 95 530 :M f0_12 sf .769 .077(Let )J 115 530 :M f4_12 sf (D)S 124 532 :M f0_10 sf .2(1)A f0_12 sf 0 -2 rm .644 .064( be a shortest directed acyclic path from )J 0 2 rm 332 530 :M f4_12 sf .46(C)A f0_12 sf .698 .07( to a member of )J 424 530 :M f2_12 sf .276(Z)A f0_12 sf .517 .052( that)J 95 549 :M .466 .047(intersects )J f4_12 sf (D)S 153 551 :M f0_10 sf .205(2)A f0_12 sf 0 -2 rm .683 .068(, a shortest directed acyclic path from )J 0 2 rm f4_12 sf 0 -2 rm (D)S 0 2 rm 358 549 :M f0_12 sf .887 .089( to a member of )J 443 549 :M f2_12 sf (Z)S f0_12 sf (.)S 95 568 :M .902 .09(Let the vertex on )J 185 568 :M f4_12 sf (D)S 194 570 :M f0_10 sf .344(1)A f0_12 sf 0 -2 rm .857 .086( closest to )J 0 2 rm 254 568 :M f4_12 sf .578(C)A f0_12 sf .78 .078( that is also on )J f4_12 sf (D)S 349 570 :M f0_10 sf .576(2)A f0_12 sf 0 -2 rm .868 .087( be )J 0 2 rm 374 568 :M f4_12 sf .46(R)A f0_12 sf .603 .06(. Let )J f4_12 sf (U)S 417 568 :M f0_12 sf .952 .095(' be the)J 95 587 :M .083 .008(concatenation of )J 179 587 :M f4_12 sf (U)S 188 587 :M f0_12 sf <28>S 192 587 :M f4_12 sf (X)S f0_12 sf (,)S f4_12 sf (C)S f0_12 sf (\), )S f4_12 sf (D)S 229 589 :M f0_10 sf (1)S f0_12 sf 0 -2 rm <28>S 0 2 rm 238 587 :M f4_12 sf (C)S f0_12 sf (,)S f4_12 sf (R)S f0_12 sf (\), )S f4_12 sf (D)S 275 589 :M f0_10 sf (2)S f0_12 sf 0 -2 rm <28>S 0 2 rm 284 587 :M f4_12 sf (C)S f0_12 sf (,)S f4_12 sf (R)S f0_12 sf .058 .006(\), and )J f4_12 sf (U)S 341 587 :M f0_12 sf <28>S 345 587 :M f4_12 sf (Y)S 352 587 :M f0_12 sf (,)S f4_12 sf (D)S 364 587 :M f0_12 sf .108 .011(\). It is now easy to)J 95 606 :M .108 .011(show that)J f4_12 sf (U)S 150 606 :M f0_12 sf .152 .015(' d-connects )J 211 606 :M f4_12 sf .095(X)A f0_12 sf .121 .012( and )J f4_12 sf (Y)S 248 606 :M f0_12 sf .187 .019( given )J 282 606 :M f2_12 sf .079(Z)A f0_12 sf .092 .009( and )J f4_12 sf .059(size)A 331 606 :M f0_12 sf <28>S 335 606 :M f4_12 sf (U)S 344 606 :M f0_12 sf .202 .02('\) < )J 364 606 :M f4_12 sf (size)S 382 606 :M f0_12 sf <28>S 386 606 :M f4_12 sf (U)S 395 606 :M f0_12 sf .137 .014(\) because )J f4_12 sf (U)S 452 606 :M f0_12 sf (')S endp %%Page: 27 27 %%BeginPageSetup initializepage (peter; page: 27 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (27)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .552 .055(contains fewer colliders than )J f4_12 sf (U)S 249 104 :M f0_12 sf .759 .076(. Hence )J 291 104 :M f4_12 sf (U)S 300 104 :M f0_12 sf .676 .068( is not minimal, contrary to the)J 95 122 :M (assumption.)S 95 152 :M .253 .025(For each collider )J f4_12 sf .115(C)A f0_12 sf .197 .02( on a minimal path )J 283 152 :M f4_12 sf (U)S 292 152 :M f0_12 sf .217 .022( that d-connects )J 372 152 :M f4_12 sf .131(X)A f0_12 sf .166 .017( and )J f4_12 sf (Y)S 409 152 :M f0_12 sf .256 .026( given )J 443 152 :M f2_12 sf (Z)S f0_12 sf (,)S 95 170 :M 1.378 .138(a shortest directed path from )J f4_12 sf .629(C)A f0_12 sf .955 .095( to a member of )J 340 170 :M f2_12 sf .536(Z)A f0_12 sf 1.004 .1( does not intersect )J f4_12 sf (U)S 95 188 :M f0_12 sf .294 .029(except at )J 142 188 :M f4_12 sf .122(C)A f0_12 sf .264 .026(, and does not intersect a shortest directed path from any other)J 95 206 :M .318 .032(collider )J 136 206 :M f4_12 sf (D)S 145 206 :M f0_12 sf .304 .03( to a member of )J f2_12 sf .203(Z)A f0_12 sf .421 .042(. It follows that the subgraph consisting of )J 445 206 :M f4_12 sf (U)S 95 224 :M f0_12 sf .486 .049(and a shortest directed acyclic path from each collider on )J 380 224 :M f4_12 sf (U)S 389 224 :M f0_12 sf .561 .056( to a member)J 95 242 :M (of )S 108 242 :M f2_12 sf (Z)S f0_12 sf ( is acyclic. )S 170 233 9 9 rC gS 1.286 1 scale 132.223 242 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 95 272 :M f2_12 sf .553 .055(Theorem 2:)J 156 272 :M f0_12 sf .795 .08( In a linear SEM )J 244 272 :M f4_12 sf (L)S 251 272 :M f0_12 sf .63 .063( with jointly independent error terms and)J 95 290 :M (directed \(cyclic or acyclic\) graph )S 257 290 :M f4_12 sf (G)S 266 290 :M f0_12 sf ( containing disjoint sets of variables )S 442 290 :M f2_12 sf (X)S 451 290 :M f0_12 sf (,)S 95 308 :M f2_12 sf (Y)S 104 308 :M f0_12 sf .488 .049( and )J f2_12 sf .419(Z)A f0_12 sf .356 .036(, if )J 154 308 :M f2_12 sf (X)S 163 308 :M f0_12 sf .387 .039( is not d-separated from )J f2_12 sf (Y)S 292 308 :M f0_12 sf .479 .048( given )J 326 308 :M f2_12 sf .158(Z)A f0_12 sf .333 .033( then)J 359 308 :M f4_12 sf (L)S 366 308 :M f0_12 sf .43 .043( does not linearly)J 95 326 :M (entail that )S 146 326 :M f2_12 sf (X)S 155 326 :M f0_12 sf ( is independent of )S 244 326 :M f2_12 sf (Y)S 253 326 :M f0_12 sf ( given )S 286 326 :M f2_12 sf (Z)S f0_12 sf (.)S 95 356 :M .272 .027(Proof. Suppose then that )J 218 356 :M f2_12 sf (X)S 227 356 :M f0_12 sf .27 .027( is not d-separated from )J f2_12 sf (Y)S 355 356 :M f0_12 sf .335 .033( given )J 389 356 :M f2_12 sf .109(Z)A f0_12 sf .262 .026(. By lemma)J 95 374 :M .021 .002(4, if )J f2_12 sf (X)S 126 374 :M f0_12 sf .025 .003( is not d-separated from )J 244 374 :M f2_12 sf (Y)S 253 374 :M f0_12 sf .029 .003( given )J 286 374 :M f2_12 sf (Z)S f0_12 sf .025 .003( in a cyclic graph )J 380 374 :M f4_12 sf (G,)S 392 374 :M f0_12 sf .026 .003( then there is)J 95 392 :M 1.153 .115(some acyclic subgraph )J f4_12 sf (G)S 222 392 :M f0_12 sf 1.795 .179(' of )J 245 392 :M f4_12 sf (G)S 254 392 :M f0_12 sf 1.367 .137( in which )J f2_12 sf (X)S 317 392 :M f0_12 sf 1.455 .146( is not d-separated from )J 445 392 :M f2_12 sf (Y)S 95 410 :M f0_12 sf .391 .039(given )J 126 410 :M f2_12 sf .182(Z)A f0_12 sf .363 .036(. Geiger and Pearl \(1988\) have shown that if )J f2_12 sf (X)S 364 410 :M f0_12 sf .367 .037( is not d-separated)J 95 428 :M .508 .051(from )J 121 428 :M f2_12 sf (Y)S 130 428 :M f0_12 sf .547 .055( given )J 165 428 :M f2_12 sf .202(Z)A f0_12 sf .443 .044( in a DAG, then there is some distribution represented by)J 95 446 :M 1.126 .113(the DAG in which )J 193 446 :M f2_12 sf (X)S 202 446 :M f0_12 sf 1.094 .109( is not independent of )J 317 446 :M f2_12 sf (Y)S 326 446 :M f0_12 sf 1.212 .121( given )J 362 446 :M f2_12 sf .589(Z)A f0_12 sf 1.021 .102(, and it has been)J 95 464 :M .469 .047(shown \(Spirtes, Glymour and Scheines, 1993\) that there is in particular a)J 95 482 :M .371 .037(linear normal distribution )J 223 482 :M f4_12 sf .277(P)A f0_12 sf .461 .046( in which )J 280 482 :M f2_12 sf (X)S 289 482 :M f0_12 sf .481 .048( is not independent of )J 400 482 :M f2_12 sf (Y)S 409 482 :M f0_12 sf .383 .038( given )J f2_12 sf .254(Z)A f0_12 sf (.)S 95 500 :M .189 .019(If )J f4_12 sf .164(P)A f0_12 sf .411 .041( satisfies the global directed Markov property for )J 357 500 :M f4_12 sf .152(G')A f0_12 sf .376 .038( it also satisfies it)J 95 518 :M .651 .065(for )J 113 518 :M f4_12 sf (G)S 122 518 :M f0_12 sf .504 .05( because every d-connecting path in )J f4_12 sf (G)S 311 518 :M f0_12 sf .54 .054(' is a d-connecting path in )J f4_12 sf (G)S 451 518 :M f0_12 sf (.)S 95 536 :M .306 .031(Hence there is some linear normal distribution represented by )J 399 536 :M f4_12 sf (G)S 408 536 :M f0_12 sf .367 .037( in which)J 95 554 :M f2_12 sf (X)S 104 554 :M f0_12 sf ( is not independent of )S 211 554 :M f2_12 sf (Y)S 220 554 :M f0_12 sf ( given )S 253 554 :M f2_12 sf (Z)S f0_12 sf (. )S 270 545 9 9 rC gS 1.286 1 scale 210.001 554 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 95 584 :M f2_12 sf .553 .055(Theorem 3:)J 156 584 :M f0_12 sf .795 .08( In a linear SEM )J 244 584 :M f4_12 sf (L)S 251 584 :M f0_12 sf .63 .063( with jointly independent error terms and)J 95 606 :M .687 .069(\(cyclic or acyclic\) directed graph )J 263 606 :M f4_12 sf (G)S 272 606 :M f0_12 sf .744 .074( containing )J 331 606 :M f4_12 sf .423(X)A f0_12 sf .288 .029(, )J f4_12 sf (Y)S 352 606 :M f0_12 sf .961 .096( and )J 378 606 :M f2_12 sf .438(Z)A f0_12 sf .749 .075(, where )J f4_12 sf .401(X)A f0_12 sf .149 .015( )J f1_12 sf S 444 606 :M f0_12 sf S f4_12 sf (Y)S endp %%Page: 28 28 %%BeginPageSetup initializepage (peter; page: 28 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (28)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .037 .004(and )J f2_12 sf (Z)S f0_12 sf .038 .004( does not contain )J 208 104 :M f4_12 sf (X)S f0_12 sf .026 .003( or )J f4_12 sf (Y)S 238 104 :M f0_12 sf (, )S f4_12 sf (X)S f0_12 sf .043 .004( is d-separated from )J 351 104 :M f4_12 sf (Y)S 358 104 :M f0_12 sf .045 .004( given )J 391 104 :M f2_12 sf (Z)S f0_12 sf .037 .004( if and only)J 95 126 :M f4_12 sf (L)S 102 126 :M f0_12 sf ( linearly entails that )S 200 126 :M f5_12 sf (r)S 207 128 :M f4_10 sf (XY)S 219 128 :M f0_10 sf (.)S 222 128 :M f2_10 sf (Z)S 229 126 :M f0_12 sf ( = 0.)S 95 160 :M f2_12 sf (Proof.)S 127 160 :M f0_12 sf .808 .081( \(This proof for cyclic or acyclic graphs is based on the proof for)J 95 178 :M .127 .013(acyclic graphs in Verma and Pearl, 1990.\) Let )J 321 178 :M f4_12 sf (L)S 328 178 :M f0_12 sf .14 .014(' be a linear SEM with the)J 95 196 :M .09 .009(same directed graph )J f4_12 sf (G)S 204 196 :M f0_12 sf .121 .012( and that is the same as )J 319 196 :M f4_12 sf (L)S 326 196 :M f0_12 sf .1 .01( except that the exogenous)J 95 214 :M .786 .079(variables are jointly normally distributed with the same variances as the)J 95 232 :M .297 .03(corresponding variables in )J f4_12 sf (L)S 234 232 :M f0_12 sf .405 .04(. By theorems 1 and 2, )J f4_12 sf (L)S 355 232 :M f0_12 sf .342 .034(' linearly entails that)J 95 250 :M f4_12 sf .07(X)A f0_12 sf .146 .015( is independent of )J 192 250 :M f4_12 sf (Y)S 199 250 :M f0_12 sf .176 .018( given )J 233 250 :M f2_12 sf .09(Z)A f0_12 sf .117 .012( if and only if )J f4_12 sf .083(X)A f0_12 sf .186 .019( is d-separated from )J 417 250 :M f4_12 sf (Y)S 424 250 :M f0_12 sf .152 .015( given)J 95 268 :M f2_12 sf .233(Z)A f0_12 sf .186 .019( in )J f4_12 sf (G)S 128 268 :M f0_12 sf .295 .03(. Hence for all values of the linear coefficients and all joint normal)J 95 286 :M 2.794 .279(distributions over the exogenous variables in which the exogenous)J 95 308 :M .174 .017(variables have positive variance and )J f5_12 sf (r)S 280 310 :M f4_10 sf (XY)S 292 310 :M f0_10 sf (.)S 295 310 :M f2_10 sf (Z)S 302 308 :M f0_12 sf .224 .022( exists, )J 339 308 :M f5_12 sf (r)S 346 310 :M f4_10 sf (XY)S 358 310 :M f0_10 sf (.)S 361 310 :M f2_10 sf (Z)S 368 308 :M f0_12 sf .241 .024( = 0 if and only if)J 95 330 :M f4_12 sf .644(X)A f0_12 sf 1.444 .144( is d-separated from )J 211 330 :M f4_12 sf (Y)S 218 330 :M f0_12 sf 2.011 .201( given )J f2_12 sf 1.331(Z)A f0_12 sf 1.108 .111( in )J 283 330 :M f4_12 sf (G)S 292 330 :M f0_12 sf 1.535 .153(. Because the value of a partial)J 95 348 :M 1.794 .179(correlation in a linear SEM depends only on the values of the linear)J 95 366 :M 1.92 .192(coefficients and the variances of the exogenous variables, )J 402 366 :M f4_12 sf (L)S 409 366 :M f0_12 sf 1.743 .174(' linearly)J 95 388 :M .31 .031(entails )J f5_12 sf (r)S 136 390 :M f4_10 sf (XY)S 148 390 :M f0_10 sf (.)S 151 390 :M f2_10 sf (Z)S 158 388 :M f0_12 sf .524 .052( = 0 if and only if )J 250 388 :M f4_12 sf .184(X)A f0_12 sf .414 .041( is d-separated from )J 359 388 :M f4_12 sf (Y)S 366 388 :M f0_12 sf .512 .051( given )J 400 388 :M f2_12 sf .333(Z)A f0_12 sf .266 .027( in )J f4_12 sf (G)S 433 388 :M f0_12 sf .512 .051( and)J 95 414 :M .318 .032(hence )J f4_12 sf (L)S 133 414 :M f0_12 sf .376 .038( also linearly entails that )J f5_12 sf (r)S 263 416 :M f4_10 sf (XY)S 275 416 :M f0_10 sf (.)S 278 416 :M f2_10 sf (Z)S 285 414 :M f0_12 sf .487 .049( = 0 if and only if )J 377 414 :M f4_12 sf .119(X)A f0_12 sf .33 .033( is d-separated)J 95 436 :M (from )S f4_12 sf (Y)S 128 436 :M f0_12 sf ( given )S 161 436 :M f2_12 sf (Z)S f0_12 sf ( in )S f4_12 sf (G)S 193 436 :M f0_12 sf (. )S 199 427 9 9 rC gS 1.286 1 scale 154.779 436 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 95 470 :M f0_12 sf .78 .078(Note that for an SEM with graph )J 264 470 :M f4_12 sf (G)S 273 470 :M f0_12 sf .828 .083(, if )J f4_12 sf .929(V)A f0_12 sf .38 .038( )J 303 470 :M f1_12 sf S 310 470 :M f0_12 sf .169A f4_12 sf .413(X)A f0_12 sf .642 .064(, then )J 352 460 47 15 rC 399 475 :M psb currentpoint pse 352 460 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1504 div 480 3 -1 roll exch div scale currentpoint translate 64 34 translate -9 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 199 286 moveto (e) show 385 382 moveto 320 /Times-Italic f1 (V) show 727 286 moveto 384 /Times-Roman f1 (/) show 929 286 moveto 384 /Symbol f2 (\266) show 1137 286 moveto 384 /Times-Italic f1 (X) show end pse gR gS 0 0 552 730 rC 399 470 :M f0_12 sf .563 .056(is non-zero)J 95 497 :M .3 .03(only if there is an edge from )J 237 497 :M f4_12 sf .231(X)A f0_12 sf .201 .02( to )J f4_12 sf .231(V)A f0_12 sf .201 .02( in )J f4_12 sf (G)S 292 497 :M f0_12 sf .293 .029( \(because )J 341 497 :M f5_12 sf .101(e)A f4_10 sf 0 2 rm .117(V)A 0 -2 rm f0_12 sf .268 .027( is a function only of)J 95 519 :M f4_12 sf .586(V)A f0_12 sf .746 .075( and )J f4_12 sf .586(V)A f0_12 sf 1.112 .111('s parents in )J 200 519 :M f4_12 sf (G)S 209 519 :M f0_12 sf .861 .086(.\) Associate with each non-zero partial derivative)J 95 528 43 15 rC 138 543 :M psb currentpoint pse 95 528 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 480 3 -1 roll exch div scale currentpoint translate 64 34 translate -9 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 199 286 moveto (e) show 385 382 moveto 320 /Times-Italic f1 (V) show 819 286 moveto 384 /Symbol f2 (\266) show 1027 286 moveto 384 /Times-Italic f1 (X) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -125 388 782 0 vb end pse gR gS 0 0 552 730 rC 138 538 :M f0_12 sf .366 .037( the edge from )J 213 538 :M f4_12 sf .268(X)A f0_12 sf .234 .023( to )J f4_12 sf .268(V)A f0_12 sf .234 .023( in )J f4_12 sf (G)S 268 538 :M f0_12 sf .302 .03(. A product of partial derivatives form)J 95 558 :M .142 .014(a )J f2_12 sf .113(loop)A f0_12 sf .137 .014( in )J 141 558 :M f4_12 sf (G)S 150 558 :M f0_12 sf .214 .021( if and only if the corresponding edges form a cycle in )J 417 558 :M f4_12 sf (G)S 426 558 :M f0_12 sf .223 .022(. Two)J 95 576 :M (loops )S f2_12 sf (intersect)S 168 576 :M f0_12 sf ( if and only if their corresponding cycles intersect.)S 95 606 :M .433 .043(Let )J f4_12 sf .17(J)A f2_10 sf 0 2 rm .248(Err)A 0 -2 rm 136 608 :M f0_10 sf .14<28>A f2_10 sf .303(V)A f0_10 sf .172(\)->)A f2_10 sf .303(V)A f0_12 sf 0 -2 rm .725 .073( be the Jacobean of the transformation from )J 0 2 rm 388 606 :M f2_12 sf (Err)S 407 606 :M f0_12 sf <28>S 411 606 :M f2_12 sf (V)S 420 606 :M f0_12 sf 1.06 .106(\) to )J 442 606 :M f2_12 sf (V)S 451 606 :M f0_12 sf (,)S 95 625 :M (and )S f4_12 sf (J)S f2_10 sf 0 2 rm (V)S 0 -2 rm f0_10 sf 0 2 rm (->)S 0 -2 rm 136 627 :M f2_10 sf (Err)S 152 627 :M f0_10 sf <28>S f2_10 sf (V)S f0_10 sf <29>S f0_12 sf 0 -2 rm .033 .003( be the Jacobean of the transformation from )J 0 2 rm 379 625 :M f2_12 sf (V)S 388 625 :M f0_12 sf ( to )S f2_12 sf .03(Err)A 422 625 :M f0_12 sf <28>S 426 625 :M f2_12 sf (V)S 435 625 :M f0_12 sf .042 .004(\). A)J endp %%Page: 29 29 %%BeginPageSetup initializepage (peter; page: 29 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (29)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf 1.428 .143(product of partial derivatives )J 247 104 :M f4_12 sf .68(S)A f0_12 sf 1.558 .156( occurring in a term )J f4_12 sf (T)S 370 104 :M f0_12 sf .882 .088( in )J f4_12 sf .736(J)A f2_10 sf 0 2 rm 1.073(Err)A 0 -2 rm 411 106 :M f0_10 sf .203<28>A f2_10 sf .44(V)A f0_10 sf .25(\)->)A f2_10 sf .44(V)A f0_12 sf 0 -2 rm .558 .056( is)J 0 2 rm 95 123 :M f2_12 sf 1.666 .167(minimally sufficient)J f0_12 sf .312 .031( in )J f4_12 sf (T)S 225 123 :M f0_12 sf 1.456 .146( if for each variable occurring in )J f4_12 sf .58(S)A f0_12 sf 1.068 .107(, all of its)J 95 141 :M 1.707 .171(occurrences in )J 174 141 :M f4_12 sf (T)S 181 141 :M f0_12 sf 2.354 .235( are in )J 223 141 :M f4_12 sf .765(S)A f0_12 sf 1.613 .161(, and no subset of )J f4_12 sf .765(S)A f0_12 sf 2.172 .217( has this property. For)J 95 159 :M (example, in)S 95 198 :M ( )S 209 180 163 29 rC 372 209 :M psb currentpoint pse 209 180 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5216 div 928 3 -1 roll exch div scale currentpoint translate 64 41 translate 23 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 231 286 moveto (e) show 420 382 moveto 320 /Times-Italic f1 (W) show 152 836 moveto 384 /Symbol f2 (\266) show 360 836 moveto 384 /Times-Italic f1 (X) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 436 moveto 764 0 rlineto stroke 854 535 moveto 384 /Symbol f1 (\264) show 1177 288 moveto 384 /Symbol f2 (\266) show 1385 288 moveto (e) show 1599 384 moveto 320 /Times-Italic f1 (X) show 1286 836 moveto 384 /Symbol f2 (\266) show 1494 836 moveto 384 /Times-Italic f1 (Y) show 1154 436 moveto 711 0 rlineto stroke 1955 535 moveto 384 /Symbol f1 (\264) show 2278 288 moveto 384 /Symbol f2 (\266) show 2486 288 moveto (e) show 2672 384 moveto 320 /Times-Italic f1 (Y) show 2315 836 moveto 384 /Symbol f2 (\266) show 2523 836 moveto 384 /Times-Italic f1 (W) show 2255 436 moveto 673 0 rlineto stroke 3018 535 moveto 384 /Symbol f1 (\264) show 3341 286 moveto 384 /Symbol f2 (\266) show 3549 286 moveto (e) show 3725 382 moveto 320 /Times-Italic f1 (U) show 3422 836 moveto 384 /Symbol f2 (\266) show 3630 836 moveto 384 /Times-Italic f1 (U) show 3318 436 moveto 710 0 rlineto stroke 4118 535 moveto 384 /Symbol f1 (\264) show 4441 286 moveto 384 /Symbol f2 (\266) show 4649 286 moveto (e) show 4835 382 moveto 320 /Times-Italic f1 (V) show 4528 836 moveto 384 /Symbol f2 (\266) show 4736 836 moveto 384 /Times-Italic f1 (V) show 4418 436 moveto 691 0 rlineto stroke end pse gR gS 0 0 552 730 rC 244 230 :M f2_12 sf (Equation 12)S 95 260 :M f0_12 sf (the three minimally sufficient products are)S 95 299 :M ( )S 209 281 187 29 rC 396 310 :M psb currentpoint pse 209 281 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5984 div 928 3 -1 roll exch div scale currentpoint translate 64 41 translate 23 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 231 286 moveto (e) show 420 382 moveto 320 /Times-Italic f1 (W) show 152 836 moveto 384 /Symbol f2 (\266) show 360 836 moveto 384 /Times-Italic f1 (X) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 436 moveto 764 0 rlineto stroke 854 535 moveto 384 /Symbol f1 (\264) show 1177 288 moveto 384 /Symbol f2 (\266) show 1385 288 moveto (e) show 1599 384 moveto 320 /Times-Italic f1 (X) show 1286 836 moveto 384 /Symbol f2 (\266) show 1494 836 moveto 384 /Times-Italic f1 (Y) show 1154 436 moveto 711 0 rlineto stroke 1955 535 moveto 384 /Symbol f1 (\264) show 2278 288 moveto 384 /Symbol f2 (\266) show 2486 288 moveto (e) show 2672 384 moveto 320 /Times-Italic f1 (Y) show 2315 836 moveto 384 /Symbol f2 (\266) show 2523 836 moveto 384 /Times-Italic f1 (W) show 2255 436 moveto 673 0 rlineto stroke 2968 535 moveto 384 /Times-Roman f1 (,) show 3452 286 moveto 384 /Symbol f2 (\266) show 3660 286 moveto (e) show 3836 382 moveto 320 /Times-Italic f1 (U) show 3533 836 moveto 384 /Symbol f2 (\266) show 3741 836 moveto 384 /Times-Italic f1 (U) show 3429 436 moveto 710 0 rlineto stroke 4179 535 moveto 384 /Times-Roman f1 (,) show 4498 535 moveto 384 /Times-Roman f1 (and) show 5199 286 moveto 384 /Symbol f2 (\266) show 5407 286 moveto (e) show 5593 382 moveto 320 /Times-Italic f1 (V) show 5286 836 moveto 384 /Symbol f2 (\266) show 5494 836 moveto 384 /Times-Italic f1 (V) show 5176 436 moveto 691 0 rlineto stroke end pse gR gS 0 0 552 730 rC 244 331 :M f2_12 sf (Equation 13)S 95 361 :M f4_12 sf (J)S f2_10 sf 0 2 rm (Err)S 0 -2 rm 116 363 :M f0_10 sf .093<28>A f2_10 sf .203(V)A f0_10 sf .115(\)->)A f2_10 sf .248 .025(V )J 149 361 :M f0_12 sf .638 .064(is equal to 1/)J 214 361 :M f4_12 sf (J)S f2_10 sf 0 2 rm (V)S 0 -2 rm f0_10 sf 0 2 rm (->)S 0 -2 rm 235 363 :M f2_10 sf (Err)S 251 363 :M f0_10 sf .124<28>A f2_10 sf .268(V)A f0_10 sf .124<29>A f0_12 sf 0 -2 rm .567 .057(, but it turns out to simplify the proofs)J 0 2 rm 95 380 :M 1.078 .108(if at intermediate stages we work with )J 293 380 :M f4_12 sf (J)S f2_10 sf 0 2 rm (V)S 0 -2 rm f0_10 sf 0 2 rm (->)S 0 -2 rm 314 382 :M f2_10 sf (Err)S 330 382 :M f0_10 sf .237<28>A f2_10 sf .514(V)A f0_10 sf .237<29>A f0_12 sf 0 -2 rm 1.047 .105( than if we work with)J 0 2 rm 95 399 :M f4_12 sf (J)S f2_10 sf 0 2 rm (Err)S 0 -2 rm 116 401 :M f0_10 sf <28>S f2_10 sf .089(V)A f0_10 sf .051(\)->)A f2_10 sf .089(V)A f0_12 sf 0 -2 rm .067 .007(. )J 0 2 rm 152 399 :M f4_12 sf (J)S f2_10 sf 0 2 rm (V)S 0 -2 rm f0_10 sf 0 2 rm (->)S 0 -2 rm 173 401 :M f2_10 sf (Err)S 189 401 :M f0_10 sf .057<28>A f2_10 sf .123(V)A f0_10 sf .057<29>A f0_12 sf 0 -2 rm .28 .028( is the determinant of a matrix in which the element)J 0 2 rm 95 421 :M (in the )S 125 421 :M f4_12 sf (i)S f0_10 sf 0 -3 rm (th)S 0 3 rm 136 421 :M f0_12 sf ( row and )S 181 421 :M f4_12 sf (j)S f0_10 sf 0 -3 rm (th)S 0 3 rm 192 421 :M f0_12 sf ( column is )S 245 411 50 16 rC 295 427 :M psb currentpoint pse 245 411 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1600 div 512 3 -1 roll exch div scale currentpoint translate 64 34 translate -9 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 199 286 moveto (e) show 390 382 moveto 224 /Times-Italic f1 (V) show 544 438 moveto 160 ns (i) show 750 286 moveto 384 /Times-Roman f1 (/) show 952 286 moveto 384 /Symbol f2 (\266) show 1160 286 moveto 384 /Times-Italic f1 (V) show 1436 382 moveto 224 ns (j) show end pse gR gS 0 0 552 730 rC 295 421 :M f0_12 sf (.)S 95 454 :M f2_12 sf (X)S 104 454 :M f0_12 sf .152 .015( is an )J f2_12 sf .575 .057(ancestral set)J 198 454 :M f0_12 sf .292 .029( for a directed graph )J f4_12 sf (G)S 309 454 :M f0_12 sf .27 .027( with vertices )J f2_12 sf (V)S 387 454 :M f0_12 sf .339 .034( if and only if)J 95 472 :M f2_12 sf (X)S 104 472 :M f0_12 sf ( = )S 117 472 :M f2_12 sf (An)S f0_12 sf <28>S 136 472 :M f2_12 sf (Y)S 145 472 :M f0_12 sf (,)S f4_12 sf (G)S 157 472 :M f0_12 sf (\) for some )S f2_12 sf (Y)S 218 472 :M f0_12 sf ( included in )S 278 472 :M f2_12 sf (V)S 287 472 :M f0_12 sf (.)S 95 502 :M f2_12 sf .439 .044(Theorem 4:)J 156 502 :M f0_12 sf .544 .054( In an acylic graph )J f4_12 sf (G)S 261 502 :M f0_12 sf .51 .051( containing disjoint sets of variables )J 442 502 :M f2_12 sf (X)S 451 502 :M f0_12 sf (,)S 95 520 :M f2_12 sf (Y)S 104 520 :M f0_12 sf .721 .072( and )J f2_12 sf .618(Z)A f0_12 sf .421 .042(, )J 144 520 :M f4_12 sf (G)S 153 520 :M f0_12 sf .454 .045( pseudo-indeterministically entails that )J f2_12 sf (X)S 355 520 :M f0_12 sf .589 .059( is d-separated from)J 95 538 :M f2_12 sf (Y)S 104 538 :M f0_12 sf ( given )S 137 538 :M f2_12 sf (Z)S f0_12 sf ( if and only )S 203 538 :M f4_12 sf (L)S 210 538 :M f0_12 sf ( entails that )S f2_12 sf (X)S 277 538 :M f0_12 sf ( is independent of )S 366 538 :M f2_12 sf (Y)S 375 538 :M f0_12 sf ( given )S 408 538 :M f2_12 sf (Z)S f0_12 sf (.)S 95 568 :M 2.367 .237(Proof. The first part of the proof is essentially the same as that of)J 95 586 :M (Theorems 1 and 2, and shows that)S endp %%Page: 30 30 %%BeginPageSetup initializepage (peter; page: 30 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (30)S gR gS 0 0 552 730 rC 95 107 :M f0_12 sf ( )S 185 95 251 30 rC 436 125 :M psb currentpoint pse 185 95 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8032 div 960 3 -1 roll exch div scale currentpoint translate 64 40 translate 56 344 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-Italic f1 (f) show 229 344 moveto 384 /Times-Roman f1 (\() show 369 344 moveto 384 /Times-Bold f1 (An) show 860 344 moveto 384 /Times-Roman f1 (\() show 1000 344 moveto 384 /Times-Bold f1 (X) show 1274 344 moveto 384 /Times-Roman f1 (,) show 1404 344 moveto 384 /Times-Italic f1 (G) show 1694 344 moveto 384 /Times-Roman f1 (\)) show 1824 344 moveto (\)) show 2057 344 moveto 384 /Symbol f1 (=) show 3542 344 moveto 384 /Times-Italic f1 (f) show 3715 344 moveto 384 /Times-Roman f1 (\() show 3860 344 moveto 384 /Times-Italic f1 (g) show 4064 440 moveto 320 ns (X) show 4312 344 moveto 384 /Times-Roman f1 (\() show 4477 344 moveto 384 /Times-Italic f1 (X) show 4728 344 moveto 384 /Times-Roman f1 (,) show 4860 344 moveto 384 /Times-Bold f1 (Parents) show 6117 344 moveto 384 /Times-Roman f1 (\() show 6282 344 moveto 384 /Times-Italic f1 (X) show 6533 344 moveto 384 /Times-Roman f1 (,) show 6663 344 moveto 384 /Times-Italic f1 (G) show 6953 344 moveto 384 /Times-Roman f1 (\)) show 7083 344 moveto (\)) show 2405 810 moveto 320 /Times-Italic f1 (X) show 2625 810 moveto 320 /Symbol f1 (\316) show 2811 810 moveto 320 /Times-Bold f1 (An) show 3226 810 moveto 320 /Times-Roman f1 (\() show 3348 810 moveto 320 /Times-Bold f1 (X) show 3582 810 moveto 320 /Times-Roman f1 (,) show 3669 810 moveto 320 /Times-Italic f1 (G) show 3916 810 moveto 320 /Times-Roman f1 (\)) show 2960 432 moveto 576 /Symbol f1 (\325) show 7282 344 moveto 384 ns (\264) show 7659 344 moveto 384 /Times-Italic f1 (J) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 7590 51 moveto 0 388 rlineto stroke 7882 51 moveto 0 388 rlineto stroke end pse gR gS 0 0 552 730 rC 244 146 :M f2_12 sf (Equation 14)S 95 176 :M f0_12 sf 1.074 .107(In an acyclic graph, the Jacobian of the transformation is a single term)J 95 194 :M 3.27 .327(consisting of the product of the terms along the diagonal of the)J 95 212 :M (transformation matrix:)S 95 251 :M ( )S 185 233 250 36 rC 435 269 :M psb currentpoint pse 185 233 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8000 div 1152 3 -1 roll exch div scale currentpoint translate 64 41 translate 13 535 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-Italic f1 (J) show 320 535 moveto 384 /Symbol f1 (=) show 2336 286 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 2544 286 moveto (e) show 2730 382 moveto 320 /Times-Italic f1 (V) show 2423 836 moveto 384 /Symbol f2 (\266) show 2631 836 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2313 436 moveto 691 0 rlineto stroke 640 1001 moveto 320 ns (V) show 868 1001 moveto 320 /Symbol f1 (\316) show 1054 1001 moveto 320 /Times-Bold f1 (An) show 1469 1001 moveto 320 /Times-Roman f1 (\() show 1591 1001 moveto 320 /Times-Bold f1 (X) show 1825 1001 moveto 320 /Times-Roman f1 (,) show 1912 1001 moveto 320 /Times-Italic f1 (G) show 2159 1001 moveto 320 /Times-Roman f1 (\)) show 1213 623 moveto 576 /Symbol f1 (\325) show 3128 535 moveto 384 ns (=) show 4543 535 moveto 384 /Times-Italic f1 (m) show 4807 631 moveto 320 ns (V) show 5063 535 moveto 384 /Times-Roman f1 (\() show 5195 535 moveto 384 /Times-Italic f1 (V) show 5456 535 moveto 384 /Times-Roman f1 (,) show 5588 535 moveto 384 /Times-Bold f1 (Parents) show 6845 535 moveto 384 /Times-Roman f1 (\() show 6977 535 moveto 384 /Times-Italic f1 (V) show 7238 535 moveto 384 /Times-Roman f1 (,) show 7368 535 moveto 384 /Times-Italic f1 (G) show 7658 535 moveto 384 /Times-Roman f1 (\)) show 7788 535 moveto (\)) show 3448 1001 moveto 320 /Times-Italic f1 (V) show 3676 1001 moveto 320 /Symbol f1 (\316) show 3862 1001 moveto 320 /Times-Bold f1 (An) show 4277 1001 moveto 320 /Times-Roman f1 (\() show 4399 1001 moveto 320 /Times-Bold f1 (X) show 4633 1001 moveto 320 /Times-Roman f1 (,) show 4720 1001 moveto 320 /Times-Italic f1 (G) show 4967 1001 moveto 320 /Times-Roman f1 (\)) show 4021 623 moveto 576 /Symbol f1 (\325) show end pse gR gS 0 0 552 730 rC 244 290 :M f2_12 sf (Equation 15)S 95 320 :M f0_12 sf 1.368 .137(\(This is because for an acyclic graph the transformation matrix can be)J 95 339 :M 2.51 .251(arranged so that it is lower triangular.\) Each term )J 369 329 45 15 rC 414 344 :M psb currentpoint pse 369 329 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1440 div 480 3 -1 roll exch div scale currentpoint translate 64 34 translate -9 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 199 286 moveto (e) show 390 382 moveto 224 /Times-Italic f1 (V) show 669 286 moveto 384 /Times-Roman f1 (/) show 871 286 moveto 384 /Symbol f2 (\266) show 1079 286 moveto 384 /Times-Italic f1 (V) show end pse gR gS 0 0 552 730 rC 414 339 :M f0_12 sf 2.458 .246(is some)J 95 363 :M 1.024 .102(function )J 140 363 :M f4_12 sf (m)S 149 365 :M f4_10 sf 1.019(V)A f0_12 sf 0 -2 rm 1.16 .116( of )J 0 2 rm 175 363 :M f4_12 sf .628(V)A f0_12 sf 1.345 .134( and its parents, because )J f5_12 sf .451(e)A f4_10 sf 0 2 rm .523(V)A 0 -2 rm f0_12 sf .987 .099( is a function of )J f4_12 sf .628(V)A f0_12 sf 1.1 .11( and its)J 95 385 :M .675 .067(parents. Hence by lemma 1, if )J f2_12 sf (X)S 257 385 :M f0_12 sf .893 .089( and )J 283 385 :M f2_12 sf (Y)S 292 385 :M f0_12 sf .691 .069( are d-separated given )J 405 385 :M f2_12 sf .455(Z)A f0_12 sf .649 .065(, then )J 445 385 :M f2_12 sf (X)S 95 403 :M f0_12 sf (and )S f2_12 sf (Y)S 124 403 :M f0_12 sf ( are independent given )S 236 403 :M f2_12 sf (Z)S f0_12 sf (.)S 95 433 :M .671 .067(Suppose that )J 162 433 :M f2_12 sf (X)S 171 433 :M f0_12 sf .627 .063( and )J f2_12 sf (Y)S 205 433 :M f0_12 sf .707 .071( are not d-separated given )J 338 433 :M f2_12 sf .307(Z)A f0_12 sf .676 .068(. Then by Theorem 2,)J 95 451 :M 1.4 .14(there is a linear SEM in which )J 258 451 :M f2_12 sf (X)S 267 451 :M f0_12 sf 1.167 .117( and )J f2_12 sf (Y)S 303 451 :M f0_12 sf 1.197 .12( are not independent given )J f2_12 sf .566(Z)A f0_12 sf (.)S 95 469 :M .168 .017(Since a linear SEM is a special case of an SEM, there is an SEM in which)J 95 487 :M f2_12 sf (X)S 104 487 :M f0_12 sf ( and )S f2_12 sf (Y)S 136 487 :M f0_12 sf ( are not independent given )S f2_12 sf (Z)S f0_12 sf (. )S 280 478 9 9 rC gS 1.286 1 scale 217.779 487 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 95 517 :M f2_12 sf 1.248 .125(Lemma 5:)J f0_12 sf .656 .066( In an SEM with directed graph )J f4_12 sf (G)S 319 517 :M f0_12 sf .683 .068( with vertices )J f2_12 sf (V)S 399 517 :M f0_12 sf .963 .096(, if )J 418 517 :M f2_12 sf (X)S 427 517 :M f0_12 sf .924 .092( is an)J 95 535 :M 2.271 .227(ancestral set for )J 185 535 :M f4_12 sf (G)S 194 535 :M f0_12 sf 2.111 .211(, then each minimally sufficient product of terms)J 95 553 :M .947 .095(occurring )J f4_12 sf (T)S 153 553 :M f0_12 sf .765 .076( of )J f4_12 sf .611(J)A f2_10 sf 0 2 rm .829(X)A 0 -2 rm f0_10 sf 0 2 rm 1.029(->)A 0 -2 rm 194 555 :M f2_10 sf (Err)S 210 555 :M f0_10 sf .357<28>A f2_10 sf .774(X)A f0_10 sf .357<29>A f0_12 sf 0 -2 rm 1.408 .141( that is non-zero is either a loop in )J 0 2 rm 410 553 :M f4_12 sf (G)S 419 553 :M f0_12 sf <28>S 423 553 :M f2_12 sf (X)S 432 553 :M f0_12 sf 1.538 .154(\), or)J 95 563 42 15 rC 137 578 :M psb currentpoint pse 95 563 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1344 div 480 3 -1 roll exch div scale currentpoint translate 64 34 translate -9 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 199 286 moveto (e) show 390 382 moveto 224 /Times-Italic f1 (V) show 760 286 moveto 384 /Symbol f2 (\266) show 968 286 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -124 386 723 0 vb end pse gR gS 0 0 552 730 rC 137 573 :M f0_12 sf ( for )S 157 573 :M f4_12 sf (V)S f0_12 sf ( in )S f2_12 sf (X)S 188 573 :M f0_12 sf (.)S 95 605 :M .966 .097(Proof. Each term in )J f4_12 sf .318(J)A f2_10 sf 0 2 rm .432(X)A 0 -2 rm f0_10 sf 0 2 rm .536(->)A 0 -2 rm 220 607 :M f2_10 sf (Err)S 236 607 :M f0_10 sf .23<28>A f2_10 sf .499(X)A f0_10 sf .23<29>A f0_12 sf 0 -2 rm 1.083 .108( is a product of partial derivatives in the)J 0 2 rm 95 624 :M 1.276 .128(transformation matrix, one from each row, and one from each column,)J endp %%Page: 31 31 %%BeginPageSetup initializepage (peter; page: 31 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (31)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .743 .074(times a variable that is either equal to 1 or -1. Hence each variable in )J f2_12 sf (X)S 95 122 :M f0_12 sf 1.176 .118(appears exactly once in the numerator of some partial derivative in the)J 95 140 :M .061 .006(term, and exactly once in the denominator of some partial derivative in the)J 95 159 :M (term. If )S 134 149 42 15 rC 176 164 :M psb currentpoint pse 134 149 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1344 div 480 3 -1 roll exch div scale currentpoint translate 64 34 translate -9 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 199 286 moveto (e) show 390 382 moveto 224 /Times-Italic f1 (V) show 760 286 moveto 384 /Symbol f2 (\266) show 968 286 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -124 386 723 0 vb end pse gR gS 0 0 552 730 rC 176 159 :M f0_12 sf ( occurs in )S 226 159 :M f4_12 sf (T)S 233 159 :M f0_12 sf (, it is minimally sufficient.)S 95 191 :M .197 .02(Suppose then that some minimally sufficient product of partial derivatives)J 95 210 :M f4_12 sf .17(S)A f0_12 sf .413 .041( occurring in )J f4_12 sf (T)S 174 210 :M f0_12 sf .549 .055( is not equal to )J 251 200 42 15 rC 293 215 :M psb currentpoint pse 251 200 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1344 div 480 3 -1 roll exch div scale currentpoint translate 64 34 translate -9 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 199 286 moveto (e) show 390 382 moveto 224 /Times-Italic f1 (V) show 760 286 moveto 384 /Symbol f2 (\266) show 968 286 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -124 386 723 0 vb end pse gR gS 0 0 552 730 rC 293 210 :M f0_12 sf .572 .057( for any )J 336 210 :M f4_12 sf .339(V)A f0_12 sf .296 .03( in )J f2_12 sf (X)S 368 210 :M f0_12 sf .558 .056(. Then )J 404 210 :M f4_12 sf .176(S)A f0_12 sf .455 .045( does not)J 95 231 :M .356 .036(contain )J 133 221 42 15 rC 175 236 :M psb currentpoint pse 133 221 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1344 div 480 3 -1 roll exch div scale currentpoint translate 64 34 translate -9 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 199 286 moveto (e) show 390 382 moveto 224 /Times-Italic f1 (V) show 760 286 moveto 384 /Symbol f2 (\266) show 968 286 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -124 386 723 0 vb end pse gR gS 0 0 552 730 rC 175 231 :M f0_12 sf .505 .05( for )J 197 231 :M f4_12 sf .284(V)A f0_12 sf .247 .025( in )J f2_12 sf (X)S 229 231 :M f0_12 sf .372 .037(, because otherwise it would not be minimally)J 95 252 :M 2.221 .222(sufficient. Hence each partial derivative in )J f4_12 sf .676(S)A f0_12 sf 1.231 .123( is of the form )J 411 242 43 15 rC 454 257 :M psb currentpoint pse 411 242 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 480 3 -1 roll exch div scale currentpoint translate 64 34 translate -9 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 199 286 moveto (e) show 385 382 moveto 320 /Times-Italic f1 (V) show 819 286 moveto 384 /Symbol f2 (\266) show 1027 286 moveto 384 /Times-Italic f1 (Y) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -125 388 782 0 vb end pse gR gS 0 0 552 730 rC 95 276 :M f0_12 sf .353 .035(where )J 128 276 :M f4_12 sf .229(V)A f0_12 sf .085 .009( )J f1_12 sf S 145 276 :M f0_12 sf S f4_12 sf (Y)S 155 276 :M f0_12 sf .372 .037(. Such a term is non-zero only if there is an edge from )J 424 276 :M f4_12 sf (Y)S 431 276 :M f0_12 sf .298 .03( to )J f4_12 sf (V)S 95 298 :M f0_12 sf .275 .027(in )J 108 298 :M f4_12 sf (G)S 117 298 :M f0_12 sf .27 .027(. Because )J f4_12 sf .121(V)A f0_12 sf .161 .016( and )J 197 298 :M f4_12 sf (Y)S 204 298 :M f0_12 sf .212 .021( are both in ancestral set )J f2_12 sf (X)S 334 298 :M f0_12 sf .233 .023(, if there is an edge from)J 95 316 :M f4_12 sf (Y)S 102 316 :M f0_12 sf .86 .086( to )J f4_12 sf .987(V)A f0_12 sf .897 .09( in )J 144 316 :M f4_12 sf (G)S 153 316 :M f0_12 sf .913 .091(, then there is an edge from )J 296 316 :M f4_12 sf (Y)S 303 316 :M f0_12 sf .86 .086( to )J f4_12 sf .987(V)A f0_12 sf .897 .09( in )J 345 316 :M f4_12 sf (G)S 354 316 :M f0_12 sf <28>S 358 316 :M f2_12 sf (X)S 367 316 :M f0_12 sf .882 .088(\). Since all of the)J 95 334 :M .158 .016(occurrences of the variables in )J 247 334 :M f4_12 sf .067(S)A f0_12 sf .098 .01( are in )J f4_12 sf .067(S)A f0_12 sf .193 .019(, each variable occurs once in the)J 95 352 :M .735 .074(numerator and once in the denominator of a partial derivative in )J f4_12 sf .249(S)A f0_12 sf .468 .047(; so in)J 95 370 :M f4_12 sf (G)S 104 370 :M f0_12 sf <28>S 108 370 :M f2_12 sf (X)S 117 370 :M f0_12 sf .865 .087(\) there is a path in which all of the variables in )J 358 370 :M f4_12 sf .325(S)A f0_12 sf .803 .08( occur once at the)J 95 388 :M (head of an edge and once at the tail. It follows that there is a cycle in )S 428 388 :M f4_12 sf (G)S 437 388 :M f0_12 sf <28>S 441 388 :M f2_12 sf (X)S 450 388 :M f0_12 sf <29>S 95 406 :M (that corresponds to the product of partial derivatives in )S f4_12 sf (S)S f0_12 sf (. )S 376 397 9 9 rC gS 1.286 1 scale 292.446 406 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 95 436 :M f0_12 sf .932 .093(A )J 108 436 :M f2_12 sf .178(cycleset)A f0_12 sf .561 .056( is a set of non-intersecting cycles. Let )J 343 436 :M f2_12 sf (Cycleset)S 386 436 :M f0_12 sf <28>S 390 436 :M f4_12 sf (G)S 399 436 :M f0_12 sf .75 .075(\) be the set)J 95 454 :M .53 .053(of all cyclesets in )J f4_12 sf (G)S 194 454 :M f0_12 sf .7 .07(. Let )J 221 454 :M f2_12 sf (Vertices)S 263 454 :M f0_12 sf <28>S 267 454 :M f2_12 sf (C)S 276 454 :M f0_12 sf .598 .06(\) be the set of vertices occuring in a)J 95 472 :M (cycleset )S 137 472 :M f2_12 sf (C)S 146 472 :M f0_12 sf (.)S 95 502 :M f2_12 sf 1.248 .125(Lemma 6:)J f0_12 sf .656 .066( In an SEM with directed graph )J f4_12 sf (G)S 319 502 :M f0_12 sf .683 .068( with vertices )J f2_12 sf (V)S 399 502 :M f0_12 sf .963 .096(, if )J 418 502 :M f2_12 sf (X)S 427 502 :M f0_12 sf .924 .092( is an)J 95 520 :M (ancestral set for )S 174 520 :M f4_12 sf (G)S 183 520 :M f0_12 sf ( then)S 95 579 384 44 rC 479 623 :M psb currentpoint pse 95 579 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 12288 div 1408 3 -1 roll exch div scale currentpoint translate 64 35 translate 13 765 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-Italic f1 (J) show 188 861 moveto 320 /Times-Bold f1 (X) show 439 861 moveto 320 /Symbol f1 (-) show 634 861 moveto (>) show 827 861 moveto 320 /Times-Bold f1 (Err) show 1332 861 moveto 320 /Times-Roman f1 (\() show 1454 861 moveto 320 /Times-Bold f1 (X) show 1699 861 moveto 320 /Times-Roman f1 (\)) show 1945 765 moveto 384 /Symbol f1 (=) show 4816 765 moveto 384 /Times-Italic f1 (d) show 5030 765 moveto 384 /Times-Roman f1 (\() show 5162 765 moveto 384 /Times-Bold f1 (C) show 5439 765 moveto 384 /Times-Roman f1 (\)) show 5638 765 moveto 384 /Symbol f1 (\264) show 7686 518 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 7894 518 moveto (e) show 8080 614 moveto 320 /Times-Italic f1 (Y) show 7723 1066 moveto 384 /Symbol f2 (\266) show 7931 1066 moveto 384 /Times-Italic f1 (W) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 7663 666 moveto 673 0 rlineto stroke 6114 1231 moveto 320 /Symbol f1 (<) show 6302 1231 moveto 320 /Times-Italic f1 (W) show 6594 1231 moveto 320 /Times-Roman f1 (,) show 6676 1231 moveto 320 /Times-Italic f1 (Y) show 6892 1231 moveto 320 /Symbol f1 (>) show 7141 1231 moveto (\316) show 7385 1231 moveto 320 /Times-Bold f1 (C) show 6622 853 moveto 576 /Symbol f1 (\325) show 5923 355 moveto 384 ns (\346) show 5923 1220 moveto (\350) show 5923 806 moveto (\347) show 5923 1001 moveto (\347) show 8348 355 moveto (\366) show 8348 1220 moveto (\370) show 8348 806 moveto (\367) show 8348 1001 moveto (\367) show 11338 516 moveto 384 /Symbol f2 (\266) show 11546 516 moveto (e) show 11732 612 moveto 320 /Times-Italic f1 (V) show 11425 1066 moveto 384 /Symbol f2 (\266) show 11633 1066 moveto 384 /Times-Italic f1 (V) show 11315 666 moveto 691 0 rlineto stroke 8722 1231 moveto 320 ns (V) show 9014 1231 moveto 320 /Symbol f1 (\316) show 9200 1231 moveto 320 /Times-Bold f1 (X) show 9480 1231 moveto 320 /Times-Roman f1 (\\) show 9684 1231 moveto 320 /Times-Bold f1 (Vertices) show 10808 1231 moveto 320 /Times-Roman f1 (\() show 10924 1231 moveto 320 /Times-Bold f1 (C) show 11161 1231 moveto 320 /Times-Roman f1 (\)) show 9755 853 moveto 576 /Symbol f1 (\325) show 8537 355 moveto 384 ns (\346) show 8537 1220 moveto (\350) show 8537 806 moveto (\347) show 8537 1001 moveto (\347) show 12018 355 moveto (\366) show 12018 1220 moveto (\370) show 12018 806 moveto (\367) show 12018 1001 moveto (\367) show 2266 1234 moveto 320 /Times-Bold f1 (C) show 2498 1234 moveto 320 /Symbol f1 (\316) show 2678 1234 moveto 320 /Times-Bold f1 (Cycleset) show 3826 1234 moveto 320 /Times-Roman f1 (\() show 3946 1234 moveto 320 /Times-Italic f1 (G) show 4187 1234 moveto 320 /Times-Roman f1 (\() show 4309 1234 moveto 320 /Times-Bold f1 (X) show 4554 1234 moveto 320 /Times-Roman f1 (\)) show 4667 1234 moveto (\)) show 3314 852 moveto 576 /Symbol f1 (\345) show end pse gR gS 0 0 552 730 rC 244 632 :M f2_12 sf (Equation 16)S endp %%Page: 32 32 %%BeginPageSetup initializepage (peter; page: 32 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (32)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf (where in each term )S 190 104 :M f4_12 sf (d)S f0_12 sf <28>S 200 104 :M f2_12 sf (C)S 209 104 :M f0_12 sf (\) is either equal to either 1 or -1.)S 95 134 :M (Proof. For each )S f2_12 sf (C)S 181 134 :M f0_12 sf ( that is a set of loops in )S 295 134 :M f4_12 sf (G)S 304 134 :M f0_12 sf <28>S 308 134 :M f2_12 sf (X)S 317 134 :M f0_12 sf (\) that do not intersect, let)S 95 180 :M ( )S 155 155 270 44 rC 425 199 :M psb currentpoint pse 155 155 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8640 div 1408 3 -1 roll exch div scale currentpoint translate 64 35 translate -3 765 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-Italic f1 (g) show 191 765 moveto 384 /Times-Roman f1 (\() show 323 765 moveto 384 /Times-Bold f1 (C) show 600 765 moveto 384 /Times-Roman f1 (\)) show 833 765 moveto 384 /Symbol f1 (=) show 1162 765 moveto 384 /Times-Italic f1 (d) show 1376 765 moveto 384 /Times-Roman f1 (\() show 1508 765 moveto 384 /Times-Bold f1 (C) show 1785 765 moveto 384 /Times-Roman f1 (\)) show 1984 765 moveto 384 /Symbol f1 (\264) show 4032 518 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 4240 518 moveto (e) show 4426 614 moveto 320 /Times-Italic f1 (Y) show 4069 1066 moveto 384 /Symbol f2 (\266) show 4277 1066 moveto 384 /Times-Italic f1 (W) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 4009 666 moveto 673 0 rlineto stroke 2460 1231 moveto 320 /Symbol f1 (<) show 2648 1231 moveto 320 /Times-Italic f1 (W) show 2940 1231 moveto 320 /Times-Roman f1 (,) show 3022 1231 moveto 320 /Times-Italic f1 (Y) show 3238 1231 moveto 320 /Symbol f1 (>) show 3487 1231 moveto (\316) show 3731 1231 moveto 320 /Times-Bold f1 (C) show 2968 853 moveto 576 /Symbol f1 (\325) show 2269 355 moveto 384 ns (\346) show 2269 1220 moveto (\350) show 2269 806 moveto (\347) show 2269 1001 moveto (\347) show 4694 355 moveto (\366) show 4694 1220 moveto (\370) show 4694 806 moveto (\367) show 4694 1001 moveto (\367) show 7684 516 moveto 384 /Symbol f2 (\266) show 7892 516 moveto (e) show 8078 612 moveto 320 /Times-Italic f1 (V) show 7771 1066 moveto 384 /Symbol f2 (\266) show 7979 1066 moveto 384 /Times-Italic f1 (V) show 7661 666 moveto 691 0 rlineto stroke 5068 1231 moveto 320 ns (V) show 5360 1231 moveto 320 /Symbol f1 (\316) show 5546 1231 moveto 320 /Times-Bold f1 (X) show 5826 1231 moveto 320 /Times-Roman f1 (\\) show 6030 1231 moveto 320 /Times-Bold f1 (Vertices) show 7154 1231 moveto 320 /Times-Roman f1 (\() show 7270 1231 moveto 320 /Times-Bold f1 (C) show 7507 1231 moveto 320 /Times-Roman f1 (\)) show 6101 853 moveto 576 /Symbol f1 (\325) show 4883 355 moveto 384 ns (\346) show 4883 1220 moveto (\350) show 4883 806 moveto (\347) show 4883 1001 moveto (\347) show 8364 355 moveto (\366) show 8364 1220 moveto (\370) show 8364 806 moveto (\367) show 8364 1001 moveto (\367) show end pse gR gS 0 0 552 730 rC 244 220 :M f2_12 sf (Equation 17)S 95 250 :M f0_12 sf .171 .017(I will show that that for each cycleset )J 280 250 :M f2_12 sf (C)S 289 250 :M f0_12 sf .135 .014( in )J f4_12 sf (G)S 313 250 :M f0_12 sf <28>S 317 250 :M f2_12 sf (X)S 326 250 :M f0_12 sf .196 .02(\) that )J 355 250 :M f4_12 sf (g)S f0_12 sf <28>S 365 250 :M f2_12 sf (C)S 374 250 :M f0_12 sf .199 .02(\) is a term in )J 439 250 :M f4_12 sf (J)S f2_10 sf 0 2 rm (X)S 0 -2 rm f0_10 sf 0 2 rm (-)S 0 -2 rm 95 271 :M (>)S 101 271 :M f2_10 sf (Err)S 117 271 :M f0_10 sf .317<28>A f2_10 sf .688(X)A f0_10 sf .317<29>A f0_12 sf 0 -2 rm 1.508 .151(, every non-zero term in )J 0 2 rm f4_12 sf 0 -2 rm .508(J)A 0 2 rm f2_10 sf .688(V)A f0_10 sf .855(->)A 284 271 :M f2_10 sf (Err)S 300 271 :M f0_10 sf .494<28>A f2_10 sf 1.071(V)A f0_10 sf .494<29>A f0_12 sf 0 -2 rm 1.548 .155( is equal to )J 0 2 rm f4_12 sf 0 -2 rm .89(g)A 0 2 rm f0_12 sf 0 -2 rm <28>S 0 2 rm 390 269 :M f2_12 sf (C)S 399 269 :M f0_12 sf 2.025 .203(\) for some)J 95 292 :M .476 .048(cycleset )J 138 292 :M f2_12 sf .779 .078(C )J 151 292 :M f0_12 sf .387 .039(in )J f4_12 sf (G)S 173 292 :M f0_12 sf <28>S 177 292 :M f2_12 sf (X)S 186 292 :M f0_12 sf .53 .053(\), and if )J f2_12 sf (C)S 238 294 :M f2_10 sf .232(1)A f0_12 sf 0 -2 rm .432 .043( and )J 0 2 rm f2_12 sf 0 -2 rm (C)S 0 2 rm 277 294 :M f2_10 sf .149(2)A f0_12 sf 0 -2 rm .486 .049( are distinct cyclesets then )J 0 2 rm f4_12 sf 0 -2 rm .179(g)A 0 2 rm f0_12 sf 0 -2 rm <28>S 0 2 rm 425 292 :M f2_12 sf (C)S 434 294 :M f2_10 sf .357(1)A f0_12 sf 0 -2 rm .454 .045(\) )J 0 2 rm 447 292 :M f1_12 sf S 95 314 :M f4_12 sf (g)S f0_12 sf <28>S 105 314 :M f2_12 sf (C)S 114 316 :M f2_10 sf (2)S f0_12 sf 0 -2 rm (\).)S 0 2 rm 95 345 :M 2.532 .253(For each )J 148 345 :M f2_12 sf (C)S 157 345 :M f0_12 sf 2.287 .229(, a variable occurs once in the denominator of a partial)J 95 363 :M .151 .015(derivative in )J 159 363 :M f4_12 sf (g)S f0_12 sf <28>S 169 363 :M f2_12 sf (C)S 178 363 :M f0_12 sf .156 .016(\), and once in the numerator of partial derivative in )J f4_12 sf .058(g)A f0_12 sf <28>S 438 363 :M f2_12 sf (C)S 447 363 :M f0_12 sf (\).)S 95 381 :M 2.097 .21(Hence one partial derivative from each row and each column of the)J 95 399 :M 2.646 .265(transformation matrix occurs in )J 269 399 :M f4_12 sf (g)S f0_12 sf <28>S 279 399 :M f2_12 sf (C)S 288 399 :M f0_12 sf 2.935 .294(\). But every product of partial)J 95 417 :M .204 .02(derivatives which consists of one partial derivative from each column and)J 95 435 :M .219 .022(each row of the transformation matrix is a term in )J f4_12 sf .073(J)A f2_10 sf 0 2 rm .099(X)A 0 -2 rm f0_10 sf 0 2 rm .123(->)A 0 -2 rm 361 437 :M f2_10 sf (Err)S 377 437 :M f0_10 sf <28>S f2_10 sf .084(X)A f0_10 sf <29>S f0_12 sf 0 -2 rm .193 .019( \(because )J 0 2 rm 439 435 :M f4_12 sf (J)S f2_10 sf 0 2 rm (X)S 0 -2 rm f0_10 sf 0 2 rm (-)S 0 -2 rm 95 456 :M (>)S 101 456 :M f2_10 sf (Err)S 117 456 :M f0_10 sf .078<28>A f2_10 sf .17(X)A f0_10 sf .078<29>A f0_12 sf 0 -2 rm .435 .043( is the determinant of the transformation matrix\). Hence)J 0 2 rm f4_12 sf 0 -2 rm .193 .019( g)J 0 2 rm 417 454 :M f0_12 sf <28>S 421 454 :M f2_12 sf (C)S 430 454 :M f0_12 sf .555 .055(\) is a)J 95 473 :M (term in )S f4_12 sf (J)S f2_10 sf 0 2 rm (X)S 0 -2 rm f0_10 sf 0 2 rm (->)S 0 -2 rm 153 475 :M f2_10 sf (Err)S 169 475 :M f0_10 sf <28>S f2_10 sf (X)S f0_10 sf <29>S f0_12 sf 0 -2 rm (.)S 0 2 rm 95 504 :M .768 .077(Let )J 115 504 :M f2_12 sf (C)S 124 506 :M f2_10 sf .277(1)A f0_12 sf 0 -2 rm .705 .071( be a set of cycles such that no pair of cycles in )J 0 2 rm 371 504 :M f2_12 sf (C)S 380 506 :M f2_10 sf .139(1)A f0_12 sf 0 -2 rm .546 .055( intersect, and)J 0 2 rm 95 527 :M (similarly for )S 158 527 :M f2_12 sf (C)S 167 529 :M f2_10 sf (2)S f0_12 sf 0 -2 rm (. Suppose that )S 0 2 rm 243 527 :M f2_12 sf (C)S 252 529 :M f2_10 sf (1)S f0_12 sf 0 -2 rm ( )S 0 2 rm 260 527 :M f1_12 sf S 267 527 :M f0_12 sf S f2_12 sf (C)S 279 529 :M f2_10 sf (2)S f0_12 sf 0 -2 rm (; then )S 0 2 rm 314 527 :M f4_12 sf (g)S f0_12 sf <28>S 324 527 :M f2_12 sf (C)S 333 529 :M f2_10 sf (1)S f0_12 sf 0 -2 rm (\) )S 0 2 rm 345 527 :M f1_12 sf S 352 527 :M f0_12 sf S f4_12 sf (g)S f0_12 sf <28>S 365 527 :M f2_12 sf (C)S 374 529 :M f2_10 sf (2)S f0_12 sf 0 -2 rm (\) unless there is)S 0 2 rm 95 549 :M .331 .033(some way to rearrange the edges in )J f2_12 sf (C)S 279 551 :M f2_10 sf .141(1)A f0_12 sf 0 -2 rm .358 .036( into the cycles in )J 0 2 rm 375 549 :M f2_12 sf (C)S 384 551 :M f2_10 sf .075(2)A f0_12 sf 0 -2 rm .303 .03(. But because)J 0 2 rm 95 568 :M .39 .039(no pair of cycles in )J 193 568 :M f2_12 sf (C)S 202 570 :M f2_10 sf .103(1)A f0_12 sf 0 -2 rm .341 .034( intersect, each vertex that appears in )J 0 2 rm 392 568 :M f2_12 sf (C)S 401 570 :M f2_10 sf .106(1)A f0_12 sf 0 -2 rm .34 .034( occurs in)J 0 2 rm 95 587 :M .211 .021(exactly two edges, once as the head, and once as the tail. Hence the edges)J 95 609 :M (in )S f2_12 sf (C)S 116 611 :M f2_10 sf (1)S f0_12 sf 0 -2 rm ( cannot be rearranged into the loops in )S 0 2 rm f2_12 sf 0 -2 rm (C)S 0 2 rm 317 611 :M f2_10 sf (2)S f0_12 sf 0 -2 rm (, and )S 0 2 rm f4_12 sf 0 -2 rm (g)S 0 2 rm f0_12 sf 0 -2 rm <28>S 0 2 rm 358 609 :M f2_12 sf (C)S 367 611 :M f2_10 sf (1)S f0_12 sf 0 -2 rm (\) )S 0 2 rm 379 609 :M f1_12 sf S 386 609 :M f0_12 sf S f4_12 sf (g)S f0_12 sf <28>S 399 609 :M f2_12 sf (C)S 408 611 :M f2_10 sf (2)S f0_12 sf 0 -2 rm (\).)S 0 2 rm endp %%Page: 33 33 %%BeginPageSetup initializepage (peter; page: 33 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (33)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .258 .026(By lemma 5, each minimally sufficient product of terms occurring in )J 434 104 :M f4_12 sf (T)S 441 104 :M f0_12 sf .347 .035( of)J 95 123 :M f4_12 sf (J)S f2_10 sf 0 2 rm (X)S 0 -2 rm f0_10 sf 0 2 rm (->)S 0 -2 rm 116 125 :M f2_10 sf (Err)S 132 125 :M f0_10 sf .537<28>A f2_10 sf 1.165(X)A f0_10 sf .537<29>A f0_12 sf 0 -2 rm 1.815 .182( is either a loop or )J 0 2 rm 251 113 42 15 rC 293 128 :M psb currentpoint pse 251 113 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1344 div 480 3 -1 roll exch div scale currentpoint translate 64 34 translate -9 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 199 286 moveto (e) show 390 382 moveto 224 /Times-Italic f1 (V) show 760 286 moveto 384 /Symbol f2 (\266) show 968 286 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -124 386 723 0 vb end pse gR gS 0 0 552 730 rC 293 123 :M f0_12 sf 2.328 .233( for )J 319 123 :M f4_12 sf 1.807(V)A f0_12 sf 1.644 .164( in )J 347 123 :M f2_12 sf (X)S 356 123 :M f0_12 sf 1.711 .171(. By definition, the)J 95 143 :M .415 .042(variables in distinct minimally sufficient product of terms do not overlap.)J 95 161 :M 1.523 .152(Hence )J 132 161 :M f4_12 sf (T)S 139 161 :M f0_12 sf 1.287 .129( consists of a product of non-intersecting minimally sufficient)J 95 179 :M .249 .025(products of terms. Hence, for every non-zero term )J f4_12 sf (T)S 349 179 :M f0_12 sf .144 .014( in )J f4_12 sf .12(J)A f2_10 sf 0 2 rm .163(X)A 0 -2 rm f0_10 sf 0 2 rm .202(->)A 0 -2 rm 386 181 :M f2_10 sf (Err)S 402 181 :M f0_10 sf .053<28>A f2_10 sf .116(X)A f0_10 sf .053<29>A f0_12 sf 0 -2 rm .217 .022( there is)J 0 2 rm 95 198 :M (a cycleset )S 145 198 :M f2_12 sf (C)S 154 198 :M f0_12 sf ( such that )S 206 198 :M f4_12 sf (T)S 213 198 :M f0_12 sf ( = )S 226 198 :M f4_12 sf (g)S f0_12 sf <28>S 236 198 :M f2_12 sf (C)S 245 198 :M f0_12 sf (\). )S 255 189 9 9 rC gS 1.286 1 scale 198.335 198 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 95 228 :M f0_12 sf (Let )S 114 228 :M f2_12 sf (Cyclegroup)S f0_12 sf <28>S 177 228 :M f4_12 sf (G)S 186 228 :M f0_12 sf -.006(\) be the set of all cyclegroups in )A 343 228 :M f4_12 sf (G)S 352 228 :M f0_12 sf -.013(. If )A 369 228 :M f2_12 sf (C)S 378 228 :M f0_12 sf -.005( is a cyclegroup)A 95 246 :M (in )S f4_12 sf (G)S 116 246 :M f0_12 sf (, let )S 137 246 :M f2_12 sf (Cycleset)S 180 246 :M f0_12 sf <28>S 184 246 :M f2_12 sf (C)S 193 246 :M f0_12 sf (\) be the set of all cyclesets included in )S f2_12 sf (C)S 388 246 :M f0_12 sf (.)S 95 276 :M f2_12 sf .722 .072(Lemma 7: )J f0_12 sf .636 .064(In an SEM with directed graph )J f4_12 sf (G)S 318 276 :M f0_12 sf .532 .053(, if )J f2_12 sf (X)S 345 276 :M f0_12 sf .697 .07( is an ancestral set for)J 95 294 :M f4_12 sf (G)S 104 294 :M f0_12 sf (, then)S 95 309 390 120 rC 485 429 :M psb currentpoint pse 95 309 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 12480 div 3840 3 -1 roll exch div scale currentpoint translate 64 62 translate 5185 261 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-Italic f1 (J) show 5360 357 moveto 320 /Times-Bold f1 (X) show 5611 357 moveto 320 /Symbol f1 (-) show 5806 357 moveto (>) show 5999 357 moveto 320 /Times-Bold f1 (Err) show 6504 357 moveto 320 /Times-Roman f1 (\() show 6626 357 moveto 320 /Times-Bold f1 (X) show 6871 357 moveto 320 /Times-Roman f1 (\)) show 7117 261 moveto 384 /Symbol f1 (=) show 6919 1161 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 7127 1161 moveto (e) show 7313 1257 moveto 320 /Times-Italic f1 (V) show 7006 1711 moveto 384 /Symbol f2 (\266) show 7214 1711 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 6896 1311 moveto 691 0 rlineto stroke 4471 1876 moveto 320 ns (V) show 4763 1876 moveto 320 /Symbol f1 (\317) show 5007 1876 moveto 320 /Times-Bold f1 (Cycles) show 5901 1876 moveto 320 /Times-Roman f1 (\() show 6021 1876 moveto 320 /Times-Italic f1 (G) show 6262 1876 moveto 320 /Times-Roman f1 (\() show 6384 1876 moveto 320 /Times-Bold f1 (X) show 6629 1876 moveto 320 /Times-Roman f1 (\)) show 6742 1876 moveto (\)) show 5420 1498 moveto 576 /Symbol f1 (\325) show 4286 1000 moveto 384 ns (\346) show 4286 1865 moveto (\350) show 4286 1451 moveto (\347) show 4286 1646 moveto (\347) show 7599 1000 moveto (\366) show 7599 1865 moveto (\370) show 7599 1451 moveto (\367) show 7599 1646 moveto (\367) show 7861 1410 moveto (\264) show 5844 3070 moveto 384 /Times-Italic f1 (d) show 6058 3070 moveto 384 /Times-Roman f1 (\() show 6198 3070 moveto 384 /Times-Bold f1 (D) show 6481 3070 moveto 384 /Times-Roman f1 (\)) show 6680 3070 moveto 384 /Symbol f1 (\264) show 8403 2821 moveto 384 /Symbol f2 (\266) show 8611 2821 moveto (e) show 8797 2917 moveto 320 /Times-Italic f1 (V) show 8490 3371 moveto 384 /Symbol f2 (\266) show 8698 3371 moveto 384 /Times-Italic f1 (V) show 8380 2971 moveto 691 0 rlineto stroke 7150 3536 moveto 320 ns (V) show 7442 3536 moveto 320 /Symbol f1 (\316) show 7686 3536 moveto 320 /Times-Bold f1 (C) show 7958 3536 moveto 320 /Times-Roman f1 (\\) show 8098 3536 moveto 320 /Times-Bold f1 (D) show 7502 3158 moveto 576 /Symbol f1 (\325) show 6965 2660 moveto 384 ns (\346) show 6965 3525 moveto (\350) show 6965 3111 moveto (\347) show 6965 3306 moveto (\347) show 9083 2660 moveto (\366) show 9083 3525 moveto (\370) show 9083 3111 moveto (\367) show 9083 3306 moveto (\367) show 10981 2823 moveto 384 /Symbol f2 (\266) show 11189 2823 moveto (e) show 11375 2919 moveto 320 /Times-Italic f1 (Y) show 11018 3371 moveto 384 /Symbol f2 (\266) show 11226 3371 moveto 384 /Times-Italic f1 (W) show 10958 2971 moveto 673 0 rlineto stroke 9463 3536 moveto 320 /Symbol f1 (<) show 9651 3536 moveto 320 /Times-Italic f1 (W) show 9943 3536 moveto 320 /Times-Roman f1 (,) show 10025 3536 moveto 320 /Times-Italic f1 (Y) show 10241 3536 moveto 320 /Symbol f1 (>) show 10490 3536 moveto (\316) show 10676 3536 moveto 320 /Times-Bold f1 (D) show 9944 3158 moveto 576 /Symbol f1 (\325) show 9272 2660 moveto 384 ns (\346) show 9272 3525 moveto (\350) show 9272 3111 moveto (\347) show 9272 3306 moveto (\347) show 11643 2660 moveto (\366) show 11643 3525 moveto (\370) show 11643 3111 moveto (\367) show 11643 3306 moveto (\367) show 5650 2628 moveto (\346) show 5650 3557 moveto (\350) show 5650 3079 moveto (\347) show 5650 3338 moveto (\347) show 11827 2628 moveto (\366) show 11827 3557 moveto (\370) show 11827 3079 moveto (\367) show 11827 3338 moveto (\367) show 3466 3539 moveto 320 /Times-Bold f1 (D) show 3766 3539 moveto 320 /Symbol f1 (\316) show 4010 3539 moveto 320 /Times-Bold f1 (Cycleset) show 5158 3539 moveto 320 /Times-Roman f1 (\() show 5274 3539 moveto 320 /Times-Bold f1 (C) show 5511 3539 moveto 320 /Times-Roman f1 (\)) show 4333 3157 moveto 576 /Symbol f1 (\345) show 3274 2596 moveto 384 ns (\346) show 3274 3589 moveto (\350) show 3274 3047 moveto (\347) show 3274 3370 moveto (\347) show 12011 2596 moveto (\366) show 12011 3589 moveto (\370) show 12011 3047 moveto (\367) show 12011 3370 moveto (\367) show 171 3536 moveto 320 /Times-Bold f1 (C) show 467 3536 moveto 320 /Symbol f1 (\316) show 711 3536 moveto 320 /Times-Bold f1 (Cyclegroup) show 2294 3536 moveto 320 /Times-Roman f1 (\() show 2414 3536 moveto 320 /Times-Italic f1 (G) show 2655 3536 moveto 320 /Times-Roman f1 (\() show 2777 3536 moveto 320 /Times-Bold f1 (X) show 3022 3536 moveto 320 /Times-Roman f1 (\)) show 3135 3536 moveto (\)) show 1466 3158 moveto 576 /Symbol f1 (\325) show -15 2564 moveto 384 ns (\346) show -15 3621 moveto (\350) show -15 3015 moveto (\347) show -15 3386 moveto (\347) show 12195 2564 moveto (\366) show 12195 3621 moveto (\370) show 12195 3015 moveto (\367) show 12195 3386 moveto (\367) show end pse gR gS 0 0 552 730 rC 242 438 :M f2_12 sf ( Equation 18)S 95 468 :M f0_12 sf (where )S f4_12 sf (d)S f0_12 sf <28>S 137 468 :M f2_12 sf (D)S 146 468 :M f0_12 sf (\) is a variable equal either to 1 or -1.)S 95 498 :M (Proof. By lemma 6,)S 95 513 384 44 rC 479 557 :M psb currentpoint pse 95 513 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 12288 div 1408 3 -1 roll exch div scale currentpoint translate 64 35 translate 13 765 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-Italic f1 (J) show 188 861 moveto 320 /Times-Bold f1 (X) show 439 861 moveto 320 /Symbol f1 (-) show 634 861 moveto (>) show 827 861 moveto 320 /Times-Bold f1 (Err) show 1332 861 moveto 320 /Times-Roman f1 (\() show 1454 861 moveto 320 /Times-Bold f1 (X) show 1699 861 moveto 320 /Times-Roman f1 (\)) show 1945 765 moveto 384 /Symbol f1 (=) show 4816 765 moveto 384 /Times-Italic f1 (d) show 5030 765 moveto 384 /Times-Roman f1 (\() show 5162 765 moveto 384 /Times-Bold f1 (C) show 5439 765 moveto 384 /Times-Roman f1 (\)) show 5638 765 moveto 384 /Symbol f1 (\264) show 7686 518 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 7894 518 moveto (e) show 8080 614 moveto 320 /Times-Italic f1 (Y) show 7723 1066 moveto 384 /Symbol f2 (\266) show 7931 1066 moveto 384 /Times-Italic f1 (W) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 7663 666 moveto 673 0 rlineto stroke 6114 1231 moveto 320 /Symbol f1 (<) show 6302 1231 moveto 320 /Times-Italic f1 (W) show 6594 1231 moveto 320 /Times-Roman f1 (,) show 6676 1231 moveto 320 /Times-Italic f1 (Y) show 6892 1231 moveto 320 /Symbol f1 (>) show 7141 1231 moveto (\316) show 7385 1231 moveto 320 /Times-Bold f1 (C) show 6622 853 moveto 576 /Symbol f1 (\325) show 5923 355 moveto 384 ns (\346) show 5923 1220 moveto (\350) show 5923 806 moveto (\347) show 5923 1001 moveto (\347) show 8348 355 moveto (\366) show 8348 1220 moveto (\370) show 8348 806 moveto (\367) show 8348 1001 moveto (\367) show 11338 516 moveto 384 /Symbol f2 (\266) show 11546 516 moveto (e) show 11732 612 moveto 320 /Times-Italic f1 (V) show 11425 1066 moveto 384 /Symbol f2 (\266) show 11633 1066 moveto 384 /Times-Italic f1 (V) show 11315 666 moveto 691 0 rlineto stroke 8722 1231 moveto 320 ns (V) show 9014 1231 moveto 320 /Symbol f1 (\316) show 9200 1231 moveto 320 /Times-Bold f1 (X) show 9480 1231 moveto 320 /Times-Roman f1 (\\) show 9684 1231 moveto 320 /Times-Bold f1 (Vertices) show 10808 1231 moveto 320 /Times-Roman f1 (\() show 10924 1231 moveto 320 /Times-Bold f1 (C) show 11161 1231 moveto 320 /Times-Roman f1 (\)) show 9755 853 moveto 576 /Symbol f1 (\325) show 8537 355 moveto 384 ns (\346) show 8537 1220 moveto (\350) show 8537 806 moveto (\347) show 8537 1001 moveto (\347) show 12018 355 moveto (\366) show 12018 1220 moveto (\370) show 12018 806 moveto (\367) show 12018 1001 moveto (\367) show 2266 1234 moveto 320 /Times-Bold f1 (C) show 2498 1234 moveto 320 /Symbol f1 (\316) show 2678 1234 moveto 320 /Times-Bold f1 (Cycleset) show 3826 1234 moveto 320 /Times-Roman f1 (\() show 3946 1234 moveto 320 /Times-Italic f1 (G) show 4187 1234 moveto 320 /Times-Roman f1 (\() show 4309 1234 moveto 320 /Times-Bold f1 (X) show 4554 1234 moveto 320 /Times-Roman f1 (\)) show 4667 1234 moveto (\)) show 3314 852 moveto 576 /Symbol f1 (\345) show end pse gR gS 0 0 552 730 rC 244 566 :M f2_12 sf (Equation 19)S endp %%Page: 34 34 %%BeginPageSetup initializepage (peter; page: 34 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (34)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf 1.402 .14(If )J 108 104 :M f4_12 sf .855(V)A f0_12 sf 1.2 .12( is not in a cycle in )J 221 104 :M f4_12 sf (G)S 230 104 :M f0_12 sf <28>S 234 104 :M f2_12 sf (X)S 243 104 :M f0_12 sf 1.147 .115(\) then it is not in any cycleset. Hence, by)J 95 122 :M .508 .051(lemma 5, every occurrence of )J f4_12 sf .202(V)A f0_12 sf .399 .04( in each non-zero term in )J 379 122 :M f4_12 sf (J)S f2_10 sf 0 2 rm (X)S 0 -2 rm f0_10 sf 0 2 rm (->)S 0 -2 rm 400 124 :M f2_10 sf (Err)S 416 124 :M f0_10 sf .122<28>A f2_10 sf .264(X)A f0_10 sf .122<29>A f0_12 sf 0 -2 rm .351 .035( is of)J 0 2 rm 95 142 :M (the form )S 139 132 42 15 rC 181 147 :M psb currentpoint pse 139 132 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1344 div 480 3 -1 roll exch div scale currentpoint translate 64 34 translate -9 286 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 199 286 moveto (e) show 390 382 moveto 224 /Times-Italic f1 (V) show 760 286 moveto 384 /Symbol f2 (\266) show 968 286 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -124 386 723 0 vb end pse gR gS 0 0 552 730 rC 181 142 :M f0_12 sf (. Hence it is possible to factor)S 218 165 112 44 rC 330 209 :M psb currentpoint pse 218 165 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3584 div 1408 3 -1 roll exch div scale currentpoint translate 64 35 translate 2618 516 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 2826 516 moveto (e) show 3012 612 moveto 320 /Times-Italic f1 (V) show 2705 1066 moveto 384 /Symbol f2 (\266) show 2913 1066 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2595 666 moveto 691 0 rlineto stroke 170 1231 moveto 320 ns (V) show 462 1231 moveto 320 /Symbol f1 (\317) show 706 1231 moveto 320 /Times-Bold f1 (Cycles) show 1600 1231 moveto 320 /Times-Roman f1 (\() show 1720 1231 moveto 320 /Times-Italic f1 (G) show 1961 1231 moveto 320 /Times-Roman f1 (\() show 2083 1231 moveto 320 /Times-Bold f1 (X) show 2328 1231 moveto 320 /Times-Roman f1 (\)) show 2441 1231 moveto (\)) show 1119 853 moveto 576 /Symbol f1 (\325) show -15 355 moveto 384 ns (\346) show -15 1220 moveto (\350) show -15 806 moveto (\347) show -15 1001 moveto (\347) show 3298 355 moveto (\366) show 3298 1220 moveto (\370) show 3298 806 moveto (\367) show 3298 1001 moveto (\367) show end pse gR gS 0 0 552 730 rC 244 230 :M f2_12 sf (Equation 20)S 95 260 :M f0_12 sf .746 .075(from each non-zero term in the previous equation, because if )J 402 260 :M f4_12 sf .346(V)A f0_12 sf .729 .073( does not)J 95 278 :M (occur in a cycle, it does not occur in any cycleset. This leads to)S 95 293 388 77 rC 483 370 :M psb currentpoint pse 95 293 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 12416 div 2464 3 -1 roll exch div scale currentpoint translate 64 39 translate 5306 261 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-Italic f1 (J) show 5483 357 moveto 224 /Times-Bold f1 (X) show 5668 357 moveto 224 /Symbol f1 (-) show 5814 357 moveto (>) show 5959 357 moveto 224 /Times-Bold f1 (Err) show 6322 357 moveto 224 /Times-Roman f1 (\() show 6417 357 moveto 224 /Times-Bold f1 (X) show 6598 357 moveto 224 /Times-Roman f1 (\)) show 6817 261 moveto 384 /Symbol f1 (=) show 1935 1637 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 2143 1637 moveto (e) show 2334 1733 moveto 224 /Times-Italic f1 (V) show 1993 2185 moveto 384 /Symbol f2 (\266) show 2201 2185 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1912 1785 moveto 633 0 rlineto stroke 175 2254 moveto 224 ns (V) show 376 2254 moveto 224 /Symbol f1 (\317) show 512 2254 moveto 224 /Times-Bold f1 (Cycles) show 1147 2254 moveto 224 /Times-Roman f1 (\() show 1241 2254 moveto 224 /Times-Italic f1 (G) show 1420 2254 moveto 224 /Times-Roman f1 (\() show 1515 2254 moveto 224 /Times-Bold f1 (X) show 1696 2254 moveto 224 /Times-Roman f1 (\)) show 1785 2254 moveto (\)) show 778 1972 moveto 576 /Symbol f1 (\325) show -15 1591 moveto 384 ns (\346) show -15 2222 moveto (\350) show -15 2003 moveto (\347) show 2557 1591 moveto (\366) show 2557 2222 moveto (\370) show 2557 2003 moveto (\367) show 4815 1884 moveto 384 /Times-Italic f1 (d) show 5029 1884 moveto 384 /Times-Roman f1 (\() show 5161 1884 moveto 384 /Times-Bold f1 (C) show 5438 1884 moveto 384 /Times-Roman f1 (\)) show 5637 1884 moveto 384 /Symbol f1 (\264) show 7276 1637 moveto 384 /Symbol f2 (\266) show 7484 1637 moveto (e) show 7675 1733 moveto 224 /Times-Italic f1 (Y) show 7287 2185 moveto 384 /Symbol f2 (\266) show 7495 2185 moveto 384 /Times-Italic f1 (W) show 7253 1785 moveto 620 0 rlineto stroke 6116 2254 moveto 224 /Symbol f1 (<) show 6258 2254 moveto 224 /Times-Italic f1 (W) show 6472 2254 moveto 224 /Times-Roman f1 (,) show 6540 2254 moveto 224 /Times-Italic f1 (Y) show 6701 2254 moveto 224 /Symbol f1 (>) show 6872 2254 moveto (\316) show 7040 2254 moveto 224 /Times-Bold f1 (C) show 6417 1972 moveto 576 /Symbol f1 (\325) show 5922 1591 moveto 384 ns (\346) show 5922 2222 moveto (\350) show 5922 2003 moveto (\347) show 7885 1591 moveto (\366) show 7885 2222 moveto (\370) show 7885 2003 moveto (\367) show 11338 1637 moveto 384 /Symbol f2 (\266) show 11546 1637 moveto (e) show 11737 1733 moveto 224 /Times-Italic f1 (V) show 11396 2185 moveto 384 /Symbol f2 (\266) show 11604 2185 moveto 384 /Times-Italic f1 (V) show 11315 1785 moveto 633 0 rlineto stroke 8264 2254 moveto 224 ns (V) show 8465 2254 moveto 224 /Symbol f1 (\316) show 8601 2254 moveto 224 /Times-Bold f1 (Cycles) show 9236 2254 moveto 224 /Times-Roman f1 (\() show 9330 2254 moveto 224 /Times-Italic f1 (G) show 9509 2254 moveto 224 /Times-Roman f1 (\() show 9604 2254 moveto 224 /Times-Bold f1 (X) show 9785 2254 moveto 224 /Times-Roman f1 (\)) show 9874 2254 moveto (\)) show 9987 2254 moveto (\\) show 10127 2254 moveto 224 /Times-Bold f1 (Vertices) show 10922 2254 moveto 224 /Times-Roman f1 (\() show 11013 2254 moveto 224 /Times-Bold f1 (C) show 11188 2254 moveto 224 /Times-Roman f1 (\)) show 9524 1972 moveto 576 /Symbol f1 (\325) show 8074 1591 moveto 384 ns (\346) show 8074 2222 moveto (\350) show 8074 2003 moveto (\347) show 11960 1591 moveto (\366) show 11960 2222 moveto (\370) show 11960 2003 moveto (\367) show 2936 2257 moveto 224 /Times-Bold f1 (C) show 3107 2257 moveto 224 /Symbol f1 (\316) show 3243 2257 moveto 224 /Times-Bold f1 (Cycleset) show 4055 2257 moveto 224 /Times-Roman f1 (\() show 4149 2257 moveto 224 /Times-Italic f1 (G) show 4328 2257 moveto 224 /Times-Roman f1 (\() show 4423 2257 moveto 224 /Times-Bold f1 (X) show 4604 2257 moveto 224 /Times-Roman f1 (\)) show 4693 2257 moveto (\)) show 3646 1971 moveto 576 /Symbol f1 (\345) show 2746 1559 moveto 384 ns (\346) show 2746 2254 moveto (\350) show 2746 2010 moveto (\347) show 12144 1559 moveto (\366) show 12144 2254 moveto (\370) show 12144 2010 moveto (\367) show end pse gR gS 0 0 552 730 rC 244 379 :M f2_12 sf (Equation 21)S 95 409 :M f0_12 sf .467 .047(The set of cyclegroups in )J f4_12 sf (G)S 232 409 :M f0_12 sf .52 .052( partitions the set of cycles in )J 381 409 :M f4_12 sf (G)S 390 409 :M f0_12 sf .49 .049(. Hence each)J 95 427 :M 2.197 .22(cycleset in )J 156 427 :M f4_12 sf (G)S 165 427 :M f0_12 sf 2.211 .221( can be partitioned into a set of cyclesets, where each)J 95 445 :M 1.987 .199(cycleset contains only cycles from the same cyclegroup. In addition,)J 95 463 :M 1.105 .111(suppose that )J 162 463 :M f2_12 sf (C)S 171 463 :M f0_12 sf 1.156 .116( is a set of cyclesets, where each cycleset in )J f2_12 sf (C)S 409 463 :M f0_12 sf .982 .098( contains)J 95 481 :M .266 .027(cycles from only one cyclegroup, and each pair of cyclesets in )J 402 481 :M f2_12 sf (C)S 411 481 :M f0_12 sf .231 .023( contains)J 95 499 :M .34 .034(cycles from different cyclegroups. Then the union of any two cyclesets in)J 95 517 :M f2_12 sf (C)S 104 517 :M f0_12 sf ( is also a cycleset. Hence)S endp %%Page: 35 35 %%BeginPageSetup initializepage (peter; page: 35 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (35)S gR gS 95 95 359 96 rC 454 191 :M psb currentpoint pse 95 95 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11488 div 3072 3 -1 roll exch div scale currentpoint translate 64 53 translate 2978 765 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-Italic f1 (d) show 3192 765 moveto 384 /Times-Roman f1 (\() show 3332 765 moveto 384 /Times-Bold f1 (D) show 3615 765 moveto 384 /Times-Roman f1 (\)) show 3814 765 moveto 384 /Symbol f1 (\264) show 5862 518 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 6070 518 moveto (e) show 6256 614 moveto 320 /Times-Italic f1 (Y) show 5899 1066 moveto 384 /Symbol f2 (\266) show 6107 1066 moveto 384 /Times-Italic f1 (W) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 5839 666 moveto 673 0 rlineto stroke 4290 1231 moveto 320 /Symbol f1 (<) show 4478 1231 moveto 320 /Times-Italic f1 (W) show 4770 1231 moveto 320 /Times-Roman f1 (,) show 4852 1231 moveto 320 /Times-Italic f1 (Y) show 5068 1231 moveto 320 /Symbol f1 (>) show 5317 1231 moveto (\316) show 5561 1231 moveto 320 /Times-Bold f1 (C) show 4798 853 moveto 576 /Symbol f1 (\325) show 4099 355 moveto 384 ns (\346) show 4099 1220 moveto (\350) show 4099 806 moveto (\347) show 4099 1001 moveto (\347) show 6524 355 moveto (\366) show 6524 1220 moveto (\370) show 6524 806 moveto (\367) show 6524 1001 moveto (\367) show 9752 516 moveto 384 /Symbol f2 (\266) show 9960 516 moveto (e) show 10146 612 moveto 320 /Times-Italic f1 (V) show 9839 1066 moveto 384 /Symbol f2 (\266) show 10047 1066 moveto 384 /Times-Italic f1 (V) show 9729 666 moveto 691 0 rlineto stroke 6898 1231 moveto 320 ns (V) show 7190 1231 moveto 320 /Symbol f1 (\316) show 7370 1231 moveto 320 /Times-Bold f1 (Cycles) show 8264 1231 moveto 320 /Times-Roman f1 (\() show 8384 1231 moveto 320 /Times-Italic f1 (G) show 8625 1231 moveto 320 /Times-Roman f1 (\() show 8747 1231 moveto 320 /Times-Bold f1 (X) show 8992 1231 moveto 320 /Times-Roman f1 (\)) show 9105 1231 moveto (\)) show 9253 1231 moveto (\\) show 9451 1231 moveto 320 /Times-Bold f1 (C) show 8050 853 moveto 576 /Symbol f1 (\325) show 6713 355 moveto 384 ns (\346) show 6713 1220 moveto (\350) show 6713 806 moveto (\347) show 6713 1001 moveto (\347) show 10432 355 moveto (\366) show 10432 1220 moveto (\370) show 10432 806 moveto (\367) show 10432 1001 moveto (\367) show 428 1234 moveto 320 /Times-Bold f1 (C) show 660 1234 moveto 320 /Symbol f1 (\316) show 840 1234 moveto 320 /Times-Bold f1 (Cycleset) show 1988 1234 moveto 320 /Times-Roman f1 (\() show 2108 1234 moveto 320 /Times-Italic f1 (G) show 2349 1234 moveto 320 /Times-Roman f1 (\() show 2471 1234 moveto 320 /Times-Bold f1 (X) show 2716 1234 moveto 320 /Times-Roman f1 (\)) show 2829 1234 moveto (\)) show 1476 852 moveto 576 /Symbol f1 (\345) show 10728 765 moveto 384 ns (=) show 5220 2361 moveto 384 /Times-Italic f1 (d) show 5434 2361 moveto 384 /Times-Roman f1 (\() show 5574 2361 moveto 384 /Times-Bold f1 (D) show 5857 2361 moveto 384 /Times-Roman f1 (\)) show 6056 2361 moveto 384 /Symbol f1 (\264) show 7715 2112 moveto 384 /Symbol f2 (\266) show 7923 2112 moveto (e) show 8109 2208 moveto 320 /Times-Italic f1 (V) show 7802 2662 moveto 384 /Symbol f2 (\266) show 8010 2662 moveto 384 /Times-Italic f1 (V) show 7692 2262 moveto 691 0 rlineto stroke 6526 2827 moveto 320 ns (V) show 6818 2827 moveto 320 /Symbol f1 (\316) show 6998 2827 moveto 320 /Times-Bold f1 (C) show 7270 2827 moveto 320 /Times-Roman f1 (\\) show 7410 2827 moveto 320 /Times-Bold f1 (D) show 6846 2449 moveto 576 /Symbol f1 (\325) show 6341 1951 moveto 384 ns (\346) show 6341 2816 moveto (\350) show 6341 2402 moveto (\347) show 6341 2597 moveto (\347) show 8395 1951 moveto (\366) show 8395 2816 moveto (\370) show 8395 2402 moveto (\367) show 8395 2597 moveto (\367) show 10357 2114 moveto 384 /Symbol f2 (\266) show 10565 2114 moveto (e) show 10751 2210 moveto 320 /Times-Italic f1 (Y) show 10394 2662 moveto 384 /Symbol f2 (\266) show 10602 2662 moveto 384 /Times-Italic f1 (W) show 10334 2262 moveto 673 0 rlineto stroke 8775 2827 moveto 320 /Symbol f1 (<) show 8963 2827 moveto 320 /Times-Italic f1 (W) show 9255 2827 moveto 320 /Times-Roman f1 (,) show 9337 2827 moveto 320 /Times-Italic f1 (Y) show 9553 2827 moveto 320 /Symbol f1 (>) show 9802 2827 moveto (\316) show 10052 2827 moveto 320 /Times-Bold f1 (D) show 9288 2449 moveto 576 /Symbol f1 (\325) show 8584 1951 moveto 384 ns (\346) show 8584 2816 moveto (\350) show 8584 2402 moveto (\347) show 8584 2597 moveto (\347) show 11019 1951 moveto (\366) show 11019 2816 moveto (\370) show 11019 2402 moveto (\367) show 11019 2597 moveto (\367) show 3154 2830 moveto 320 /Times-Bold f1 (D) show 3390 2830 moveto 320 /Symbol f1 (\316) show 3570 2830 moveto 320 /Times-Bold f1 (Cycleset) show 4718 2830 moveto 320 /Times-Roman f1 (\() show 4834 2830 moveto 320 /Times-Bold f1 (C) show 5071 2830 moveto 320 /Times-Roman f1 (\)) show 3957 2448 moveto 576 /Symbol f1 (\345) show 2962 1919 moveto 384 ns (\346) show 2962 2848 moveto (\350) show 2962 2370 moveto (\347) show 2962 2629 moveto (\347) show 11203 1919 moveto (\366) show 11203 2848 moveto (\370) show 11203 2370 moveto (\367) show 11203 2629 moveto (\367) show -13 2827 moveto 320 /Times-Bold f1 (C) show 219 2827 moveto 320 /Symbol f1 (\316) show 399 2827 moveto 320 /Times-Bold f1 (Cyclegroup) show 1982 2827 moveto 320 /Times-Roman f1 (\() show 2102 2827 moveto 320 /Times-Italic f1 (G) show 2343 2827 moveto 320 /Times-Roman f1 (\() show 2465 2827 moveto 320 /Times-Bold f1 (X) show 2710 2827 moveto 320 /Times-Roman f1 (\)) show 2823 2827 moveto (\)) show 1218 2449 moveto 576 /Symbol f1 (\325) show end pse gR gS 0 0 552 730 rC 244 200 :M f2_12 sf (Equation 22)S 95 224 9 9 rC gS 1.286 1 scale 73.889 233 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 95 260 :M f2_12 sf .751 .075(Lemma 8:)J f0_12 sf .412 .041( For an SEM with directed graph )J 313 260 :M f4_12 sf (G)S 322 260 :M f0_12 sf .469 .047( with vertices )J 392 260 :M f2_12 sf (V)S 401 260 :M f0_12 sf .574 .057(, if )J 419 260 :M f2_12 sf (X)S 428 260 :M f0_12 sf .551 .055( is an)J 95 278 :M (ancestral set for )S 174 278 :M f4_12 sf (G)S 183 278 :M f0_12 sf ( then)S 95 293 369 64 rC 464 357 :M psb currentpoint pse 95 293 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11808 div 2048 3 -1 roll exch div scale currentpoint translate 64 58 translate 5240 262 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-Italic f1 (f) show 5390 358 moveto 320 /Times-Bold f1 (V) show 5658 262 moveto 384 /Times-Roman f1 (\() show 5798 262 moveto 384 /Times-Bold f1 (X) show 6085 262 moveto 384 /Times-Roman f1 (\)) show 6318 262 moveto 384 /Symbol f1 (=) show 1642 1360 moveto 384 /Times-Italic f1 (g) show 1818 1456 moveto 320 ns (V) show 2074 1360 moveto 384 /Times-Roman f1 (\() show 2206 1360 moveto 384 /Times-Italic f1 (V) show 2467 1360 moveto 384 /Times-Roman f1 (,) show 2599 1360 moveto 384 /Times-Bold f1 (Parents) show 3856 1360 moveto 384 /Times-Roman f1 (\() show 3988 1360 moveto 384 /Times-Italic f1 (V) show 4249 1360 moveto 384 /Times-Roman f1 (,) show 4379 1360 moveto 384 /Times-Italic f1 (G) show 4662 1360 moveto 384 /Times-Roman f1 (\() show 4802 1360 moveto 384 /Times-Bold f1 (X) show 5089 1360 moveto 384 /Times-Roman f1 (\)) show 5219 1360 moveto (\)) show 170 1826 moveto 320 /Times-Italic f1 (V) show 462 1826 moveto 320 /Symbol f1 (\317) show 706 1826 moveto 320 /Times-Bold f1 (Cycles) show 1600 1826 moveto 320 /Times-Roman f1 (\() show 1720 1826 moveto 320 /Times-Italic f1 (G) show 1961 1826 moveto 320 /Times-Roman f1 (\() show 2083 1826 moveto 320 /Times-Bold f1 (X) show 2328 1826 moveto 320 /Times-Roman f1 (\)) show 2441 1826 moveto (\)) show 1119 1448 moveto 576 /Symbol f1 (\325) show -15 950 moveto 384 ns (\346) show -15 1815 moveto (\350) show -15 1401 moveto (\347) show -15 1596 moveto (\347) show 5340 950 moveto (\366) show 5340 1815 moveto (\370) show 5340 1401 moveto (\367) show 5340 1596 moveto (\367) show 5602 1360 moveto (\264) show 7827 1360 moveto 384 /Times-Italic f1 (g) show 8004 1459 moveto 320 /Times-Bold f1 (C) show 8264 1360 moveto 384 /Times-Roman f1 (\() show 8396 1360 moveto 384 /Times-Bold f1 (C) show 8660 1360 moveto 384 /Times-Roman f1 (,) show 8792 1360 moveto 384 /Times-Bold f1 (Parents) show 10049 1360 moveto 384 /Times-Roman f1 (\() show 10181 1360 moveto 384 /Times-Bold f1 (C) show 10445 1360 moveto 384 /Times-Roman f1 (,) show 10575 1360 moveto 384 /Times-Italic f1 (G) show 10858 1360 moveto 384 /Times-Roman f1 (\() show 10998 1360 moveto 384 /Times-Bold f1 (X) show 11285 1360 moveto 384 /Times-Roman f1 (\)) show 11415 1360 moveto (\)) show 6073 1826 moveto 320 /Times-Bold f1 (C) show 6305 1826 moveto 320 /Symbol f1 (\316) show 6485 1826 moveto 320 /Times-Bold f1 (Cyclegroup) show 8068 1826 moveto 320 /Times-Roman f1 (\() show 8188 1826 moveto 320 /Times-Italic f1 (G) show 8429 1826 moveto 320 /Times-Roman f1 (\() show 8551 1826 moveto 320 /Times-Bold f1 (X) show 8796 1826 moveto 320 /Times-Roman f1 (\)) show 8909 1826 moveto (\)) show 7304 1448 moveto 576 /Symbol f1 (\325) show 5887 952 moveto 384 ns (\346) show 5887 1813 moveto (\350) show 5887 1403 moveto (\347) show 5887 1594 moveto (\347) show 11536 952 moveto (\366) show 11536 1813 moveto (\370) show 11536 1403 moveto (\367) show 11536 1594 moveto (\367) show end pse gR gS 0 0 552 730 rC 244 366 :M f2_12 sf (Equation 23)S 95 396 :M f0_12 sf (where each )S f4_12 sf (g)S f0_12 sf ( is a non-negative function.)S 95 426 :M (Proof. The transformed density function of )S 304 426 :M f2_12 sf (Err)S 323 426 :M f0_12 sf <28>S 327 426 :M f2_12 sf (X)S 336 426 :M f0_12 sf (\) is equal to)S 95 467 :M ( )S 170 447 233 34 rC 403 481 :M psb currentpoint pse 170 447 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7456 div 1088 3 -1 roll exch div scale currentpoint translate 64 40 translate 762 600 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-Italic f1 (f) show 915 696 moveto 224 /Times-Bold f1 (Err) show 1302 600 moveto 384 /Times-Roman f1 (\() show 1442 600 moveto 384 /Times-Italic f1 (h) show 1631 696 moveto 224 ns (X) show 1818 600 moveto 384 /Times-Roman f1 (\() show 1983 600 moveto 384 /Times-Italic f1 (X) show 2234 600 moveto 384 /Times-Roman f1 (,) show 2366 600 moveto 384 /Times-Bold f1 (Parents) show 3623 600 moveto 384 /Times-Roman f1 (\() show 3788 600 moveto 384 /Times-Italic f1 (X) show 4039 600 moveto 384 /Times-Roman f1 (,) show 4169 600 moveto 384 /Times-Italic f1 (G) show 4452 600 moveto 384 /Times-Roman f1 (\() show 4592 600 moveto 384 /Times-Bold f1 (X) show 4879 600 moveto 384 /Times-Roman f1 (\)) show 5009 600 moveto (\)) show 5139 600 moveto (\)) show 194 970 moveto 224 /Times-Italic f1 (X) show 357 970 moveto 224 /Symbol f1 (\316) show 497 970 moveto 224 /Times-Bold f1 (X) show 180 688 moveto 576 /Symbol f1 (\325) show -15 355 moveto 384 ns (\346) show -15 890 moveto (\350) show -15 671 moveto (\347) show 5260 355 moveto (\366) show 5260 890 moveto (\370) show 5260 671 moveto (\367) show 5522 600 moveto (\264) show 5899 600 moveto 384 /Times-Italic f1 (J) show 6077 696 moveto 224 /Times-Bold f1 (Err) show 6440 696 moveto 224 /Times-Roman f1 (\() show 6535 696 moveto 224 /Times-Bold f1 (X) show 6716 696 moveto 224 /Times-Roman f1 (\)) show 6809 696 moveto 224 /Symbol f1 (-) show 6955 696 moveto (>) show 7099 696 moveto 224 /Times-Bold f1 (X) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 5830 235 moveto 0 532 rlineto stroke 7326 235 moveto 0 532 rlineto stroke end pse gR gS 0 0 552 730 rC 244 502 :M f2_12 sf (Equation 24)S 95 536 :M f0_12 sf (where )S f5_12 sf (e)S f4_10 sf 0 2 rm (X)S 0 -2 rm f0_12 sf ( = )S 151 536 :M f4_12 sf (h)S f4_10 sf 0 2 rm (X)S 0 -2 rm f0_12 sf <28>S 167 536 :M f4_12 sf (X)S f0_12 sf (,)S f2_12 sf (Parents)S f0_12 sf <28>S 220 536 :M f4_12 sf (X)S f0_12 sf (,)S f4_12 sf (G)S 239 536 :M f0_12 sf <28>S 243 536 :M f2_12 sf (X)S 252 536 :M f0_12 sf (\)\). By lemma 7,)S endp %%Page: 36 36 %%BeginPageSetup initializepage (peter; page: 36 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (36)S gR gS 0 0 552 730 rC 95 136 :M f2_12 sf ( )S 122 95 318 104 rC 440 199 :M psb currentpoint pse 122 95 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 10176 div 3328 3 -1 roll exch div scale currentpoint translate 64 36 translate 4182 261 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-Italic f1 (J) show 4359 357 moveto 224 /Times-Bold f1 (X) show 4544 357 moveto 224 /Symbol f1 (-) show 4690 357 moveto (>) show 4835 357 moveto 224 /Times-Bold f1 (Err) show 5198 357 moveto 224 /Times-Roman f1 (\() show 5293 357 moveto 224 /Times-Bold f1 (X) show 5474 357 moveto 224 /Times-Roman f1 (\)) show 5693 261 moveto 384 /Symbol f1 (=) show 5474 1029 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 5682 1029 moveto (e) show 5873 1125 moveto 224 /Times-Italic f1 (V) show 5532 1577 moveto 384 /Symbol f2 (\266) show 5740 1577 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 5451 1177 moveto 633 0 rlineto stroke 3682 1646 moveto 224 ns (V) show 3883 1646 moveto 224 /Symbol f1 (\317) show 4051 1646 moveto 224 /Times-Bold f1 (Cycles) show 4686 1646 moveto 224 /Times-Roman f1 (\() show 4780 1646 moveto 224 /Times-Italic f1 (G) show 4959 1646 moveto 224 /Times-Roman f1 (\() show 5054 1646 moveto 224 /Times-Bold f1 (X) show 5235 1646 moveto 224 /Times-Roman f1 (\)) show 5324 1646 moveto (\)) show 4301 1364 moveto 576 /Symbol f1 (\325) show 3492 983 moveto 384 ns (\346) show 3492 1614 moveto (\350) show 3492 1395 moveto (\347) show 6096 983 moveto (\366) show 6096 1614 moveto (\370) show 6096 1395 moveto (\367) show 6358 1276 moveto (\264) show 4366 2702 moveto 384 /Times-Italic f1 (d) show 4580 2702 moveto 384 /Times-Roman f1 (\() show 4720 2702 moveto 384 /Times-Bold f1 (D) show 5003 2702 moveto 384 /Times-Roman f1 (\)) show 5202 2702 moveto 384 /Symbol f1 (\264) show 6593 2455 moveto 384 /Symbol f2 (\266) show 6801 2455 moveto (e) show 6992 2551 moveto 224 /Times-Italic f1 (V) show 6651 3003 moveto 384 /Symbol f2 (\266) show 6859 3003 moveto 384 /Times-Italic f1 (V) show 6570 2603 moveto 633 0 rlineto stroke 5677 3072 moveto 224 ns (V) show 5878 3072 moveto 224 /Symbol f1 (\316) show 6046 3072 moveto 224 /Times-Bold f1 (C) show 6245 3072 moveto 224 /Times-Roman f1 (\\) show 6353 3072 moveto 224 /Times-Bold f1 (D) show 5858 2790 moveto 576 /Symbol f1 (\325) show 5487 2409 moveto 384 ns (\346) show 5487 3040 moveto (\350) show 5487 2821 moveto (\347) show 7215 2409 moveto (\366) show 7215 3040 moveto (\370) show 7215 2821 moveto (\367) show 8735 2455 moveto 384 /Symbol f2 (\266) show 8943 2455 moveto (e) show 9134 2551 moveto 224 /Times-Italic f1 (Y) show 8746 3003 moveto 384 /Symbol f2 (\266) show 8954 3003 moveto 384 /Times-Italic f1 (W) show 8712 2603 moveto 620 0 rlineto stroke 7598 3072 moveto 224 /Symbol f1 (<) show 7740 3072 moveto 224 /Times-Italic f1 (W) show 7954 3072 moveto 224 /Times-Roman f1 (,) show 8022 3072 moveto 224 /Times-Italic f1 (Y) show 8183 3072 moveto 224 /Symbol f1 (>) show 8354 3072 moveto (\316) show 8495 3072 moveto 224 /Times-Bold f1 (D) show 7887 2790 moveto 576 /Symbol f1 (\325) show 7404 2409 moveto 384 ns (\346) show 7404 3040 moveto (\350) show 7404 2821 moveto (\347) show 9344 2409 moveto (\366) show 9344 3040 moveto (\370) show 9344 2821 moveto (\367) show 4172 2377 moveto (\346) show 4172 3072 moveto (\350) show 4172 2828 moveto (\347) show 9528 2377 moveto (\366) show 9528 3072 moveto (\370) show 9528 2828 moveto (\367) show 2607 3075 moveto 224 /Times-Bold f1 (D) show 2814 3075 moveto 224 /Symbol f1 (\316) show 2982 3075 moveto 224 /Times-Bold f1 (Cycleset) show 3794 3075 moveto 224 /Times-Roman f1 (\() show 3885 3075 moveto 224 /Times-Bold f1 (C) show 4060 3075 moveto 224 /Times-Roman f1 (\)) show 3163 2789 moveto 576 /Symbol f1 (\345) show 2412 2345 moveto 384 ns (\346) show 2412 3104 moveto (\350) show 2412 2796 moveto (\347) show 9712 2345 moveto (\366) show 9712 3104 moveto (\370) show 9712 2796 moveto (\367) show 175 3072 moveto 224 /Times-Bold f1 (C) show 378 3072 moveto 224 /Symbol f1 (\316) show 546 3072 moveto 224 /Times-Bold f1 (Cyclegroup) show 1662 3072 moveto 224 /Times-Roman f1 (\() show 1756 3072 moveto 224 /Times-Italic f1 (G) show 1935 3072 moveto 224 /Times-Roman f1 (\() show 2030 3072 moveto 224 /Times-Bold f1 (X) show 2211 3072 moveto 224 /Times-Roman f1 (\)) show 2300 3072 moveto (\)) show 1035 2790 moveto 576 /Symbol f1 (\325) show -15 2313 moveto 384 ns (\346) show -15 3136 moveto (\350) show -15 2764 moveto (\347) show -15 2917 moveto (\347) show 9896 2313 moveto (\366) show 9896 3136 moveto (\370) show 9896 2764 moveto (\367) show 9896 2917 moveto (\367) show end pse gR gS 0 0 552 730 rC 244 220 :M f2_12 sf (Equation 25)S 95 250 :M f0_12 sf (Each term in)S 219 265 112 44 rC 331 309 :M psb currentpoint pse 219 265 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3584 div 1408 3 -1 roll exch div scale currentpoint translate 64 35 translate 2618 516 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 /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 2826 516 moveto (e) show 3012 612 moveto 320 /Times-Italic f1 (V) show 2705 1066 moveto 384 /Symbol f2 (\266) show 2913 1066 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2595 666 moveto 691 0 rlineto stroke 170 1231 moveto 320 ns (V) show 462 1231 moveto 320 /Symbol f1 (\317) show 706 1231 moveto 320 /Times-Bold f1 (Cycles) show 1600 1231 moveto 320 /Times-Roman f1 (\() show 1720 1231 moveto 320 /Times-Italic f1 (G) show 1961 1231 moveto 320 /Times-Roman f1 (\() show 2083 1231 moveto 320 /Times-Bold f1 (X) show 2328 1231 moveto 320 /Times-Roman f1 (\)) show 2441 1231 moveto (\)) show 1119 853 moveto 576 /Symbol f1 (\325) show -15 355 moveto 384 ns (\346) show -15 1220 moveto (\350) show -15 806 moveto (\347) show -15 1001 moveto (\347) show 3298 355 moveto (\366) show 3298 1220 moveto (\370) show 3298 806 moveto (\367) show 3298 1001 moveto (\367) show end pse gR gS 0 0 552 730 rC 244 330 :M f2_12 sf (Equation 26)S 95 360 :M f0_12 sf (is a function of )S f4_12 sf (V)S f0_12 sf ( and )S f2_12 sf (Parents)S f0_12 sf <28>S 243 360 :M f4_12 sf (V)S f0_12 sf (,)S f4_12 sf (G)S 262 360 :M f0_12 sf <28>S 266 360 :M f2_12 sf (X)S 275 360 :M f0_12 sf (\)\). Each term in)S 95 375 370 48 rC 465 423 :M psb currentpoint pse 95 375 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11840 div 1536 3 -1 roll exch div scale currentpoint translate 64 35 translate 5404 829 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-Italic f1 (d) show 5618 829 moveto 384 /Times-Roman f1 (\() show 5758 829 moveto 384 /Times-Bold f1 (D) show 6041 829 moveto 384 /Times-Roman f1 (\)) show 6240 829 moveto 384 /Symbol f1 (\264) show 7899 580 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 8107 580 moveto (e) show 8293 676 moveto 320 /Times-Italic f1 (V) show 7986 1130 moveto 384 /Symbol f2 (\266) show 8194 1130 moveto 384 /Times-Italic f1 (V) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 7876 730 moveto 691 0 rlineto stroke 6710 1295 moveto 320 ns (V) show 7002 1295 moveto 320 /Symbol f1 (\316) show 7182 1295 moveto 320 /Times-Bold f1 (C) show 7454 1295 moveto 320 /Times-Roman f1 (\\) show 7594 1295 moveto 320 /Times-Bold f1 (D) show 7030 917 moveto 576 /Symbol f1 (\325) show 6525 419 moveto 384 ns (\346) show 6525 1284 moveto (\350) show 6525 870 moveto (\347) show 6525 1065 moveto (\347) show 8579 419 moveto (\366) show 8579 1284 moveto (\370) show 8579 870 moveto (\367) show 8579 1065 moveto (\367) show 10541 582 moveto 384 /Symbol f2 (\266) show 10749 582 moveto (e) show 10935 678 moveto 320 /Times-Italic f1 (Y) show 10578 1130 moveto 384 /Symbol f2 (\266) show 10786 1130 moveto 384 /Times-Italic f1 (W) show 10518 730 moveto 673 0 rlineto stroke 8959 1295 moveto 320 /Symbol f1 (<) show 9147 1295 moveto 320 /Times-Italic f1 (W) show 9439 1295 moveto 320 /Times-Roman f1 (,) show 9521 1295 moveto 320 /Times-Italic f1 (Y) show 9737 1295 moveto 320 /Symbol f1 (>) show 9986 1295 moveto (\316) show 10236 1295 moveto 320 /Times-Bold f1 (D) show 9472 917 moveto 576 /Symbol f1 (\325) show 8768 419 moveto 384 ns (\346) show 8768 1284 moveto (\350) show 8768 870 moveto (\347) show 8768 1065 moveto (\347) show 11203 419 moveto (\366) show 11203 1284 moveto (\370) show 11203 870 moveto (\367) show 11203 1065 moveto (\367) show 5210 387 moveto (\346) show 5210 1316 moveto (\350) show 5210 838 moveto (\347) show 5210 1097 moveto (\347) show 11387 387 moveto (\366) show 11387 1316 moveto (\370) show 11387 838 moveto (\367) show 11387 1097 moveto (\367) show 3154 1298 moveto 320 /Times-Bold f1 (D) show 3390 1298 moveto 320 /Symbol f1 (\316) show 3570 1298 moveto 320 /Times-Bold f1 (Cycleset) show 4718 1298 moveto 320 /Times-Roman f1 (\() show 4834 1298 moveto 320 /Times-Bold f1 (C) show 5071 1298 moveto 320 /Times-Roman f1 (\)) show 3957 916 moveto 576 /Symbol f1 (\345) show 2962 355 moveto 384 ns (\346) show 2962 1348 moveto (\350) show 2962 806 moveto (\347) show 2962 1129 moveto (\347) show 11571 355 moveto (\366) show 11571 1348 moveto (\370) show 11571 806 moveto (\367) show 11571 1129 moveto (\367) show -13 1295 moveto 320 /Times-Bold f1 (C) show 219 1295 moveto 320 /Symbol f1 (\316) show 399 1295 moveto 320 /Times-Bold f1 (Cyclegroup) show 1982 1295 moveto 320 /Times-Roman f1 (\() show 2102 1295 moveto 320 /Times-Italic f1 (G) show 2343 1295 moveto 320 /Times-Roman f1 (\() show 2465 1295 moveto 320 /Times-Bold f1 (X) show 2710 1295 moveto 320 /Times-Roman f1 (\)) show 2823 1295 moveto (\)) show 1218 917 moveto 576 /Symbol f1 (\325) show end pse gR gS 0 0 552 730 rC 244 432 :M f2_12 sf (Equation 27)S 95 462 :M f0_12 sf (contains only error terms associated with variables in )S 354 462 :M f2_12 sf (C)S 363 462 :M f0_12 sf (, and hence is a)S 95 480 :M (function of )S 151 480 :M f2_12 sf (C)S 160 480 :M f0_12 sf ( and )S f2_12 sf (Parents)S f0_12 sf <28>S 226 480 :M f2_12 sf (C)S 235 480 :M f0_12 sf (,)S f4_12 sf (G)S 247 480 :M f0_12 sf <28>S 251 480 :M f2_12 sf (X)S 260 480 :M f0_12 sf (\)\). Hence, there exist functions )S 411 480 :M f4_12 sf (m)S 420 482 :M f4_10 sf (V)S f0_12 sf 0 -2 rm ( such)S 0 2 rm 95 499 :M (that)S 95 508 354 75 rC 449 583 :M psb currentpoint pse 95 508 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11328 div 2400 3 -1 roll exch div scale currentpoint translate 64 39 translate 4748 261 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-Italic f1 (J) show 4925 357 moveto 224 /Times-Bold f1 (X) show 5110 357 moveto 224 /Symbol f1 (-) show 5256 357 moveto (>) show 5401 357 moveto 224 /Times-Bold f1 (Err) show 5764 357 moveto 224 /Times-Roman f1 (\() show 5859 357 moveto 224 /Times-Bold f1 (X) show 6040 357 moveto 224 /Times-Roman f1 (\)) show 6259 261 moveto 384 /Symbol f1 (=) show 1316 1852 moveto 384 /Times-Italic f1 (m) show 1585 1948 moveto 224 ns (V) show 1778 1852 moveto 384 /Times-Roman f1 (\() show 1910 1852 moveto 384 /Times-Italic f1 (V) show 2171 1852 moveto 384 /Times-Roman f1 (,) show 2303 1852 moveto 384 /Times-Bold f1 (Parents) show 3560 1852 moveto 384 /Times-Roman f1 (\() show 3692 1852 moveto 384 /Times-Italic f1 (V) show 3953 1852 moveto 384 /Times-Roman f1 (,) show 4083 1852 moveto 384 /Times-Italic f1 (G) show 4366 1852 moveto 384 /Times-Roman f1 (\() show 4506 1852 moveto 384 /Times-Bold f1 (X) show 4793 1852 moveto 384 /Times-Roman f1 (\)) show 4923 1852 moveto (\)) show 175 2222 moveto 224 /Times-Italic f1 (V) show 376 2222 moveto 224 /Symbol f1 (\317) show 544 2222 moveto 224 /Times-Bold f1 (Cycles) show 1179 2222 moveto 224 /Times-Roman f1 (\() show 1273 2222 moveto 224 /Times-Italic f1 (G) show 1452 2222 moveto 224 /Times-Roman f1 (\() show 1547 2222 moveto 224 /Times-Bold f1 (X) show 1728 2222 moveto 224 /Times-Roman f1 (\)) show 1817 2222 moveto (\)) show 794 1940 moveto 576 /Symbol f1 (\325) show -15 1559 moveto 384 ns (\346) show -15 2190 moveto (\350) show -15 1971 moveto (\347) show 5044 1559 moveto (\366) show 5044 2190 moveto (\370) show 5044 1971 moveto (\367) show 5306 1852 moveto (\264) show 7163 1852 moveto 384 /Times-Italic f1 (m) show 7432 1950 moveto 224 /Times-Bold f1 (C) show 7627 1852 moveto 384 /Times-Roman f1 (\() show 7759 1852 moveto 384 /Times-Bold f1 (C) show 8023 1852 moveto 384 /Times-Roman f1 (,) show 8155 1852 moveto 384 /Times-Bold f1 (Parents) show 9412 1852 moveto 384 /Times-Roman f1 (\() show 9544 1852 moveto 384 /Times-Bold f1 (C) show 9808 1852 moveto 384 /Times-Roman f1 (,) show 9938 1852 moveto 384 /Times-Italic f1 (G) show 10221 1852 moveto 384 /Times-Roman f1 (\() show 10361 1852 moveto 384 /Times-Bold f1 (X) show 10648 1852 moveto 384 /Times-Roman f1 (\)) show 10778 1852 moveto (\)) show 10908 1852 moveto (\)) show 5783 2222 moveto 224 /Times-Italic f1 (C) show 5983 2222 moveto 224 /Symbol f1 (\316) show 6151 2222 moveto 224 /Times-Bold f1 (Cyclegroup) show 7267 2222 moveto 224 /Times-Roman f1 (\() show 7361 2222 moveto 224 /Times-Italic f1 (G) show 7540 2222 moveto 224 /Times-Roman f1 (\() show 7635 2222 moveto 224 /Times-Bold f1 (X) show 7816 2222 moveto 224 /Times-Roman f1 (\)) show 7905 2222 moveto (\)) show 6641 1940 moveto 576 /Symbol f1 (\325) show 5591 1559 moveto 384 ns (\346) show 5591 2190 moveto (\350) show 5591 1971 moveto (\347) show 11029 1559 moveto (\366) show 11029 2190 moveto (\370) show 11029 1971 moveto (\367) show end pse gR gS 0 0 552 730 rC 244 604 :M f2_12 sf (Equation 28)S endp %%Page: 37 37 %%BeginPageSetup initializepage (peter; page: 37 of 37)setjob %%EndPageSetup gS 0 0 552 730 rC 516 5 29 24 rC 533 26 :M f0_12 sf (37)S gR gS 0 0 552 730 rC 95 104 :M f0_12 sf .521 .052(Because )J f4_12 sf .116(J)A f2_10 sf 0 2 rm .17(Err)A 0 -2 rm 160 106 :M f0_10 sf .178<28>A f2_10 sf .386(X)A f0_10 sf .219(\)->)A f2_10 sf .386(X)A f0_12 sf 0 -2 rm .492 .049( = 1/)J 0 2 rm 214 104 :M f4_12 sf (J)S f2_10 sf 0 2 rm (X)S 0 -2 rm f0_10 sf 0 2 rm (->)S 0 -2 rm 235 106 :M f2_10 sf (Err)S 251 106 :M f0_10 sf .233<28>A f2_10 sf .506(X)A f0_10 sf .233<29>A f0_12 sf 0 -2 rm .382 .038(, )J 0 2 rm 272 104 :M f4_12 sf (J)S f2_10 sf 0 2 rm (Err)S 0 -2 rm 293 106 :M f0_10 sf .181<28>A f2_10 sf .392(X)A f0_10 sf .222(\)->)A f2_10 sf .392(X)A f0_12 sf 0 -2 rm .782 .078( can also be factored as in)J 0 2 rm 95 123 :M .364 .036(Equation 28)J 153 123 :M .478 .048(. Combining this with )J 264 123 :M .414 .041(Equation 24, there exist functions non-)J 95 141 :M (negative functions )S f4_12 sf (g)S f4_10 sf 0 2 rm (V)S 0 -2 rm f0_12 sf ( such that)S 95 157 421 44 rC 516 201 :M psb currentpoint pse 95 157 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 13472 div 1408 3 -1 roll exch div scale currentpoint translate 64 35 translate 56 765 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-Italic f1 (f) show 206 861 moveto 320 /Times-Bold f1 (V) show 474 765 moveto 384 /Times-Roman f1 (\() show 614 765 moveto 384 /Times-Bold f1 (X) show 901 765 moveto 384 /Times-Roman f1 (\)) show 1134 765 moveto 384 /Symbol f1 (=) show 3110 765 moveto 384 /Times-Italic f1 (g) show 3286 861 moveto 320 ns (V) show 3542 765 moveto 384 /Times-Roman f1 (\() show 3674 765 moveto 384 /Times-Italic f1 (V) show 3935 765 moveto 384 /Times-Roman f1 (,) show 4067 765 moveto 384 /Times-Bold f1 (Parents) show 5324 765 moveto 384 /Times-Roman f1 (\() show 5456 765 moveto 384 /Times-Italic f1 (V) show 5717 765 moveto 384 /Times-Roman f1 (,) show 5847 765 moveto 384 /Times-Italic f1 (G) show 6130 765 moveto 384 /Times-Roman f1 (\() show 6270 765 moveto 384 /Times-Bold f1 (X) show 6557 765 moveto 384 /Times-Roman f1 (\)) show 6687 765 moveto (\)) show 1638 1231 moveto 320 /Times-Italic f1 (V) show 1930 1231 moveto 320 /Symbol f1 (\317) show 2174 1231 moveto 320 /Times-Bold f1 (Cycles) show 3068 1231 moveto 320 /Times-Roman f1 (\() show 3188 1231 moveto 320 /Times-Italic f1 (G) show 3429 1231 moveto 320 /Times-Roman f1 (\() show 3551 1231 moveto 320 /Times-Bold f1 (X) show 3796 1231 moveto 320 /Times-Roman f1 (\)) show 3909 1231 moveto (\)) show 2587 853 moveto 576 /Symbol f1 (\325) show 1453 355 moveto 384 ns (\346) show 1453 1220 moveto (\350) show 1453 806 moveto (\347) show 1453 1001 moveto (\347) show 6808 355 moveto (\366) show 6808 1220 moveto (\370) show 6808 806 moveto (\367) show 6808 1001 moveto (\367) show 7070 765 moveto (\264) show 9359 765 moveto 384 /Times-Italic f1 (g) show 9536 864 moveto 320 /Times-Bold f1 (C) show 9796 765 moveto 384 /Times-Roman f1 (\() show 9928 765 moveto 384 /Times-Bold f1 (C) show 10192 765 moveto 384 /Times-Roman f1 (,) show 10324 765 moveto 384 /Times-Bold f1 (Parents) show 11581 765 moveto 384 /Times-Roman f1 (\() show 11713 765 moveto 384 /Times-Bold f1 (C) show 11977 765 moveto 384 /Times-Roman f1 (,) show 12107 765 moveto 384 /Times-Italic f1 (G) show 12390 765 moveto 384 /Times-Roman f1 (\() show 12530 765 moveto 384 /Times-Bold f1 (X) show 12817 765 moveto 384 /Times-Roman f1 (\)) show 12947 765 moveto (\)) show 13077 765 moveto (\)) show 7541 1231 moveto 320 /Times-Bold f1 (C) show 7837 1231 moveto 320 /Symbol f1 (\316) show 8081 1231 moveto 320 /Times-Bold f1 (Cyclegroup) show 9664 1231 moveto 320 /Times-Roman f1 (\() show 9784 1231 moveto 320 /Times-Italic f1 (G) show 10025 1231 moveto 320 /Times-Roman f1 (\() show 10147 1231 moveto 320 /Times-Bold f1 (X) show 10392 1231 moveto 320 /Times-Roman f1 (\)) show 10505 1231 moveto (\)) show 8836 853 moveto 576 /Symbol f1 (\325) show 7355 355 moveto 384 ns (\346) show 7355 1220 moveto (\350) show 7355 806 moveto (\347) show 7355 1001 moveto (\347) show 13198 355 moveto (\366) show 13198 1220 moveto (\370) show 13198 806 moveto (\367) show 13198 1001 moveto (\367) show end pse gR gS 0 0 552 730 rC 244 210 :M f2_12 sf (Equation 29)S 95 243 :M f0_12 sf ( )S 98 234 9 9 rC gS 1.286 1 scale 76.223 243 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 95 270 :M f2_12 sf .845 .084(Theorem 5:)J f0_12 sf .329 .033( In an SEM )J 216 270 :M f4_12 sf (L)S 223 270 :M f0_12 sf .424 .042( with directed \(cyclic or acyclic\) graph )J 417 270 :M f4_12 sf (G)S 426 270 :M f0_12 sf .533 .053( with)J 95 288 :M .354 .035(vertices )J f2_12 sf (V)S 145 288 :M f0_12 sf .494 .049( and collapsed graph )J f4_12 sf (G)S 259 288 :M f0_12 sf .468 .047(' containing disjoint sets of variables )J f2_12 sf (X)S 451 288 :M f0_12 sf (,)S 95 306 :M f2_12 sf (Y)S 104 306 :M f0_12 sf .502 .05( and )J 129 306 :M f2_12 sf .292(Z)A f0_12 sf .239 .024(, if )J f2_12 sf (X)S 163 306 :M f0_12 sf .363 .036( is d-separated from )J f2_12 sf (Y)S 273 306 :M f0_12 sf .464 .046( given )J 307 306 :M f2_12 sf .414(Z)A f0_12 sf .344 .034( in )J 332 306 :M f4_12 sf (G)S 341 306 :M f0_12 sf .343 .034(' then )J f2_12 sf (X)S 380 306 :M f0_12 sf .355 .035( is independent)J 95 324 :M (of )S 108 324 :M f2_12 sf (Y)S 117 324 :M f0_12 sf ( given )S 150 324 :M f2_12 sf (Z)S f0_12 sf (.)S 95 354 :M (Proof. By lemma 8)S 100 418 :M f2_12 sf ( )S 148 369 302 112 rC 450 481 :M psb currentpoint pse 148 369 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9664 div 3584 3 -1 roll exch div scale currentpoint translate 64 48 translate 2874 262 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-Italic f1 (f) show 3024 358 moveto 320 /Times-Bold f1 (V) show 3292 262 moveto 384 /Times-Roman f1 (\() show 3432 262 moveto 384 /Times-Bold f1 (An) show 3923 262 moveto 384 /Times-Roman f1 (\() show 4063 262 moveto 384 /Times-Bold f1 (X) show 4410 262 moveto 384 /Symbol f1 (\310) show 4780 262 moveto 384 /Times-Bold f1 (Y) show 5128 262 moveto 384 /Symbol f1 (\310) show 5491 262 moveto 384 /Times-Bold f1 (Z) show 5738 262 moveto 384 /Times-Roman f1 (,) show 5868 262 moveto 384 /Times-Italic f1 (G) show 6158 262 moveto 384 /Times-Roman f1 (\)) show 6288 262 moveto (\)) show 6521 262 moveto 384 /Symbol f1 (=) show 2779 1360 moveto 384 /Times-Italic f1 (g) show 2955 1456 moveto 320 ns (V) show 3211 1360 moveto 384 /Times-Roman f1 (\() show 3343 1360 moveto 384 /Times-Italic f1 (V) show 3604 1360 moveto 384 /Times-Roman f1 (,) show 3736 1360 moveto 384 /Times-Bold f1 (Parents) show 4993 1360 moveto 384 /Times-Roman f1 (\() show 5125 1360 moveto 384 /Times-Italic f1 (V) show 5386 1360 moveto 384 /Times-Roman f1 (,) show 5516 1360 moveto 384 /Times-Italic f1 (G) show 5799 1360 moveto 384 /Times-Roman f1 (\() show 5939 1360 moveto 384 /Times-Bold f1 (An) show 6430 1360 moveto 384 /Times-Roman f1 (\() show 6570 1360 moveto 384 /Times-Bold f1 (X) show 6917 1360 moveto 384 /Symbol f1 (\310) show 7287 1360 moveto 384 /Times-Bold f1 (Y) show 7635 1360 moveto 384 /Symbol f1 (\310) show 7998 1360 moveto 384 /Times-Bold f1 (Z) show 8245 1360 moveto 384 /Times-Roman f1 (,) show 8375 1360 moveto 384 /Times-Italic f1 (G) show 8665 1360 moveto 384 /Times-Roman f1 (\)) show 8795 1360 moveto (\)) show 332 1826 moveto 320 /Times-Italic f1 (V) show 624 1826 moveto 320 /Symbol f1 (\317) show 868 1826 moveto 320 /Times-Bold f1 (Cycles) show 1762 1826 moveto 320 /Times-Roman f1 (\() show 1882 1826 moveto 320 /Times-Italic f1 (G) show 2123 1826 moveto 320 /Times-Roman f1 (\() show 2245 1826 moveto 320 /Times-Bold f1 (An) show 2660 1826 moveto 320 /Times-Roman f1 (\() show 2782 1826 moveto 320 /Times-Bold f1 (X) show 3024 1826 moveto 320 /Symbol f1 (\310) show 3284 1826 moveto 320 /Times-Bold f1 (Y) show 3527 1826 moveto 320 /Symbol f1 (\310) show 3782 1826 moveto 320 /Times-Bold f1 (Z) show 3993 1826 moveto 320 /Times-Roman f1 (,) show 4080 1826 moveto 320 /Times-Italic f1 (G) show 4327 1826 moveto 320 /Times-Roman f1 (\)) show 4440 1826 moveto (\)) show 4553 1826 moveto (\)) show 2256 1448 moveto 576 /Symbol f1 (\325) show 147 950 moveto 384 ns (\346) show 147 1815 moveto (\350) show 147 1401 moveto (\347) show 147 1596 moveto (\347) show 8916 950 moveto (\366) show 8916 1815 moveto (\370) show 8916 1401 moveto (\367) show 8916 1596 moveto (\367) show 9178 1360 moveto (\264) show 2964 2924 moveto 384 /Times-Italic f1 (g) show 3141 3023 moveto 320 /Times-Bold f1 (C) show 3401 2924 moveto 384 /Times-Roman f1 (\() show 3533 2924 moveto 384 /Times-Bold f1 (C) show 3797 2924 moveto 384 /Times-Roman f1 (,) show 3929 2924 moveto 384 /Times-Bold f1 (Parents) show 5186 2924 moveto 384 /Times-Roman f1 (\() show 5318 2924 moveto 384 /Times-Bold f1 (C) show 5582 2924 moveto 384 /Times-Roman f1 (,) show 5712 2924 moveto 384 /Times-Italic f1 (G) show 5995 2924 moveto 384 /Times-Roman f1 (\() show 6135 2924 moveto 384 /Times-Bold f1 (An) show 6626 2924 moveto 384 /Times-Roman f1 (\() show 6766 2924 moveto 384 /Times-Bold f1 (X) show 7113 2924 moveto 384 /Symbol f1 (\310) show 7483 2924 moveto 384 /Times-Bold f1 (Y) show 7831 2924 moveto 384 /Symbol f1 (\310) show 8194 2924 moveto 384 /Times-Bold f1 (Z) show 8441 2924 moveto 384 /Times-Roman f1 (,) show 8571 2924 moveto 384 /Times-Italic f1 (G) show 8861 2924 moveto 384 /Times-Roman f1 (\)) show 8991 2924 moveto (\)) show 9121 2924 moveto (\)) show 9251 2924 moveto (\)) show 171 3390 moveto 320 /Times-Bold f1 (C) show 467 3390 moveto 320 /Symbol f1 (\316) show 711 3390 moveto 320 /Times-Bold f1 (Cyclegroup) show 2294 3390 moveto 320 /Times-Roman f1 (\() show 2414 3390 moveto 320 /Times-Italic f1 (G) show 2655 3390 moveto 320 /Times-Roman f1 (\() show 2777 3390 moveto 320 /Times-Bold f1 (An) show 3192 3390 moveto 320 /Times-Roman f1 (\() show 3314 3390 moveto 320 /Times-Bold f1 (X) show 3556 3390 moveto 320 /Symbol f1 (\310) show 3816 3390 moveto 320 /Times-Bold f1 (Y) show 4059 3390 moveto 320 /Symbol f1 (\310) show 4314 3390 moveto 320 /Times-Bold f1 (Z) show 4525 3390 moveto 320 /Times-Roman f1 (,) show 4612 3390 moveto 320 /Times-Italic f1 (G) show 4859 3390 moveto 320 /Times-Roman f1 (\)) show 4972 3390 moveto (\)) show 5085 3390 moveto (\)) show 2441 3012 moveto 576 /Symbol f1 (\325) show -15 2514 moveto 384 ns (\346) show -15 3379 moveto (\350) show -15 2965 moveto (\347) show -15 3160 moveto (\347) show 9372 2514 moveto (\366) show 9372 3379 moveto (\370) show 9372 2965 moveto (\367) show 9372 3160 moveto (\367) show end pse gR gS 0 0 552 730 rC 244 502 :M f2_12 sf (Equation 30)S 95 532 :M f0_12 sf .665 .067(This is a factorization according to the collapsed graph )J f4_12 sf (G)S 380 532 :M f0_12 sf .765 .076(', and hence by)J 95 550 :M .175 .017(lemma 1, for three disjoint sets of variables )J 309 550 :M f2_12 sf (X)S 318 550 :M f0_12 sf .093 .009(, )J f2_12 sf (Y)S 333 550 :M f0_12 sf .214 .021( and )J f2_12 sf .183(Z)A f0_12 sf .156 .016(, if )J 381 550 :M f2_12 sf (X)S 390 550 :M f0_12 sf .159 .016( and )J f2_12 sf (Y)S 422 550 :M f0_12 sf .21 .021( are d-)J 95 568 :M (separated given )S 173 568 :M f2_12 sf (Z)S f0_12 sf ( in )S f4_12 sf (G)S 205 568 :M f0_12 sf (', then )S 237 568 :M f2_12 sf (X)S 246 568 :M f0_12 sf ( and )S f2_12 sf (Y)S 278 568 :M f0_12 sf ( are independent given )S 390 568 :M f2_12 sf (Z)S f0_12 sf (. )S 407 559 9 9 rC gS 1.286 1 scale 316.558 568 :M f1_10 sf <5C>S gR endp %%Trailer end %%EOF