%!PS-Adobe-3.0 %%Title: (Microsoft Word - correrr7) %%Creator: (Microsoft Word: LaserWriter 8 8.3.4) %%CreationDate: (10:42 PM Tuesday, March 25, 1997) %%For: (peter) %%Pages: 17 %%DocumentFonts: Times-Roman Symbol Times-Bold Times-Italic Times-BoldItalic TimesNewRomanPSMT %%DocumentNeededFonts: Times-Roman Symbol Times-Bold Times-Italic Times-BoldItalic TimesNewRomanPSMT %%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 - correrr7)def /Creator(Microsoft Word: LaserWriter 8 8.3.4)def /CreationDate(10:42 PM Tuesday, March 25, 1997)def /For(peter)def /Pages 17 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 227 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. 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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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/f8_1/TimesNewRomanPSMT :mre /f8_12 f8_1 12 scf /f8_10 f8_1 10 scf /f8_6 f8_1 6 scf /Courier findfont[10 0 0 -10 0 0]:mf setfont %%EndSetup %%Page: 1 1 %%BeginPageSetup initializepage (peter; page: 1 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (1)S gR gS 0 0 552 730 rC 186 62 :M f2_12 sf 3.345 .334(Technical Report 바카라 온라인 추천-Phil-72)J 72 101 :M f2_14 sf 3.338 .334(Using D-separation to Calculate Zero Partial Correlations in)J 147 116 :M 3.448 .345(Linear Models with Correlated Errors)J 245 134 :M f0_12 sf -.081(Peter Spirtes)A 227 152 :M -.082(Thomas Richardson)A 232 170 :M -.131(Christopher Meek)A 234 188 :M -.109(Richard Scheines)A 239 206 :M -.082(Clark Glymour)A 251 254 :M f2_12 sf .337(Abstract)A 59 282 :M f0_12 sf -.164(It )A 70 282 :M (has )S 90 282 :M -.163(been )A 116 282 :M .447 .045(shown )J 153 282 :M -.167(in )A 167 282 :M -.024(Spirtes\(1995\) )A 237 282 :M -.249(that )A 259 282 :M .306 .031(X )J 273 282 :M -.109(and )A 295 282 :M 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This result cannot be directly )A 405 314 :M -.189(applied )A 443 314 :M -.167(to )A 456 314 :M -.326(a )A 465 314 :M -.263(linear)A 59 330 :M -.199(model )A 92 330 :M -.083(with )A 117 330 :M -.196(correlated )A 167 330 :M .604 .06(errors, )J 203 330 :M .384 .038(however, )J 252 330 :M -.139(because )A 293 330 :M -.22(the )A 311 330 :M -.081(standard )A 355 330 :M -.182(graphical )A 402 330 :M -.14(representation )A 472 330 :M (of )S 486 330 :M (a)S 59 346 :M -.117(linear model with correlated errors is not )A 254 346 :M -.326(a )A 263 346 :M -.205(directed )A 304 346 :M .425 .043(graph. )J 339 346 :M -.219(The )A 361 346 :M -.249(main )A 388 346 :M -.109(result )A 418 346 :M (of )S 432 346 :M -.084(this )A 453 346 :M -.129(paper )A 483 346 :M (is)S 59 362 :M -.104(to show how to associate a directed graph with a linear model L )A 363 362 :M -.083(with )A 388 362 :M -.196(correlated )A 438 362 :M .604 .06(errors, )J 474 362 :M -.163(and)A 59 378 :M -.109(then use d-separation in the associated )A 243 378 :M -.205(directed )A 284 378 :M -.064(graph )A 315 378 :M -.167(to )A 328 378 :M -.219(determine )A 378 378 :M -.092(whether )A 420 378 :M -.33(L )A 431 378 :M -.189(entails )A 465 378 :M -.249(that )A 486 378 :M (a)S 59 394 :M -.138(particular partial correlation is zero.)A 59 454 :M -.137(In a linear structural equation )A 200 454 :M -.199(model )A 233 454 :M -.131(\(SEM\) )A 269 454 :M -.082(some )A 298 454 :M -.235(partial )A 331 454 :M -.136(correlations )A 390 454 :M -.22(may )A 414 454 :M -.163(be )A 429 454 :M -.197(equal )A 458 454 :M -.167(to )A 471 454 :M -.215(zero)A 59 472 :M -.215(for )A f4_12 sf -.214(all)A f0_12 sf ( )S 91 472 :M -.109(values )A 125 472 :M (of )S 139 472 :M -.22(the )A 157 472 :M -.093(model\325s )A 199 472 :M -.16(free )A 221 472 :M -.163(parameters )A 276 472 :M (\(for )S 298 472 :M -.065(which )A 331 472 :M -.22(the )A 349 472 :M -.235(partial )A 382 472 :M -.179(correlation )A 436 472 :M (is )S 448 472 :M (defined\).)S 59 490 :M -.129(\(When )A 97 490 :M (we )S 117 490 :M -.126(refer )A 145 490 :M -.167(to )A 160 490 :M -.33(\322all )A 182 490 :M -.14(values\323 )A 223 490 :M (of )S 239 490 :M -.22(the )A 259 490 :M -.16(free )A 283 490 :M -.057(parameters, )A 344 490 :M (we )S 364 490 :M -.053(assume )A 405 490 :M -.249(that )A 428 490 :M -.196(there )A 458 490 :M -.215(are )A 479 490 :M (no)S 59 508 :M -.089(constraints )A 115 508 :M (upon )S 144 508 :M -.22(the )A 163 508 :M -.11(models )A 202 508 :M -.163(parameters )A 258 508 :M -.219(except )A 293 508 :M (for )S 313 508 :M -.22(the )A 333 508 :M -.163(coefficients )A 393 508 :M -.109(and )A 416 508 :M -.22(the )A 436 508 :M -.149(correlations)A 59 526 :M -.132(among )A 95 526 :M -.22(the )A 113 526 :M -.061(error )A 140 526 :M -.145(variables )A 186 526 :M -.249(that )A 207 526 :M -.215(are )A 226 526 :M -.131(fixed )A 255 526 :M -.33(at )A 268 526 :M .226 .023(zero.\) )J 301 526 :M (In )S 316 526 :M -.084(this )A 338 526 :M -.161(case )A 363 526 :M (we )S 382 526 :M -.166(will )A 405 526 :M (say )S 426 526 :M -.249(that )A 448 526 :M -.22(the )A 467 526 :M -.334(SEM)A 59 544 :M f2_12 sf (linearly)S f0_12 sf ( )S f2_12 sf (entails)S f0_12 sf .02 .002( that the partial correlation is zero. It has been shown in Spirtes\(1995\) that)J 59 562 :M .306 .031(X )J 72 562 :M -.109(and )A 93 562 :M -.663(Y )A 105 562 :M -.215(are )A 123 562 :M -.117(d-separated )A 181 562 :M -.132(given )A 211 562 :M f2_12 sf (Z)S f0_12 sf ( )S 223 562 :M -.167(in )A 236 562 :M -.326(a )A 245 562 :M -.205(directed )A 286 562 :M -.064(graph )A 317 562 :M -.131(associated )A 370 562 :M -.083(with )A 396 562 :M -.326(a )A 406 562 :M -.107(recursive )A 454 562 :M (or )S 469 562 :M (non-)S 59 580 :M -.117(recursive linear model with uncorrelated errors if )A 293 580 :M -.109(and )A 314 580 :M -.083(only )A 339 580 :M -.164(if )A 350 580 :M -.22(the )A 368 580 :M -.199(model )A 401 580 :M -.206(linearly )A 440 580 :M -.189(entails )A 474 580 :M -.331(that)A 59 597 :M f1_12 sf .224(r)A f0_7 sf 0 3 rm .134(XY.)A 0 -3 rm f2_7 sf 0 3 rm .159(Z)A 0 -3 rm f0_12 sf .102 .01( )J 87 597 :M .211 .021(= )J 98 597 :M .833 .083(0. )J 112 597 :M -.083(This )A 137 597 :M -.109(result )A 167 597 :M -.164(cannot )A 202 597 :M -.163(be )A 217 597 :M -.206(directly )A 256 597 :M -.189(applied )A 294 597 :M -.167(to )A 307 597 :M -.326(a )A 317 597 :M -.219(linear )A 348 597 :M -.199(model )A 382 597 :M -.083(with )A 408 597 :M -.196(correlated )A 459 597 :M .171(errors,)A 59 616 :M .384 .038(however, )J 108 616 :M -.139(because )A 149 616 :M -.22(the )A 167 616 :M -.081(standard )A 211 616 :M -.182(graphical )A 258 616 :M -.14(representation )A 329 616 :M (of )S 344 616 :M -.326(a )A 354 616 :M -.219(linear )A 385 616 :M -.199(model )A 419 616 :M -.083(with )A 445 616 :M -.218(correlated)A 59 634 :M -.095(errors is not a directed )A 168 634 :M .425 .043(graph. )J 203 634 :M -.219(The )A 225 634 :M -.249(main )A 252 634 :M -.109(result )A 282 634 :M (of )S 296 634 :M -.084(this )A 317 634 :M -.129(paper )A 347 634 :M (is )S 359 634 :M -.167(to )A 372 634 :M .479 .048(show )J 402 634 :M .259 .026(how )J 427 634 :M -.167(to )A 440 634 :M -.145(associate )A 486 634 :M (a)S 59 652 :M -.102(directed graph with a linear model L with correlated errors, and then use d-separation )A 464 652 :M -.167(in )A 477 652 :M -.33(the)A 59 670 :M -.131(associated )A 112 670 :M -.205(directed )A 154 670 :M -.064(graph )A 186 670 :M -.167(to )A 200 670 :M -.219(determine )A 251 670 :M -.092(whether )A 294 670 :M -.33(L )A 306 670 :M -.206(linearly )A 346 670 :M -.189(entails )A 381 670 :M -.249(that )A 403 670 :M -.326(a )A 413 670 :M -.197(particular )A 462 670 :M -.275(partial)A endp %%Page: 2 2 %%BeginPageSetup initializepage (peter; page: 2 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (2)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.097(correlation is zero. The standard graph terminology in )A 317 56 :M -.084(this )A 338 56 :M .222 .022(paper, )J 372 56 :M -.22(the )A 390 56 :M -.081(standard )A 434 56 :M -.165(terminology)A 59 74 :M (for )S 77 74 :M -.219(linear )A 107 74 :M -.131(structural )A 155 74 :M -.165(equation )A 199 74 :M .199 .02(models, )J 241 74 :M -.109(and )A 262 74 :M -.22(the )A 281 74 :M -.137(relationship )A 341 74 :M -.139(between )A 385 74 :M -.22(the )A 404 74 :M (two )S 427 74 :M -.165(terminologies)A 59 92 :M -.101(are described in the Appendix.)A 59 122 :M (If )S 72 122 :M .306 .031(G )J 86 122 :M (is )S 99 122 :M -.22(the )A 118 122 :M -.064(graph )A 150 122 :M (of )S 165 122 :M -.223(SEM )A 195 122 :M -.33(L )A 208 122 :M -.083(with )A 235 122 :M -.196(correlated )A 287 122 :M .604 .06(errors, )J 325 122 :M -.331(let )A 342 122 :M f2_12 sf .505(Transform\()A f6_12 sf .73(G)A f2_12 sf .336<29>A f0_12 sf .253 .025( )J 427 122 :M -.163(be )A 444 122 :M -.22(the )A 464 122 :M -.08(graph)A 59 139 :M -.11(resulting )A 104 139 :M -.08(from )A 131 139 :M -.182(replacing )A 178 139 :M -.326(a )A 187 139 :M -.11(double )A 223 139 :M -.163(headed )A 260 139 :M (arrow )S 292 139 :M -.139(between )A 335 139 :M -.196(correlated )A 385 139 :M (errors )S 417 139 :M f1_12 sf -.11(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 428 139 :M -.109(and )A 449 139 :M f1_12 sf -.11(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( )S 460 139 :M -.083(with )A 486 139 :M (a)S 59 158 :M -.137(new latent variable T)A f0_7 sf 0 3 rm -.055(ij)A 0 -3 rm f0_12 sf -.118( \(i < j\) )A 195 158 :M -.109(and )A 216 158 :M -.064(edges )A 247 158 :M -.08(from )A 274 158 :M -.113(T)A f0_7 sf 0 3 rm -.03(ij)A 0 -3 rm f0_12 sf ( )S 289 158 :M -.167(to )A 302 158 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 317 158 :M -.109(and )A 338 158 :M .926(X)A f0_7 sf 0 3 rm .208(j)A 0 -3 rm f0_12 sf .321(,)A f0_7 sf 0 3 rm .187 .019( )J 0 -3 rm 356 158 :M f0_12 sf -.109(and )A 377 158 :M -.165(then )A 401 158 :M -.123(removing )A 450 158 :M -.22(the )A 468 158 :M -.076(error)A 59 176 :M -.131(terms )A 90 176 :M -.08(from )A 118 176 :M -.22(the )A 137 176 :M .425 .043(graph. )J 173 176 :M -.108(See )A 195 176 :M -.054(Figure )A 231 176 :M .833 .083(1. )J 246 176 :M -.663(A )A 259 176 :M f2_12 sf .15(trek)A f0_12 sf .084 .008( )J 286 176 :M -.139(between )A 330 176 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 346 176 :M -.109(and )A 368 176 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 385 176 :M (is )S 399 176 :M -.163(an )A 416 176 :M -.164(undirected )A 471 176 :M -.22(path)A 59 193 :M -.139(between )A 104 193 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 121 193 :M -.109(and )A 144 193 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 161 193 :M -.249(that )A 184 193 :M -.123(contains )A 229 193 :M (no )S 247 193 :M -.031(colliders. )A 297 193 :M (If )S 311 193 :M -.196(there )A 340 193 :M (is )S 354 193 :M -.326(a )A 365 193 :M -.163(trek )A 390 193 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 408 193 :M f1_12 sf .126A f0_12 sf ( )S 427 193 :M -.113(T)A f0_7 sf 0 3 rm -.03(ij)A 0 -3 rm f0_12 sf ( )S 445 193 :M f1_12 sf .126A f0_12 sf ( )S 464 193 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 482 193 :M -.334(in)A 59 211 :M -.033(Transform\()A f4_12 sf -.053(G)A f0_12 sf -.028(\), we will say that )A 211 211 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 226 211 :M -.109(and )A 247 211 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 262 211 :M -.215(are )A 280 211 :M f2_12 sf .318(d-adjacent)A f0_12 sf .175 .017( )J 342 211 :M -.167(in )A 355 211 :M .053(Transform\()A f4_12 sf .084(G)A f0_12 sf .081 .008(\). )J 430 211 :M -.663(A )A 442 211 :M -.163(trek )A 464 211 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 479 211 :M f1_12 sf S 59 229 :M f0_12 sf -.113(T)A f0_7 sf 0 3 rm -.03(ij)A 0 -3 rm f0_12 sf ( )S 76 229 :M f1_12 sf .126A f0_12 sf ( )S 94 229 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 111 229 :M (is )S 125 229 :M -.274(called )A 158 229 :M -.326(a )A 170 229 :M f2_12 sf 2.186 .219(correlated )J 234 229 :M .178(error)A f2_7 sf 0 3 rm .057 .006( )J 0 -3 rm 268 229 :M f2_12 sf .15(trek)A f0_12 sf .084 .008( )J 297 229 :M -.167(in )A 313 229 :M .053(Transform\()A f4_12 sf .084(G)A f0_12 sf .081 .008(\). )J 391 229 :M (In )S 408 229 :M .053(Transform\()A f4_12 sf .084(G)A f0_12 sf .081 .008(\), )J 486 229 :M (a)S 59 248 :M f2_12 sf 1.166 .117(correlated error trek sequence)J f0_12 sf .694 .069( is a sequence of vertices such )J 436 248 :M -.249(that )A 457 248 :M (no )S 473 248 :M -.218(pair)A 59 266 :M (of )S 73 266 :M -.163(vertices )A 113 266 :M -.247(adjacent )A 155 266 :M -.167(in )A 168 266 :M -.22(the )A 186 266 :M -.121(sequence )A 233 266 :M -.215(are )A 251 266 :M -.131(identical, )A 298 266 :M -.109(and )A 319 266 :M (for )S 337 266 :M -.245(each )A 362 266 :M -.163(pair )A 385 266 :M (of )S 400 266 :M -.163(vertices )A 441 266 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 457 266 :M -.109(and )A 479 266 :M .468(X)A f0_7 sf 0 3 rm (s)S 0 -3 rm 59 284 :M f0_12 sf -.118(adjacent in the sequence, there is a correlated error trek between X)A f0_7 sf 0 3 rm -.058(r)A 0 -3 rm f0_12 sf ( )S 377 284 :M -.109(and )A 398 284 :M .805(X)A f0_7 sf 0 3 rm .253(s)A 0 -3 rm f0_12 sf .506 .051(. )J 418 284 :M .258 .026(For )J 439 284 :M -.235(example )A 482 284 :M -.334(in)A 59 302 :M -.054(Figure )A 95 302 :M .833 .083(1, )J 110 302 :M -.22(the )A 129 302 :M -.121(sequence )A 177 302 :M (of )S 192 302 :M -.163(vertices )A 233 302 :M 1.654 .165( )J 321 302 :M (is )S 334 302 :M -.326(a )A 344 302 :M -.196(correlated )A 395 302 :M -.061(error )A 424 302 :M -.163(trek )A 448 302 :M -.139(sequence)A 59 320 :M -.042(between X and Y.)A 126 336 297 62 rC 126 349 :M f1_12 sf (e)S 131 352 :M f0_7 sf (X)S 136 349 :M f0_12 sf ( )S 139 349 :M ( )S 142 349 :M ( )S 145 349 :M ( )S 148 349 :M ( )S 151 349 :M ( )S 154 349 :M ( )S 157 349 :M ( )S 160 349 :M ( )S 163 349 :M ( )S 166 349 :M ( )S 169 349 :M ( )S 172 349 :M ( )S 175 349 :M ( )S 178 349 :M f1_12 sf (e)S 183 352 :M f0_7 sf (A)S 188 349 :M f0_12 sf ( )S 191 349 :M ( )S 194 349 :M ( )S 197 349 :M ( )S 200 349 :M ( )S 203 349 :M ( )S 206 349 :M ( )S 209 349 :M ( )S 212 349 :M ( )S 215 349 :M ( )S 218 349 :M ( )S 221 349 :M ( )S 224 349 :M ( )S 227 349 :M ( )S 230 349 :M ( )S 233 349 :M f1_12 sf (e)S 238 352 :M f0_7 sf (B)S 243 349 :M f0_12 sf ( )S 246 349 :M ( )S 249 349 :M ( )S 252 349 :M ( )S 255 349 :M ( )S 258 349 :M ( )S 261 349 :M ( )S 264 349 :M ( )S 267 349 :M ( )S 270 349 :M ( )S 273 349 :M ( )S 276 349 :M ( )S 279 349 :M ( )S 282 349 :M f1_12 sf (e)S 287 352 :M f0_7 sf (C)S 292 349 :M f0_12 sf ( )S 295 349 :M ( )S 298 349 :M ( )S 301 349 :M ( )S 304 349 :M ( )S 307 349 :M ( )S 310 349 :M ( )S 313 349 :M ( )S 316 349 :M ( )S 319 349 :M ( )S 322 349 :M ( )S 325 349 :M ( )S 328 349 :M ( )S 331 349 :M ( )S 334 349 :M ( )S 337 349 :M f1_12 sf (e)S 342 352 :M f0_7 sf (D)S 347 349 :M f0_12 sf ( )S 350 349 :M ( )S 353 349 :M ( )S 356 349 :M ( )S 359 349 :M ( )S 362 349 :M ( )S 365 349 :M ( )S 368 349 :M ( )S 371 349 :M ( )S 374 349 :M ( )S 377 349 :M ( )S 380 349 :M ( )S 383 349 :M ( )S 386 349 :M ( )S 389 349 :M ( )S 392 349 :M f1_12 sf (e)S 397 352 :M f0_7 sf (Y)S 126 386 :M f0_12 sf (X)S 135 386 :M ( )S 138 386 :M ( )S 141 386 :M ( )S 144 386 :M ( )S 147 386 :M ( )S 150 386 :M ( )S 153 386 :M ( )S 156 386 :M ( )S 159 386 :M ( )S 162 386 :M ( )S 165 386 :M ( )S 168 386 :M ( )S 171 386 :M ( )S 174 386 :M ( )S 177 386 :M (A)S 185 386 :M ( )S 188 386 :M ( )S 191 386 :M ( )S 194 386 :M ( )S 197 386 :M ( )S 200 386 :M ( )S 203 386 :M ( )S 206 386 :M ( )S 209 386 :M ( )S 212 386 :M ( )S 215 386 :M ( )S 218 386 :M ( )S 221 386 :M ( )S 224 386 :M ( )S 227 386 :M ( )S 230 386 :M (B)S 238 386 :M ( )S 241 386 :M ( )S 244 386 :M ( )S 247 386 :M ( )S 250 386 :M ( )S 253 386 :M ( )S 256 386 :M ( )S 259 386 :M ( )S 262 386 :M ( )S 265 386 :M ( )S 268 386 :M ( )S 271 386 :M ( )S 274 386 :M ( )S 277 386 :M ( )S 280 386 :M (C)S 288 386 :M ( )S 291 386 :M ( )S 294 386 :M ( )S 297 386 :M ( )S 300 386 :M ( )S 303 386 :M ( )S 306 386 :M ( )S 309 386 :M ( )S 312 386 :M ( )S 315 386 :M ( )S 318 386 :M ( )S 321 386 :M ( )S 324 386 :M ( )S 327 386 :M ( )S 330 386 :M ( )S 333 386 :M ( )S 336 386 :M (D)S 344 386 :M ( )S 347 386 :M ( )S 350 386 :M ( )S 353 386 :M ( )S 356 386 :M ( )S 359 386 :M ( )S 362 386 :M ( )S 365 386 :M ( )S 368 386 :M ( )S 371 386 :M ( )S 374 386 :M ( )S 377 386 :M ( )S 380 386 :M ( )S 383 386 :M ( )S 386 386 :M ( )S 389 386 :M ( )S 392 386 :M (Y)S gR gS 125 335 300 94 rC 146 348.75 -.75 .75 168.75 348 .75 146 348 @a np 166 344 :M 166 352 :L 174 348 :L 166 344 :L .75 lw eofill -.75 -.75 166.75 352.75 .75 .75 166 344 @b -.75 -.75 166.75 352.75 .75 .75 174 348 @b 166 344.75 -.75 .75 174.75 348 .75 166 344 @a np 148 352 :M 148 344 :L 140 348 :L 148 352 :L eofill -.75 -.75 148.75 352.75 .75 .75 148 344 @b -.75 -.75 140.75 348.75 .75 .75 148 344 @b 140 348.75 -.75 .75 148.75 352 .75 140 348 @a 201 348.75 -.75 .75 223.75 348 .75 201 348 @a np 221 344 :M 221 352 :L 229 348 :L 221 344 :L eofill -.75 -.75 221.75 352.75 .75 .75 221 344 @b -.75 -.75 221.75 352.75 .75 .75 229 348 @b 221 344.75 -.75 .75 229.75 348 .75 221 344 @a np 203 352 :M 203 344 :L 195 348 :L 203 352 :L eofill -.75 -.75 203.75 352.75 .75 .75 203 344 @b -.75 -.75 195.75 348.75 .75 .75 203 344 @b 195 348.75 -.75 .75 203.75 352 .75 195 348 @a 250 348.75 -.75 .75 272.75 348 .75 250 348 @a np 270 344 :M 270 352 :L 278 348 :L 270 344 :L eofill -.75 -.75 270.75 352.75 .75 .75 270 344 @b -.75 -.75 270.75 352.75 .75 .75 278 348 @b 270 344.75 -.75 .75 278.75 348 .75 270 344 @a np 252 352 :M 252 344 :L 244 348 :L 252 352 :L eofill -.75 -.75 252.75 352.75 .75 .75 252 344 @b -.75 -.75 244.75 348.75 .75 .75 252 344 @b 244 348.75 -.75 .75 252.75 352 .75 244 348 @a 305 348.75 -.75 .75 327.75 348 .75 305 348 @a np 325 344 :M 325 352 :L 333 348 :L 325 344 :L eofill -.75 -.75 325.75 352.75 .75 .75 325 344 @b -.75 -.75 325.75 352.75 .75 .75 333 348 @b 325 344.75 -.75 .75 333.75 348 .75 325 344 @a np 307 352 :M 307 344 :L 299 348 :L 307 352 :L eofill -.75 -.75 307.75 352.75 .75 .75 307 344 @b -.75 -.75 299.75 348.75 .75 .75 307 344 @b 299 348.75 -.75 .75 307.75 352 .75 299 348 @a 361 348.75 -.75 .75 383.75 348 .75 361 348 @a np 381 344 :M 381 352 :L 389 348 :L 381 344 :L eofill -.75 -.75 381.75 352.75 .75 .75 381 344 @b -.75 -.75 381.75 352.75 .75 .75 389 348 @b 381 344.75 -.75 .75 389.75 348 .75 381 344 @a np 363 352 :M 363 344 :L 355 348 :L 363 352 :L eofill -.75 -.75 363.75 352.75 .75 .75 363 344 @b -.75 -.75 355.75 348.75 .75 .75 363 344 @b 355 348.75 -.75 .75 363.75 352 .75 355 348 @a -.75 -.75 131.75 369.75 .75 .75 131 358 @b np 135 367 :M 127 367 :L 131 375 :L 135 367 :L eofill 127 367.75 -.75 .75 135.75 367 .75 127 367 @a 127 367.75 -.75 .75 131.75 375 .75 127 367 @a -.75 -.75 131.75 375.75 .75 .75 135 367 @b -.75 -.75 181.75 368.75 .75 .75 181 357 @b np 185 366 :M 177 366 :L 181 374 :L 185 366 :L eofill 177 366.75 -.75 .75 185.75 366 .75 177 366 @a 177 366.75 -.75 .75 181.75 374 .75 177 366 @a -.75 -.75 181.75 374.75 .75 .75 185 366 @b -.75 -.75 235.75 367.75 .75 .75 235 356 @b np 239 365 :M 231 365 :L 235 373 :L 239 365 :L eofill 231 365.75 -.75 .75 239.75 365 .75 231 365 @a 231 365.75 -.75 .75 235.75 373 .75 231 365 @a -.75 -.75 235.75 373.75 .75 .75 239 365 @b -.75 -.75 285.75 367.75 .75 .75 285 356 @b np 289 365 :M 281 365 :L 285 373 :L 289 365 :L eofill 281 365.75 -.75 .75 289.75 365 .75 281 365 @a 281 365.75 -.75 .75 285.75 373 .75 281 365 @a -.75 -.75 285.75 373.75 .75 .75 289 365 @b -.75 -.75 341.75 368.75 .75 .75 341 357 @b np 345 366 :M 337 366 :L 341 374 :L 345 366 :L eofill 337 366.75 -.75 .75 345.75 366 .75 337 366 @a 337 366.75 -.75 .75 341.75 374 .75 337 366 @a -.75 -.75 341.75 374.75 .75 .75 345 366 @b -.75 -.75 395.75 367.75 .75 .75 395 356 @b np 399 365 :M 391 365 :L 395 373 :L 399 365 :L eofill 391 365.75 -.75 .75 399.75 365 .75 391 365 @a 391 365.75 -.75 .75 395.75 373 .75 391 365 @a -.75 -.75 395.75 373.75 .75 .75 399 365 @b 148 383.75 -.75 .75 171.75 383 .75 148 383 @a np 150 387 :M 150 379 :L 142 383 :L 150 387 :L eofill -.75 -.75 150.75 387.75 .75 .75 150 379 @b -.75 -.75 142.75 383.75 .75 .75 150 379 @b 142 383.75 -.75 .75 150.75 387 .75 142 383 @a 352 383.75 -.75 .75 380.75 383 .75 352 383 @a np 378 379 :M 378 387 :L 386 383 :L 378 379 :L eofill -.75 -.75 378.75 387.75 .75 .75 378 379 @b -.75 -.75 378.75 387.75 .75 .75 386 383 @b 378 379.75 -.75 .75 386.75 383 .75 378 379 @a -.75 -.75 129.75 400.75 .75 .75 129 398 @b np 125 400 :M 133 400 :L 129 392 :L 125 400 :L eofill 125 400.75 -.75 .75 133.75 400 .75 125 400 @a 129 392.75 -.75 .75 133.75 400 .75 129 392 @a -.75 -.75 125.75 400.75 .75 .75 129 392 @b -.75 -.75 141.75 400.75 .75 .75 141 398 @b np 137 400 :M 145 400 :L 141 392 :L 137 400 :L eofill 137 400.75 -.75 .75 145.75 400 .75 137 400 @a 141 392.75 -.75 .75 145.75 400 .75 141 392 @a -.75 -.75 137.75 400.75 .75 .75 141 392 @b 0 90 100 48 180.5 393.5 @n 90 180 94 36 188.5 399.5 @n 90 180 160 56 210.5 399.5 @n 0 90 148 70 209.5 392.5 @n gR gS 0 0 552 730 rC 270 456 :M f4_12 sf (G)S 126 472 297 62 rC 126 485 :M f1_12 sf ( )S 129 485 :M ( )S 132 485 :M ( )S 135 485 :M ( )S 138 485 :M ( )S 141 485 :M ( )S 144 485 :M ( )S 147 485 :M ( )S 150 485 :M ( )S 153 485 :M ( )S 156 485 :M (T)S 164 488 :M f0_7 sf (X)S 169 488 :M (A)S 174 485 :M f1_12 sf ( )S 177 485 :M ( )S 180 485 :M ( )S 183 485 :M ( )S 186 485 :M ( )S 189 485 :M ( )S 192 485 :M ( )S 195 485 :M ( )S 198 485 :M ( )S 201 485 :M ( )S 204 485 :M (T)S 212 488 :M f0_7 sf (A)S 217 488 :M (B)S 222 485 :M f1_12 sf ( )S 225 485 :M ( )S 228 485 :M ( )S 231 485 :M ( )S 234 485 :M ( )S 237 485 :M ( )S 240 485 :M ( )S 243 485 :M ( )S 246 485 :M ( )S 249 485 :M (T)S 257 488 :M f0_7 sf (B)S 262 488 :M (C)S 267 485 :M f1_12 sf ( )S 270 485 :M ( )S 273 485 :M ( )S 276 485 :M ( )S 279 485 :M ( )S 282 485 :M ( )S 285 485 :M ( )S 288 485 :M ( )S 291 485 :M ( )S 294 485 :M ( )S 297 485 :M (T)S 305 488 :M f0_7 sf (C)S 310 488 :M (D)S 315 485 :M f1_12 sf ( )S 318 485 :M ( )S 321 485 :M ( )S 324 485 :M ( )S 327 485 :M ( )S 330 485 :M ( )S 333 485 :M ( )S 336 485 :M ( )S 339 485 :M ( )S 342 485 :M ( )S 345 485 :M ( )S 348 485 :M ( )S 351 485 :M (T)S 359 488 :M f0_7 sf (D)S 364 488 :M (Y)S 126 522 :M f0_12 sf (X)S 135 522 :M ( )S 138 522 :M ( )S 141 522 :M ( )S 144 522 :M ( )S 147 522 :M ( )S 150 522 :M ( )S 153 522 :M ( )S 156 522 :M ( )S 159 522 :M ( )S 162 522 :M ( )S 165 522 :M ( )S 168 522 :M ( )S 171 522 :M ( )S 174 522 :M ( )S 177 522 :M ( )S 180 522 :M ( )S 183 522 :M ( )S 186 522 :M (A)S 194 522 :M ( )S 197 522 :M ( )S 200 522 :M ( )S 203 522 :M ( )S 206 522 :M ( )S 209 522 :M ( )S 212 522 :M ( )S 215 522 :M ( )S 218 522 :M ( )S 221 522 :M ( )S 224 522 :M ( )S 227 522 :M ( )S 230 522 :M (B)S 238 522 :M ( )S 241 522 :M ( )S 244 522 :M ( )S 247 522 :M ( )S 250 522 :M ( )S 253 522 :M ( )S 256 522 :M ( )S 259 522 :M ( )S 262 522 :M ( )S 265 522 :M ( )S 268 522 :M ( )S 271 522 :M ( )S 274 522 :M ( )S 277 522 :M ( )S 280 522 :M (C)S 288 522 :M ( )S 291 522 :M ( )S 294 522 :M ( )S 297 522 :M ( )S 300 522 :M ( )S 303 522 :M ( )S 306 522 :M ( )S 309 522 :M ( )S 312 522 :M ( )S 315 522 :M ( )S 318 522 :M ( )S 321 522 :M ( )S 324 522 :M ( )S 327 522 :M ( )S 330 522 :M ( )S 333 522 :M (D)S 341 522 :M ( )S 344 522 :M ( )S 347 522 :M ( )S 350 522 :M ( )S 353 522 :M ( )S 356 522 :M ( )S 359 522 :M ( )S 362 522 :M ( )S 365 522 :M ( )S 368 522 :M ( )S 371 522 :M ( )S 374 522 :M ( )S 377 522 :M ( )S 380 522 :M ( )S 383 522 :M ( )S 386 522 :M ( )S 389 522 :M ( )S 392 522 :M (Y)S gR gS 125 471 299 93 rC 148 519.75 -.75 .75 171.75 519 .75 148 519 @a np 150 523 :M 150 515 :L 142 519 :L 150 523 :L .75 lw eofill -.75 -.75 150.75 523.75 .75 .75 150 515 @b -.75 -.75 142.75 519.75 .75 .75 150 515 @b 142 519.75 -.75 .75 150.75 523 .75 142 519 @a 352 519.75 -.75 .75 380.75 519 .75 352 519 @a np 378 515 :M 378 523 :L 386 519 :L 378 515 :L eofill -.75 -.75 378.75 523.75 .75 .75 378 515 @b -.75 -.75 378.75 523.75 .75 .75 386 519 @b 378 515.75 -.75 .75 386.75 519 .75 378 515 @a -.75 -.75 129.75 536.75 .75 .75 129 534 @b np 125 536 :M 133 536 :L 129 528 :L 125 536 :L eofill 125 536.75 -.75 .75 133.75 536 .75 125 536 @a 129 528.75 -.75 .75 133.75 536 .75 129 528 @a -.75 -.75 125.75 536.75 .75 .75 129 528 @b -.75 -.75 141.75 536.75 .75 .75 141 534 @b np 137 536 :M 145 536 :L 141 528 :L 137 536 :L eofill 137 536.75 -.75 .75 145.75 536 .75 137 536 @a 141 528.75 -.75 .75 145.75 536 .75 141 528 @a -.75 -.75 137.75 536.75 .75 .75 141 528 @b 0 90 100 48 180.5 529.5 @n 90 180 94 36 188.5 535.5 @n 90 180 160 56 210.5 535.5 @n 0 90 148 70 209.5 528.5 @n -.75 -.75 140.75 510.75 .75 .75 159 491 @b np 144 511 :M 139 506 :L 136 514 :L 144 511 :L eofill 139 506.75 -.75 .75 144.75 511 .75 139 506 @a -.75 -.75 136.75 514.75 .75 .75 139 506 @b -.75 -.75 136.75 514.75 .75 .75 144 511 @b 160 490.75 -.75 .75 178.75 507 .75 160 490 @a np 179 503 :M 174 508 :L 182 511 :L 179 503 :L eofill -.75 -.75 174.75 508.75 .75 .75 179 503 @b 174 508.75 -.75 .75 182.75 511 .75 174 508 @a 179 503.75 -.75 .75 182.75 511 .75 179 503 @a -.75 -.75 196.75 506.75 .75 .75 214 492 @b np 199 508 :M 195 502 :L 191 510 :L 199 508 :L eofill 195 502.75 -.75 .75 199.75 508 .75 195 502 @a -.75 -.75 191.75 510.75 .75 .75 195 502 @b -.75 -.75 191.75 510.75 .75 .75 199 508 @b 214 491.75 -.75 .75 228.75 506 .75 214 491 @a np 229 502 :M 224 507 :L 232 510 :L 229 502 :L eofill -.75 -.75 224.75 507.75 .75 .75 229 502 @b 224 507.75 -.75 .75 232.75 510 .75 224 507 @a 229 502.75 -.75 .75 232.75 510 .75 229 502 @a -.75 -.75 241.75 507.75 .75 .75 258 490 @b np 245 508 :M 240 503 :L 237 511 :L 245 508 :L eofill 240 503.75 -.75 .75 245.75 508 .75 240 503 @a -.75 -.75 237.75 511.75 .75 .75 240 503 @b -.75 -.75 237.75 511.75 .75 .75 245 508 @b 259 490.75 -.75 .75 276.75 508 .75 259 490 @a np 277 504 :M 272 509 :L 280 512 :L 277 504 :L eofill -.75 -.75 272.75 509.75 .75 .75 277 504 @b 272 509.75 -.75 .75 280.75 512 .75 272 509 @a 277 504.75 -.75 .75 280.75 512 .75 277 504 @a -.75 -.75 289.75 508.75 .75 .75 307 490 @b np 293 509 :M 288 504 :L 285 512 :L 293 509 :L eofill 288 504.75 -.75 .75 293.75 509 .75 288 504 @a -.75 -.75 285.75 512.75 .75 .75 288 504 @b -.75 -.75 285.75 512.75 .75 .75 293 509 @b 307 490.75 -.75 .75 326.75 509 .75 307 490 @a np 327 505 :M 322 510 :L 330 513 :L 327 505 :L eofill -.75 -.75 322.75 510.75 .75 .75 327 505 @b 322 510.75 -.75 .75 330.75 513 .75 322 510 @a 327 505.75 -.75 .75 330.75 513 .75 327 505 @a -.75 -.75 345.75 507.75 .75 .75 361 490 @b np 349 508 :M 344 503 :L 341 511 :L 349 508 :L eofill 344 503.75 -.75 .75 349.75 508 .75 344 503 @a -.75 -.75 341.75 511.75 .75 .75 344 503 @b -.75 -.75 341.75 511.75 .75 .75 349 508 @b 362 490.75 -.75 .75 381.75 509 .75 362 490 @a np 382 505 :M 377 510 :L 385 513 :L 382 505 :L eofill -.75 -.75 377.75 510.75 .75 .75 382 505 @b 377 510.75 -.75 .75 385.75 513 .75 377 510 @a 382 505.75 -.75 .75 385.75 513 .75 382 505 @a gR gS 0 0 552 730 rC 241 591 :M f0_12 sf -.023(Transform\()A f4_12 sf (G)S f0_12 sf <29>S 251 615 :M f2_12 sf 2.949 .295(Figure 1)J endp %%Page: 3 3 %%BeginPageSetup initializepage (peter; page: 3 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (3)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.164(It )A 74 56 :M -.2(might )A 109 56 :M -.33(at )A 125 56 :M -.064(first )A 152 56 :M -.219(glance )A 190 56 :M -.162(appear )A 229 56 :M -.249(that )A 254 56 :M (for )S 276 56 :M -.129(every )A 310 56 :M -.226(parameterization )A 396 56 :M (of )S 415 56 :M f4_12 sf .79(G)A f0_12 sf .497 .05(, )J 437 56 :M -.196(there )A 469 56 :M (is )S 486 56 :M (a)S 59 74 :M -.12(parameterization of Transform\()A f4_12 sf -.206(G)A f0_12 sf -.118(\) with the same covariance matrix. )A 381 74 :M -.219(The )A 403 74 :M -.073(following )A 453 74 :M -.219(theorem)A 59 92 :M -.031(shows that this is not the case.)A 59 122 :M f2_12 sf 1.973 .197(Theorem )J 113 122 :M .387(1:)A f0_12 sf .232 .023( )J 128 122 :M -.195(There )A 159 122 :M -.055(exists )A 190 122 :M -.326(a )A 199 122 :M -.223(SEM )A 227 122 :M -.33(L )A 238 122 :M -.083(with )A 263 122 :M -.122(measured )A 312 122 :M -.145(variables )A 358 122 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(, )J 376 122 :M -.196(correlated )A 427 122 :M .604 .06(errors, )J 464 122 :M -.08(graph)A 59 139 :M f4_12 sf .79(G)A f0_12 sf .497 .05(, )J 76 139 :M -.109(and )A 97 139 :M -.179(correlation )A 151 139 :M -.22(matrix )A 185 139 :M f1_12 sf .597(S)A f0_12 sf .336<28>A f2_12 sf .728(X)A f0_12 sf .534 .053(\) )J 215 139 :M (such )S 241 139 :M -.249(that )A 262 139 :M (no )S 278 139 :M -.219(linear )A 308 139 :M -.226(parameterization )A 389 139 :M (of )S 403 139 :M -.022(Transform\()A f4_12 sf (G)S f0_12 sf (\) )S 475 139 :M (has)S 59 157 :M -.05(marginal correlation matrix )A f1_12 sf -.074(S)A f0_12 sf <28>S f2_12 sf -.09(X)A f0_12 sf -.073(\).)A 59 188 :M f2_12 sf .035(Proof.)A f0_12 sf .112 .011( Assume that L has no structural equations, but )J 322 188 :M -.129(every )A 352 188 :M -.163(pair )A 374 188 :M (of )S 388 188 :M (errors )S 420 188 :M (is )S 432 188 :M -.196(correlated )A 482 188 :M -.334(in)A 59 206 :M (L. )S f4_12 sf (G)S f0_12 sf .063 .006( and Transform\(G\) are shown in Figure 2.)J 59 251 217 102 rC 59 297 :M f1_12 sf (e)S 64 300 :M f0_7 sf (1)S 95 297 :M f0_12 sf ( )S 98 297 :M ( )S 101 297 :M ( )S 104 297 :M ( )S 107 297 :M f1_12 sf (e)S 112 300 :M f0_7 sf (2)S 131 297 :M f0_12 sf ( )S 134 297 :M ( )S 137 297 :M ( )S 140 297 :M ( )S 143 297 :M ( )S 146 297 :M ( )S 149 297 :M ( )S 152 297 :M ( )S 155 297 :M f1_12 sf (e)S 160 300 :M f0_7 sf (3)S 59 334 :M f0_12 sf (X)S 68 337 :M f0_7 sf (1)S 95 334 :M f0_12 sf ( )S 98 334 :M ( )S 101 334 :M ( )S 104 334 :M ( )S 107 334 :M (X)S 116 337 :M f0_7 sf (2)S 131 334 :M f0_12 sf ( )S 134 334 :M ( )S 137 334 :M ( )S 140 334 :M ( )S 143 334 :M ( )S 146 334 :M ( )S 149 334 :M ( )S 152 334 :M ( )S 155 334 :M (X)S 164 337 :M f0_7 sf (3)S 77 295 -1 1 96 294 1 77 294 @a np 93 290 :M 93 298 :L 101 294 :L 93 290 :L eofill -1 -1 94 299 1 1 93 290 @b -1 -1 94 299 1 1 101 294 @b 93 291 -1 1 102 294 1 93 290 @a np 79 298 :M 79 290 :L 71 294 :L 79 298 :L eofill -1 -1 80 299 1 1 79 290 @b -1 -1 72 295 1 1 79 290 @b 71 295 -1 1 80 298 1 71 294 @a 126 294 -1 1 145 293 1 126 293 @a np 142 289 :M 142 297 :L 150 293 :L 142 289 :L eofill -1 -1 143 298 1 1 142 289 @b -1 -1 143 298 1 1 150 293 @b 142 290 -1 1 151 293 1 142 289 @a np 128 297 :M 128 289 :L 120 293 :L 128 297 :L eofill -1 -1 129 298 1 1 128 289 @b -1 -1 121 294 1 1 128 289 @b 120 294 -1 1 129 297 1 120 293 @a -1 -1 66 318 1 1 65 305 @b np 69 315 :M 61 315 :L 65 323 :L 69 315 :L eofill 61 316 -1 1 70 315 1 61 315 @a 61 316 -1 1 66 323 1 61 315 @a -1 -1 66 324 1 1 69 315 @b -1 -1 160 315 1 1 159 302 @b np 163 312 :M 155 312 :L 159 320 :L 163 312 :L eofill 155 313 -1 1 164 312 1 155 312 @a 155 313 -1 1 160 320 1 155 312 @a -1 -1 160 321 1 1 163 312 @b -1 -1 112 315 1 1 111 302 @b np 115 312 :M 107 312 :L 111 320 :L 115 312 :L eofill 107 313 -1 1 116 312 1 107 312 @a 107 313 -1 1 112 320 1 107 312 @a -1 -1 112 321 1 1 115 312 @b -1 -1 159 281 1 1 158 279 @b np 162 278 :M 154 278 :L 158 286 :L 162 278 :L eofill 154 279 -1 1 163 278 1 154 278 @a 154 279 -1 1 159 286 1 154 278 @a -1 -1 159 287 1 1 162 278 @b -1 -1 63 283 1 1 62 281 @b np 66 280 :M 58 280 :L 62 288 :L 66 280 :L eofill 58 281 -1 1 67 280 1 58 280 @a 58 281 -1 1 63 288 1 58 280 @a -1 -1 63 289 1 1 66 280 @b -180 -90 70 38 97.5 280.5 @n -90 0 120 34 97.5 278.5 @n gR gS 0 0 552 730 rC 59 368 :M f0_12 sf .016 .002( )J f4_12 sf (G)S 293 251 217 108 rC 293 271 :M f0_12 sf ( )S 296 271 :M ( )S 299 271 :M ( )S 302 271 :M ( )S 305 271 :M ( )S 308 271 :M ( )S 311 271 :M ( )S 314 271 :M ( )S 317 271 :M (T)S 324 274 :M f0_7 sf (1)S 328 274 :M .25 .025( )J 330 271 :M f0_12 sf ( )S 333 271 :M ( )S 336 271 :M ( )S 339 271 :M ( )S 342 271 :M ( )S 345 271 :M ( )S 348 271 :M ( )S 351 271 :M ( )S 354 271 :M ( )S 357 271 :M ( )S 360 271 :M ( )S 363 271 :M ( )S 366 271 :M ( )S 369 271 :M ( )S 372 271 :M (T)S 379 274 :M f0_7 sf (2)S 293 307 :M f0_12 sf (X)S 302 310 :M f0_7 sf (1)S 329 307 :M f0_12 sf ( )S 332 307 :M ( )S 335 307 :M ( )S 338 307 :M ( )S 341 307 :M (X)S 350 310 :M f0_7 sf (2)S 365 307 :M f0_12 sf ( )S 368 307 :M ( )S 371 307 :M ( )S 374 307 :M ( )S 377 307 :M ( )S 380 307 :M ( )S 383 307 :M ( )S 386 307 :M ( )S 389 307 :M (X)S 398 310 :M f0_7 sf (3)S 293 343 :M f0_12 sf ( )S 296 343 :M ( )S 299 343 :M ( )S 302 343 :M ( )S 305 343 :M ( )S 308 343 :M ( )S 311 343 :M ( )S 314 343 :M ( )S 317 343 :M ( )S 320 343 :M ( )S 323 343 :M ( )S 326 343 :M ( )S 329 343 :M ( )S 332 343 :M ( )S 335 343 :M (T)S 342 346 :M f0_7 sf (3)S -1 -1 307 293 1 1 321 277 @b np 310 293 :M 305 288 :L 302 296 :L 310 293 :L eofill 305 289 -1 1 311 293 1 305 288 @a -1 -1 303 297 1 1 305 288 @b -1 -1 303 297 1 1 310 293 @b 323 278 -1 1 339 292 1 323 277 @a np 339 288 :M 334 293 :L 342 296 :L 339 288 :L eofill -1 -1 335 294 1 1 339 288 @b 334 294 -1 1 343 296 1 334 293 @a 339 289 -1 1 343 296 1 339 288 @a -1 -1 358 293 1 1 372 277 @b np 361 293 :M 356 288 :L 353 296 :L 361 293 :L eofill 356 289 -1 1 362 293 1 356 288 @a -1 -1 354 297 1 1 356 288 @b -1 -1 354 297 1 1 361 293 @b 374 278 -1 1 390 292 1 374 277 @a np 390 288 :M 385 293 :L 393 296 :L 390 288 :L eofill -1 -1 386 294 1 1 390 288 @b 385 294 -1 1 394 296 1 385 293 @a 390 289 -1 1 394 296 1 390 288 @a -1 -1 344 332 1 1 382 313 @b np 378 311 :M 381 318 :L 387 311 :L 378 311 :L eofill 378 312 -1 1 382 318 1 378 311 @a -1 -1 382 319 1 1 387 311 @b 378 312 -1 1 388 311 1 378 311 @a 311 316 -1 1 343 331 1 311 315 @a np 311 319 :M 315 313 :L 306 312 :L 311 319 :L eofill -1 -1 312 320 1 1 315 313 @b 306 313 -1 1 316 313 1 306 312 @a 306 313 -1 1 312 319 1 306 312 @a gR gS 0 0 552 730 rC 293 374 :M f0_12 sf -.016( Transform\()A f4_12 sf (G)S f0_12 sf <29>S 251 422 :M f2_12 sf 2.949 .295(Figure 2)J 59 457 :M f0_12 sf -.085(Suppose that the marginal correlation matrix )A f1_12 sf -.125(S)A f0_12 sf -.07<28>A f2_12 sf -.152(X)A f0_12 sf -.083(\) is the following:)A 206 473 138 52 rC 344 525 :M psb currentpoint pse 206 473 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4416 div 1664 3 -1 roll exch div scale currentpoint translate 64 39 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (S) -1 889 sh 384 /Times-Roman f1 (\() 236 889 sh (\)) 659 889 sh (.) 1666 313 sh (.) 2621 313 sh (.) 3652 313 sh (.) 1590 889 sh (.) 2697 889 sh (.) 3652 889 sh (.) 1590 1465 sh (.) 2621 1465 sh (.) 3728 1465 sh 384 /Times-Italic f1 (X) 398 889 sh 384 /Symbol f1 (=) 892 889 sh (\346) 1211 355 sh (\350) 1211 1468 sh (\347) 1211 806 sh (\347) 1211 1177 sh (\366) 4125 355 sh (\370) 4125 1468 sh (\367) 4125 806 sh (\367) 4125 1177 sh 384 /Times-Roman f1 (1) 1474 313 sh (0) 1762 313 sh (0) 2429 313 sh (99) 2717 313 sh (0) 3460 313 sh (99) 3748 313 sh (0) 1398 889 sh (99) 1686 889 sh (1) 2505 889 sh (0) 2793 889 sh (0) 3460 889 sh (99) 3748 889 sh (0) 1398 1465 sh (99) 1686 1465 sh (0) 2429 1465 sh (99) 2717 1465 sh (1) 3536 1465 sh (0) 3824 1465 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 552 :M f0_12 sf -.104(Every parameterization of Transform\()A f4_12 sf -.178(G)A f0_12 sf -.099(\) is of the following form:)A 59 595 :M ( )S 212 567 125 51 rC 337 618 :M psb currentpoint pse 212 567 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4000 div 1632 3 -1 roll exch div scale currentpoint translate 64 43 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (X) 17 261 sh (a) 808 261 sh (T) 1196 261 sh (a) 1864 261 sh (T) 2260 261 sh (b) 2947 261 sh (X) 17 853 sh (a) 844 853 sh (T) 1250 853 sh (a) 1918 853 sh (T) 2342 853 sh (b) 3042 853 sh (X) 17 1442 sh (a) 831 1442 sh (T) 1252 1442 sh (a) 1956 1442 sh (T) 2367 1442 sh (b) 3054 1442 sh 224 /Times-Roman f1 (1) 235 357 sh (11) 976 357 sh (1) 1354 357 sh (13) 2032 357 sh (3) 2433 357 sh (11) 3103 357 sh (1) 3594 357 sh (2) 253 949 sh (21) 1030 949 sh (1) 1408 949 sh (22) 2104 949 sh (2) 2518 949 sh (22) 3216 949 sh (2) 3743 949 sh (3) 250 1538 sh (32) 1014 1538 sh (2) 1428 1538 sh (33) 2139 1538 sh (3) 2540 1538 sh (33) 3225 1538 sh (3) 3739 1538 sh 384 /Symbol f1 (=) 479 261 sh (+) 1567 261 sh (+) 2654 261 sh (=) 515 853 sh (+) 1621 853 sh (+) 2749 853 sh (=) 502 1442 sh (+) 1659 1442 sh (+) 2761 1442 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 3319 261 sh (e) 3450 853 sh (e) 3449 1442 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 3494 261 sh (\251) 3625 853 sh (\251) 3624 1442 sh end MTsave restore pse gR gS 0 0 552 730 rC 337 595 :M f0_12 sf ( \(1\))S 59 645 :M -.093(Suppose first that the variance of each of the variables is equal to 1. It follows then that)A endp %%Page: 4 4 %%BeginPageSetup initializepage (peter; page: 4 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (4)S gR gS 0 0 552 730 rC 59 73 :M f0_12 sf ( )S 212 41 148 57 rC 360 98 :M psb currentpoint pse 212 41 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4736 div 1824 3 -1 roll exch div scale currentpoint translate 64 57 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (var\() 56 318 sh (\)) 1063 318 sh (var\() -3 967 sh (\)) 1040 967 sh (var\() 9 1613 sh (\)) 1039 1613 sh 384 /Times-Italic f1 (X) 707 318 sh (a) 2192 318 sh (a) 3114 318 sh (b) 4040 318 sh (X) 648 967 sh (a) 2169 967 sh (a) 3109 967 sh (b) 4063 967 sh (X) 660 1613 sh (a) 2168 1613 sh (a) 3123 1613 sh (b) 4064 1613 sh 224 /Times-Roman f1 (1) 925 414 sh (11) 2360 414 sh (2) 2586 152 sh (13) 3282 414 sh (2) 3516 152 sh (11) 4196 414 sh (2) 4422 152 sh (2) 884 1063 sh (21) 2355 1063 sh (2) 2581 801 sh (22) 3295 1063 sh (2) 3539 801 sh (22) 4237 1063 sh (2) 4481 801 sh (3) 893 1709 sh (32) 2351 1709 sh (2) 2595 1447 sh (33) 3306 1709 sh (2) 3540 1447 sh (33) 4235 1709 sh (2) 4469 1447 sh 384 ns (1) 1588 318 sh (1) 1565 967 sh (1) 1564 1613 sh 384 /Symbol f1 (=) 1296 318 sh (=) 1863 318 sh (+) 2817 318 sh (+) 3747 318 sh (=) 1273 967 sh (=) 1840 967 sh (+) 2812 967 sh (+) 3770 967 sh (=) 1272 1613 sh (=) 1839 1613 sh (+) 2826 1613 sh (+) 3771 1613 sh end MTsave restore pse gR gS 0 0 552 730 rC 360 73 :M f0_12 sf ( \(2\))S 59 138 :M ( )S 215 110 136 51 rC 351 161 :M psb currentpoint pse 215 110 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4352 div 1632 3 -1 roll exch div scale currentpoint translate 64 46 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (corr) 28 261 sh (corr) -9 850 sh (corr) 7 1442 sh 384 /Times-Roman f1 (\() 656 261 sh (,) 1161 261 sh (\)) 1711 261 sh (.) 2461 261 sh (\() 619 850 sh (,) 1160 850 sh (\)) 1697 850 sh (.) 2447 850 sh (\() 635 1442 sh (,) 1183 1442 sh (\)) 1720 1442 sh (.) 2470 1442 sh 384 /Times-Italic f1 (X) 818 261 sh (X) 1319 261 sh (a) 3378 261 sh (a) 3773 261 sh (X) 781 850 sh (X) 1318 850 sh (a) 3364 850 sh (a) 3795 850 sh (X) 797 1442 sh (X) 1341 1442 sh (a) 3387 1442 sh (a) 3790 1442 sh 224 /Times-Roman f1 (1) 1036 357 sh (2) 1555 357 sh (11) 3546 357 sh (21) 3959 357 sh (2) 1017 946 sh (3) 1551 946 sh (22) 3550 946 sh (32) 3978 946 sh (3) 1574 1538 sh (13) 3555 1538 sh (33) 3973 1538 sh 384 ns (0) 2269 261 sh (99) 2557 261 sh (0) 2255 850 sh (99) 2543 850 sh (1) 1027 1442 sh (0) 2278 1442 sh (99) 2566 1442 sh 384 /Symbol f1 (=) 1944 261 sh (=) 3049 261 sh (=) 1930 850 sh (=) 3035 850 sh (=) 1953 1442 sh (=) 3058 1442 sh end MTsave restore pse gR gS 0 0 552 730 rC 351 138 :M f0_12 sf ( \(3\))S 59 188 :M (From )S 89 188 :M .724 .072(\(1\), )J 112 188 :M -.22(the )A 131 188 :M -.123(absolute )A 175 188 :M -.109(values )A 210 188 :M (of )S 225 188 :M -.245(each )A 251 188 :M (of )S 266 188 :M -.22(the )A 285 188 :M -.163(coefficients )A 344 188 :M (is )S 357 188 :M (less )S 380 188 :M -.165(than )A 405 188 :M .481 .048(one. )J 431 188 :M (From )S 462 188 :M .724 .072(\(2\), )J 485 188 :M -.668(it)A 59 206 :M .16 .016(follows that )J f4_12 sf .053(a)A f0_7 sf 0 3 rm .031(11)A 0 -3 rm f0_12 sf .044 .004(, )J f4_12 sf .053(a)A f0_7 sf 0 3 rm .031(21)A 0 -3 rm f0_12 sf .044 .004(, )J f4_12 sf .053(a)A f0_7 sf 0 3 rm .031(22)A 0 -3 rm f0_12 sf .044 .004(, )J f4_12 sf .053(a)A f0_7 sf 0 3 rm .031(32)A 0 -3 rm f0_12 sf .044 .004(, )J f4_12 sf .053(a)A f0_7 sf 0 3 rm .031(13)A 0 -3 rm f0_12 sf .089 .009(, and )J f4_12 sf .053(a)A f0_7 sf 0 3 rm .031(33)A 0 -3 rm f0_12 sf .098 .01( all have )J 294 206 :M -.123(absolute )A 337 206 :M -.109(values )A 371 206 :M -.186(greater )A 407 206 :M -.165(than )A 431 206 :M 1.333 .133(0.99. )J 461 206 :M -.16(Hence)A 59 224 :M -.049(var\(T)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.041(\) is greater than 1, )A 179 224 :M -.065(which )A 212 224 :M (is )S 224 224 :M -.326(a )A 233 224 :M -.093(contradiction. )A 302 224 :M -.164(It )A 313 224 :M (follows )S 353 224 :M -.249(that )A 374 224 :M -.196(there )A 401 224 :M -.215(are )A 419 224 :M (no )S 435 224 :M -.037(solutions )A 482 224 :M -.334(to)A 59 242 :M .226 .023(\(2\) and \(3\).)J 59 272 :M -.016(Suppose now that we )A 165 272 :M (do )S 181 272 :M -.111(not )A 200 272 :M -.109(fix )A 217 272 :M -.22(the )A 235 272 :M -.144(variances )A 283 272 :M (of )S 297 272 :M -.22(the )A 315 272 :M -.035(exogenous )A 370 272 :M -.145(variables )A 416 272 :M -.33(at )A 428 272 :M .481 .048(one. )J 453 272 :M -.326(We )A 473 272 :M -.222(will)A 59 290 :M .479 .048(show )J 89 290 :M -.249(that )A 110 290 :M -.164(if )A 121 290 :M -.22(the )A 140 290 :M -.049(corresponding )A 213 290 :M -.109(set )A 231 290 :M (of )S 246 290 :M -.11(equations )A 296 290 :M (has )S 317 290 :M -.326(a )A 327 290 :M .174 .017(solution, )J 374 290 :M -.165(then )A 399 290 :M .277 .028(so )J 415 290 :M (do )S 432 290 :M (\(2\) )S 451 290 :M -.109(and )A 473 290 :M .338(\(3\),)A 59 308 :M -.082(which is a contradiction.)A 69 355 :M ( )S 138 323 267 57 rC 405 380 :M psb currentpoint pse 138 323 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8544 div 1824 3 -1 roll exch div scale currentpoint translate 64 57 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (var\() 80 318 sh (\)) 1087 318 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 2388 318 sh (var\() 2913 318 sh (\)) 3831 318 sh (\251) 4502 318 sh (var\() 5035 318 sh (\)) 5976 318 sh (\251) 6643 318 sh (var\() 7168 318 sh (\251) 7961 318 sh (\)) 8234 318 sh (var\() -3 967 sh (\)) 1040 967 sh (\251) 2341 967 sh (var\() 2884 967 sh (\)) 3802 967 sh (\251) 4473 967 sh (var\() 5034 967 sh (\)) 5988 967 sh (\251) 6655 967 sh (var\() 7216 967 sh (\251) 8009 967 sh (\)) 8318 967 sh (var\() 4 1613 sh (\)) 1034 1613 sh (\251) 2335 1613 sh (var\() 2893 1613 sh (\)) 3847 1613 sh (\251) 4518 1613 sh (var\() 5066 1613 sh (\)) 6007 1613 sh (\251) 6674 1613 sh (var\() 7222 1613 sh (\251) 8015 1613 sh (\)) 8311 1613 sh 384 /Times-Italic f1 (X) 731 318 sh (a) 2216 318 sh (T) 3535 318 sh (a) 4330 318 sh (T) 5657 318 sh (b) 6471 318 sh (X) 648 967 sh (a) 2169 967 sh (T) 3506 967 sh (a) 4301 967 sh (T) 5656 967 sh (b) 6483 967 sh (X) 655 1613 sh (a) 2163 1613 sh (T) 3515 1613 sh (a) 4346 1613 sh (T) 5688 1613 sh (b) 6502 1613 sh 224 /MT_Times-Roman f1 (1) 949 414 sh (11) 2488 414 sh (2) 2714 152 sh (1) 3693 414 sh (13) 4602 414 sh (2) 4836 152 sh (3) 5830 414 sh (11) 6743 414 sh (2) 6969 152 sh (1) 8061 414 sh (2) 884 1063 sh (21) 2459 1063 sh (2) 2685 801 sh (1) 3664 1063 sh (22) 4591 1063 sh (2) 4835 801 sh (2) 5832 1063 sh (22) 6773 1063 sh (2) 7017 801 sh (2) 8127 1063 sh (3) 888 1709 sh (32) 2450 1709 sh (2) 2694 1447 sh (2) 3691 1709 sh (33) 4633 1709 sh (2) 4867 1447 sh (3) 5861 1709 sh (33) 6789 1709 sh (2) 7023 1447 sh (3) 8130 1709 sh 384 ns (1) 1612 318 sh (1) 1565 967 sh (1) 1559 1613 sh 384 /Symbol f1 (=) 1320 318 sh (=) 1887 318 sh (+) 4033 318 sh (+) 6178 318 sh (=) 1273 967 sh (=) 1840 967 sh (+) 4004 967 sh (+) 6190 967 sh (=) 1267 1613 sh (=) 1834 1613 sh (+) 4049 1613 sh (+) 6209 1613 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 7786 318 sh (e) 7834 967 sh (e) 7840 1613 sh end MTsave restore pse gR gS 0 0 552 730 rC 405 355 :M f0_12 sf ( \(2\325\))S 59 420 :M ( )S 170 392 179 51 rC 349 443 :M psb currentpoint pse 170 392 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5728 div 1632 3 -1 roll exch div scale currentpoint translate 64 46 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (corr) 46 261 sh (corr) -9 850 sh (corr) 35 1442 sh 384 /Times-Roman f1 (\() 674 261 sh (,) 1179 261 sh (\)) 1729 261 sh (.) 2479 261 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 3568 261 sh (\251) 4131 261 sh (var\() 4542 261 sh (\)) 5460 261 sh (\() 619 850 sh (,) 1160 850 sh (\)) 1697 850 sh (.) 2447 850 sh (\251) 3536 850 sh (\251) 4135 850 sh (var\() 4561 850 sh (\)) 5515 850 sh (\() 663 1442 sh (,) 1168 1442 sh (\)) 1705 1442 sh (.) 2455 1442 sh (\251) 3544 1442 sh (\251) 4115 1442 sh (var\() 4531 1442 sh (\)) 5472 1442 sh 384 /Times-Italic f1 (X) 836 261 sh (X) 1337 261 sh (a) 3396 261 sh (a) 3959 261 sh (T) 5164 261 sh (X) 781 850 sh (X) 1318 850 sh (a) 3364 850 sh (a) 3963 850 sh (T) 5183 850 sh (X) 825 1442 sh (X) 1326 1442 sh (a) 3372 1442 sh (a) 3943 1442 sh (T) 5153 1442 sh 224 /MT_Times-Roman f1 (1) 1054 357 sh (2) 1573 357 sh (11) 3668 357 sh (21) 4249 357 sh (1) 5322 357 sh (2) 1017 946 sh (3) 1551 946 sh (22) 3654 946 sh (32) 4250 946 sh (2) 5359 946 sh (1) 1043 1538 sh (3) 1559 1538 sh (13) 3644 1538 sh (33) 4230 1538 sh (3) 5326 1538 sh 384 ns (0) 2287 261 sh (99) 2575 261 sh (0) 2255 850 sh (99) 2543 850 sh (0) 2263 1442 sh (99) 2551 1442 sh 384 /Symbol f1 (=) 1962 261 sh (=) 3067 261 sh (=) 1930 850 sh (=) 3035 850 sh (=) 1938 1442 sh (=) 3043 1442 sh end MTsave restore pse gR gS 0 0 552 730 rC 349 420 :M f0_12 sf ( \(3\325\))S 59 470 :M -.044(Suppose now that 2\325 and 3\325 have a solution. Then set)A 124 485 301 62 rC 425 547 :M psb currentpoint pse 124 485 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9632 div 1984 3 -1 roll exch div scale currentpoint translate 64 46 translate /thick 0 def /th { dup setlinewidth /thick exch def } def /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .34 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .34 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 1315 469 384 1625 469 16 sqr 1315 469 384 4776 469 16 sqr 1351 469 384 7963 469 16 sqr 1351 469 384 1658 1170 16 sqr 1338 472 384 4825 1173 16 sqr 1338 472 384 8009 1173 16 sqr 1463 469 384 1396 1874 16 sqr 1499 469 384 4711 1874 16 sqr 1486 472 384 8036 1877 16 sqr /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 (a) 204 341 sh (a) 1057 341 sh (T) 2503 341 sh (a) 3319 341 sh (a) 4190 341 sh (T) 5654 341 sh (a) 6470 341 sh (a) 7359 341 sh (T) 8841 341 sh (a) 171 1042 sh (a) 1057 1042 sh (T) 2536 1042 sh (a) 3388 1042 sh (a) 4249 1042 sh (T) 5703 1042 sh (a) 6542 1042 sh (a) 7418 1042 sh (T) 8887 1042 sh (b) -9 1746 sh (b) 828 1746 sh (b) 3234 1746 sh (b) 4107 1746 sh (b) 6585 1746 sh (b) 7445 1746 sh 224 /Times-Roman f1 (11) 372 437 sh (11) 1329 437 sh (1) 2661 437 sh (21) 3505 437 sh (21) 4480 437 sh (1) 5812 437 sh (22) 6656 437 sh (22) 7649 437 sh (2) 9017 437 sh (32) 354 1138 sh (32) 1344 1138 sh (2) 2712 1138 sh (13) 3556 1138 sh (13) 4521 1138 sh (3) 5876 1138 sh (33) 6725 1138 sh (33) 7705 1138 sh (3) 9060 1138 sh (11) 147 1842 sh (11) 1100 1842 sh (1) 2545 1842 sh (22) 3408 1842 sh (22) 4397 1842 sh (2) 5878 1842 sh (33) 6756 1842 sh (33) 7732 1842 sh (3) 9200 1842 sh 384 /Symbol f1 (=) 728 341 sh (=) 3861 341 sh (=) 7030 341 sh (=) 728 1042 sh (=) 3920 1042 sh (=) 7089 1042 sh (=) 503 1746 sh (=) 3782 1746 sh (=) 7120 1746 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 1229 341 sh (var\() 1881 341 sh (\)) 2799 341 sh (\251) 4362 341 sh (var\() 5032 341 sh (\)) 5950 341 sh (\251) 7531 341 sh (var\() 8219 341 sh (\)) 9173 341 sh (\251) 1229 1042 sh (var\() 1914 1042 sh (\)) 2868 1042 sh (\251) 4421 1042 sh (var\() 5081 1042 sh (\)) 6022 1042 sh (\251) 7590 1042 sh (var\() 8265 1042 sh (\)) 9206 1042 sh (\251) 1000 1746 sh (var\() 1652 1746 sh (\251) 2445 1746 sh (\)) 2718 1746 sh (\251) 4279 1746 sh (var\() 4967 1746 sh (\251) 5760 1746 sh (\)) 6069 1746 sh (\251) 7617 1746 sh (var\() 8292 1746 sh (\251) 9085 1746 sh (\)) 9381 1746 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 2270 1746 sh (e) 5585 1746 sh (e) 8910 1746 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 573 :M f0_12 sf -.051(These now form a solution to \(2\) and \(3\), which is a contradiction. )A f1_12 sf <5C>S 59 603 :M f2_12 sf .043 .004(Lemma 1:)J f0_12 sf ( If )S f1_12 sf (S)S f0_12 sf .023 .002( is a positive definite matrix, )J 274 603 :M -.165(then )A 298 603 :M -.196(there )A 325 603 :M -.055(exists )A 356 603 :M -.326(a )A 365 603 :M -.124(positive )A 406 603 :M -.206(definite )A 445 603 :M -.22(matrix )A 479 603 :M f1_12 sf .579(S)A f0_12 sf S 59 621 :M (= )S f1_12 sf (S)S f0_12 sf ( - )S f1_12 sf (d)S f0_12 sf (I, where )S f1_12 sf (d)S f0_12 sf ( is a real positive number.)S 59 651 :M f2_12 sf (Proof)S f0_12 sf .016 .002(. Suppose that )J f1_12 sf (S)S f0_12 sf .016 .002( is a positive definite matrix. It follows then )J 380 651 :M -.249(that )A 401 651 :M (for )S 419 651 :M -.331(all )A 434 651 :M -.037(solutions )A 481 651 :M (of)S 59 669 :M .128(det\()A f1_12 sf .195(S)A f0_12 sf .119 .012( - )J f1_12 sf .181(l)A f0_12 sf .237 .024(I\) = 0, )J f1_12 sf .181(l)A f0_12 sf .082 .008( )J 147 669 :M (is )S 159 669 :M (positive. )S 204 669 :M -.33(Let )A 223 669 :M -.22(the )A 241 669 :M -.165(smallest )A 283 669 :M -.084(solution )A 325 669 :M (of )S 339 669 :M .04(det\()A f1_12 sf .061(S)A f0_12 sf ( )S 369 669 :M (- )S 377 669 :M f1_12 sf .16(l)A f0_12 sf .223 .022(I\) )J 396 669 :M .211 .021(= )J 407 669 :M (0 )S 417 669 :M -.163(be )A 432 669 :M f1_12 sf .784(l)A f0_7 sf 0 3 rm .416(1)A 0 -3 rm f0_12 sf .649 .065(. )J 451 669 :M -.33(Let )A 470 669 :M f1_12 sf (d)S f0_12 sf ( )S 480 669 :M -.326(be)A endp %%Page: 5 5 %%BeginPageSetup initializepage (peter; page: 5 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (5)S gR gS 0 0 552 730 rC 59 55 :M f0_12 sf .037 .004(less than )J f1_12 sf (l)S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .034 .003( and greater than 0. Let )J f1_12 sf (S)S f0_12 sf (\325 = )S f1_12 sf (S)S f0_12 sf ( - )S f1_12 sf (d)S f0_12 sf .037 .004(I. We will now show )J 380 55 :M -.249(that )A 401 55 :M -.331(all )A 416 55 :M (of )S 430 55 :M -.22(the )A 448 55 :M -.042(solutions)A 59 73 :M .245 .024(of det\()J f1_12 sf .093(S)A f0_12 sf .077 .008(\325 - )J f1_12 sf .087(l)A f0_12 sf .174 .017(\325I\) = 0 are positive. )J f1_12 sf .093(S)A f0_12 sf .077 .008(\325 - )J f1_12 sf .087(l)A f0_12 sf .109 .011(\325I = )J f1_12 sf .093(S)A f0_12 sf .057 .006( - )J f1_12 sf .078(d)A f0_12 sf .077 .008(I - )J f1_12 sf .087(l)A f0_12 sf .114 .011(\325I = )J 331 73 :M f1_12 sf .632(S)A f0_12 sf .267 .027( )J 343 73 :M (- )S 351 73 :M .097<28>A f1_12 sf .16(l)A f0_12 sf .155 .015J 370 73 :M .211 .021(+ )J 381 73 :M f1_12 sf .323(d)A f0_12 sf .587 .059(\)I. )J 403 73 :M (If )S 415 73 :M (we )S 433 73 :M -.109(set )A 450 73 :M f1_12 sf .204(l)A f0_12 sf .197 .02J 465 73 :M .211 .021(= )J 476 73 :M f1_12 sf .285(l)A f0_12 sf .13 .013( )J 487 73 :M (-)S 59 91 :M f1_12 sf (d)S f0_12 sf .095 .009(, then for )J 113 91 :M -.245(each )A 138 91 :M -.084(solution )A 180 91 :M (of )S 194 91 :M .04(det\()A f1_12 sf .061(S)A f0_12 sf ( )S 224 91 :M (- )S 232 91 :M f1_12 sf .16(l)A f0_12 sf .223 .022(I\) )J 251 91 :M .211 .021(= )J 262 91 :M .833 .083(0, )J 276 91 :M -.196(there )A 303 91 :M (is )S 315 91 :M -.326(a )A 324 91 :M -.084(solution )A 366 91 :M (of )S 380 91 :M .04(det\()A f1_12 sf .061(S)A f0_12 sf ( )S 410 91 :M (- )S 418 91 :M .097<28>A f1_12 sf .16(l)A f0_12 sf .155 .015J 437 91 :M .211 .021(+ )J 448 91 :M f1_12 sf (d)S f0_12 sf .052 .005(\)I\) )J 470 91 :M .211 .021(= )J 481 91 :M 1(0.)A 59 109 :M .253 .025(Since )J f1_12 sf .09(l)A f0_12 sf .095 .01(\325 = )J f1_12 sf .09(l)A f0_12 sf .059 .006( - )J f1_12 sf .081(d)A f0_12 sf .138 .014(, and )J f1_12 sf .081(d)A f0_12 sf .148 .015( is less than )J f1_12 sf .09(l)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .074 .007(, )J 245 109 :M -.22(the )A 263 109 :M -.165(smallest )A 305 109 :M -.084(solution )A 347 109 :M (of )S 361 109 :M .036(det\()A f1_12 sf .055(S)A f0_12 sf .049 .005J 395 109 :M (- )S 403 109 :M f1_12 sf .133(l)A f0_12 sf .232 .023J 426 109 :M .211 .021(= )J 437 109 :M (0 )S 447 109 :M (is )S 459 109 :M -.216(greater)A 59 127 :M (than 0. )S f1_12 sf <5C>S 59 158 :M f0_12 sf -.663(A )A 71 158 :M -.219(linear )A 101 158 :M -.117(transformation )A 174 158 :M (of )S 188 158 :M -.326(a )A 197 158 :M -.109(set )A 214 158 :M (of )S 228 158 :M -.109(random )A 268 158 :M -.145(variables )A 314 158 :M (is )S 326 158 :M f2_12 sf 2.9 .29(lower )J 364 158 :M .222(triangular)A f0_12 sf .126 .013( )J 424 158 :M -.164(if )A 436 158 :M -.109(and )A 458 158 :M -.083(only )A 484 158 :M -.327(if)A 59 176 :M -.085(there is an ordering of )A 167 176 :M -.22(the )A 185 176 :M -.145(variables )A 231 176 :M (such )S 257 176 :M -.249(that )A 278 176 :M -.22(the )A 296 176 :M -.22(matrix )A 330 176 :M -.108(representing )A 392 176 :M -.22(the )A 410 176 :M -.117(transformation )A 483 176 :M (is)S 59 194 :M -.035(zero for all entries a)A f0_7 sf 0 3 rm (ij)S 0 -3 rm f0_12 sf -.038(, when j > i.)A 59 223 :M f2_12 sf 1.122 .112(Lemma )J 104 223 :M .387(2:)A f0_12 sf .232 .023( )J 119 223 :M (If )S 131 223 :M .876(X)A f0_7 sf 0 3 rm .354(1)A 0 -3 rm f0_12 sf .552 .055(, )J 152 223 :M f1_12 sf .667A f0_12 sf .303 .03(, )J 172 223 :M .478(X)A f0_7 sf 0 3 rm .193(n)A 0 -3 rm f0_12 sf .166 .017( )J 190 223 :M -.163(have )A 217 223 :M -.326(a )A 227 223 :M -.2(joint )A 253 223 :M -.165(normal )A 291 223 :M -.111(distribution )A 350 223 :M .454(N\(0,)A f1_12 sf .595(S)A f0_12 sf .698 .07(\), )J 394 223 :M -.062(where )A 428 223 :M f1_12 sf .632(S)A f0_12 sf .267 .027( )J 441 223 :M (is )S 454 223 :M -.142(positive)A 59 241 :M -.082(definite, then there is a set of n mututally independent standard normal variables T)A f0_7 sf 0 3 rm -.061(1)A 0 -3 rm f0_12 sf -.052(, )A f1_12 sf -.208A f0_12 sf -.104(, )A 476 241 :M .62(T)A f0_7 sf 0 3 rm .296(n)A 0 -3 rm f0_12 sf (,)S 59 259 :M -.049(such that X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.03(, )A f1_12 sf -.119A f0_12 sf -.049(, X)A f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf -.047( are a lower triangular linear transformation of )A 377 259 :M .51(T)A f0_7 sf 0 3 rm .243(1)A 0 -3 rm f0_12 sf .379 .038(, )J 396 259 :M f1_12 sf .667A f0_12 sf .303 .03(, )J 416 259 :M .09(T)A f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf ( )S 431 259 :M -.109(and )A 452 259 :M (for )S 470 259 :M -.326(each)A 59 278 :M -.083(i, the coefficient of T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.088( in the equation for X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.087( is not equal to zero.)A 59 307 :M f2_12 sf .494(Proof)A f0_12 sf .886 .089(. For )J 119 307 :M -.129(every )A 149 307 :M -.124(positive )A 190 307 :M -.206(definite )A 229 307 :M -.179(correlation )A 283 307 :M -.22(matrix )A 317 307 :M f1_12 sf 1.029(S)A f0_12 sf .79 .079(, )J 333 307 :M -.326(a )A 342 307 :M -.248(complete )A 388 307 :M -.205(directed )A 429 307 :M -.064(graph )A 460 307 :M -.217(can )A 480 307 :M -.326(be)A 59 325 :M -.124(given a linear parameterization that represents )A f1_12 sf -.189(S)A f0_12 sf -.122( \(Spirtes et )A 339 325 :M .261 .026(al. )J 355 325 :M .629 .063(1993\). )J 391 325 :M -.219(The )A 413 325 :M -.139(reduced )A 454 325 :M -.08(form )A 481 325 :M (of)S 59 344 :M -.326(a )A 71 344 :M -.248(complete )A 120 344 :M -.205(directed )A 164 344 :M -.064(graph )A 198 344 :M (is )S 213 344 :M -.326(a )A 225 344 :M -.063(lower )A 259 344 :M -.164(triangular )A 311 344 :M -.117(transformation )A 387 344 :M (of )S 404 344 :M -.15(independent )A 468 344 :M -.076(error)A 59 361 :M -.078(variables that is non-zero on the diagonal, because )A f1_12 sf -.118(S)A f0_12 sf -.071( is positive definite. )A f1_12 sf <5C>S 59 392 :M f2_12 sf .327 .033(Theorem 2:)J f0_12 sf .064 .006( If )J f4_12 sf .095(G)A f0_12 sf .141 .014( is the graph of SEM L with )J 282 392 :M -.122(measured )A 331 392 :M -.145(variables )A 377 392 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(, )J 395 392 :M -.165(normally )A 441 392 :M -.132(distributed)A 59 409 :M -.196(correlated )A 109 409 :M .604 .06(errors, )J 145 409 :M -.109(and )A 166 409 :M -.206(marginal )A 211 409 :M -.179(correlation )A 265 409 :M -.22(matrix )A 299 409 :M f1_12 sf .776(S)A f0_12 sf .436<28>A f2_12 sf .946(X)A f0_12 sf .909 .091(\), )J 333 409 :M 1.213 .121({X, )J 356 409 :M -.211(Y} )A 374 409 :M f1_12 sf -.161A f0_12 sf ( )S 387 409 :M f2_12 sf (Z)S f0_12 sf ( )S 399 409 :M f1_12 sf .33A f0_12 sf .116 .012( )J 413 409 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(, )J 432 409 :M -.109(and )A 454 409 :M .306 .031(X )J 468 409 :M (is )S 481 409 :M (d-)S 59 427 :M .047 .005(separated from Y given )J f2_12 sf (Z)S f0_12 sf .052 .005( in Transform\()J f4_12 sf (G)S f0_12 sf .03 .003(\), then )J f1_12 sf (r)S f0_7 sf 0 3 rm (XY.)S 0 -3 rm f2_7 sf 0 3 rm (Z)S 0 -3 rm f0_12 sf .018 .002( = 0 in )J f1_12 sf (S)S f0_12 sf <28>S f2_12 sf (X)S f0_12 sf (\).)S 59 457 :M f2_12 sf .719(Proof)A f0_12 sf .685 .068(. )J 99 457 :M (First )S 125 457 :M (we )S 143 457 :M -.166(will )A 165 457 :M -.109(construct )A 212 457 :M -.326(a )A 221 457 :M -.276(latent )A 250 457 :M -.205(variable )A 291 457 :M -.199(model )A 324 457 :M (of )S 338 457 :M f1_12 sf .884(e)A f0_7 sf 0 3 rm .587(1)A 0 -3 rm f0_12 sf .503(,...,)A f1_12 sf .884(e)A f0_7 sf 0 3 rm .587(n)A 0 -3 rm f0_12 sf .915 .092(. )J 384 457 :M -.164(Then )A 412 457 :M (we )S 430 457 :M -.166(will )A 453 457 :M (use )S 474 457 :M -.112(this)A 59 476 :M -.199(model )A 93 476 :M -.167(to )A 107 476 :M -.08(form )A 135 476 :M -.22(the )A 154 476 :M -.276(latent )A 184 476 :M -.205(variable )A 226 476 :M -.199(model )A 260 476 :M -.162<4CD520>A 276 476 :M -.083(with )A 302 476 :M -.064(graph )A 334 476 :M f4_12 sf .19(G)A f0_12 sf .139 .014J 352 476 :M -.249(that )A 374 476 :M (has )S 395 476 :M -.206(marginal )A 441 476 :M -.197(correlation)A 59 493 :M -.049(matrix )A f1_12 sf -.071(S)A f0_12 sf <28>S f2_12 sf -.086(X)A f0_12 sf -.048(\) but no correlated errors, and in which X is d-separated from Y given )A 449 493 :M f2_12 sf (Z)S f0_12 sf ( )S 461 493 :M -.167(in )A 474 493 :M f4_12 sf .744(G)A f0_12 sf .6A 59 511 :M .27 .027(It follows that )J f1_12 sf .116(r)A f0_7 sf 0 3 rm .07(XY.)A 0 -3 rm f2_7 sf 0 3 rm .082(Z)A 0 -3 rm f0_12 sf .125 .013( = 0 in )J f1_12 sf .125(S)A f0_12 sf .07<28>A f2_12 sf .153(X)A f0_12 sf .123(\).)A 59 542 :M -.195(Order )A 91 542 :M -.22(the )A 110 542 :M -.145(variables )A 157 542 :M .277 .028(so )J 173 542 :M -.249(that )A 195 542 :M .306 .031(X )J 209 542 :M (is )S 223 542 :M .424 .042(first, )J 252 542 :M -.663(Y )A 266 542 :M (is )S 280 542 :M .401 .04(second, )J 323 542 :M -.081(followed )A 371 542 :M (by )S 389 542 :M -.245(each )A 416 542 :M -.205(variable )A 459 542 :M -.083(with )A 486 542 :M (a)S 59 560 :M -.131(descendant )A 115 560 :M -.167(in )A 128 560 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 144 560 :M -.081(followed )A 190 560 :M (by )S 206 560 :M -.109(any )A 227 560 :M -.183(remaining )A 278 560 :M -.145(variables )A 324 560 :M -.249(that )A 346 560 :M -.163(have )A 373 560 :M .306 .031(X )J 387 560 :M (or )S 402 560 :M -.663(Y )A 415 560 :M (as )S 430 560 :M (descendants,)S 59 578 :M -.081(followed )A 106 578 :M (by )S 124 578 :M -.22(the )A 144 578 :M -.08(rest )A 167 578 :M (of )S 183 578 :M -.22(the )A 203 578 :M -.031(variables. )A 255 578 :M -.065(Given )A 290 578 :M -.084(this )A 313 578 :M .186 .019(ordering, )J 363 578 :M (we )S 383 578 :M -.166(will )A 407 578 :M .259 .026(now )J 434 578 :M -.126(refer )A 462 578 :M -.167(to )A 477 578 :M -.33(the)A 59 596 :M .195 .02(variables as X)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .039(,...,X)A f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf .123 .012(, where for all i, X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .088 .009( is the i)J f0_7 sf 0 -5 rm .027(th)A 0 5 rm f0_12 sf .156 .016( variable in the ordering. )J 414 596 :M .197 .02(Suppose )J 459 596 :M (for )S 477 596 :M -.33(the)A 59 613 :M -.094(graph in Figure 1 we are interested in )A 239 613 :M -.092(whether )A 281 613 :M f1_12 sf .103(r)A f0_7 sf 0 3 rm .079(XY)A 0 -3 rm f0_12 sf ( )S 302 613 :M .211 .021(= )J 313 613 :M (0 )S 323 613 :M .898 .09(\(i.e. )J 347 613 :M f2_12 sf (Z)S f0_12 sf ( )S 359 613 :M .211 .021(= )J 370 613 :M f1_12 sf .562A f0_12 sf .474 .047(\). )J 392 613 :M -.33(One )A 415 613 :M -.164(renaming )A 463 613 :M (of )S 477 613 :M -.33(the)A 59 632 :M -.115(variables for the graph in Figure 1 that is compatible with the )A 350 632 :M -.081(ordering )A 394 632 :M -.064(rules )A 421 632 :M -.132(given )A 451 632 :M -.13(above )A 483 632 :M (is)S 59 650 :M .215 .022(shown in Figure 3.)J endp %%Page: 6 6 %%BeginPageSetup initializepage (peter; page: 6 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (6)S gR gS 126 42 297 100 rC 126 55 :M f1_12 sf (e)S 131 58 :M f0_7 sf (X)S 136 58 :M (1)S 140 55 :M f0_12 sf ( )S 143 55 :M ( )S 146 55 :M ( )S 149 55 :M ( )S 152 55 :M ( )S 155 55 :M ( )S 158 55 :M ( )S 161 55 :M ( )S 164 55 :M ( )S 167 55 :M ( )S 170 55 :M ( )S 173 55 :M ( )S 176 55 :M ( )S 179 55 :M f1_12 sf (e)S 184 58 :M f0_7 sf (X)S 189 58 :M (3)S 193 55 :M f0_12 sf ( )S 196 55 :M ( )S 199 55 :M ( )S 202 55 :M ( )S 205 55 :M ( )S 208 55 :M ( )S 211 55 :M ( )S 214 55 :M ( )S 217 55 :M ( )S 220 55 :M ( )S 223 55 :M ( )S 226 55 :M ( )S 229 55 :M f1_12 sf (e)S 234 58 :M f0_7 sf (X)S 239 58 :M (5)S 243 55 :M f0_12 sf ( )S 246 55 :M ( )S 249 55 :M ( )S 252 55 :M ( )S 255 55 :M ( )S 258 55 :M ( )S 261 55 :M ( )S 264 55 :M ( )S 267 55 :M ( )S 270 55 :M ( )S 273 55 :M ( )S 276 55 :M ( )S 279 55 :M ( )S 282 55 :M f1_12 sf (e)S 287 58 :M f0_7 sf (X)S 292 58 :M (4)S 296 55 :M f0_12 sf ( )S 299 55 :M ( )S 302 55 :M ( )S 305 55 :M ( )S 308 55 :M ( )S 311 55 :M ( )S 314 55 :M ( )S 317 55 :M ( )S 320 55 :M ( )S 323 55 :M ( )S 326 55 :M ( )S 329 55 :M ( )S 332 55 :M ( )S 335 55 :M f1_12 sf (e)S 340 58 :M f0_7 sf (X)S 345 58 :M (6)S 349 55 :M f0_12 sf ( )S 352 55 :M ( )S 355 55 :M ( )S 358 55 :M ( )S 361 55 :M ( )S 364 55 :M ( )S 367 55 :M ( )S 370 55 :M ( )S 373 55 :M ( )S 376 55 :M ( )S 379 55 :M ( )S 382 55 :M ( )S 385 55 :M ( )S 388 55 :M ( )S 391 55 :M ( )S 394 55 :M f1_12 sf (e)S 399 58 :M f0_7 sf (X)S 404 58 :M (2)S 126 92 :M f0_12 sf (X)S 135 95 :M f0_7 sf (1)S 139 92 :M f0_12 sf ( )S 142 92 :M ( )S 145 92 :M ( )S 148 92 :M ( )S 151 92 :M ( )S 154 92 :M ( )S 157 92 :M ( )S 160 92 :M ( )S 163 92 :M ( )S 166 92 :M ( )S 169 92 :M ( )S 172 92 :M ( )S 175 92 :M ( )S 178 92 :M ( )S 181 92 :M (X)S 190 95 :M f0_7 sf (3)S 194 92 :M f0_12 sf ( )S 197 92 :M ( )S 200 92 :M ( )S 203 92 :M ( )S 206 92 :M ( )S 209 92 :M ( )S 212 92 :M ( )S 215 92 :M ( )S 218 92 :M ( )S 221 92 :M ( )S 224 92 :M ( )S 227 92 :M (X)S 236 95 :M f0_7 sf (5)S 240 92 :M f0_12 sf ( )S 243 92 :M ( )S 246 92 :M ( )S 249 92 :M ( )S 252 92 :M ( )S 255 92 :M ( )S 258 92 :M ( )S 261 92 :M ( )S 264 92 :M ( )S 267 92 :M ( )S 270 92 :M ( )S 273 92 :M (X)S 282 95 :M f0_7 sf (4)S 286 92 :M f0_12 sf ( )S 289 92 :M ( )S 292 92 :M ( )S 295 92 :M ( )S 298 92 :M ( )S 301 92 :M ( )S 304 92 :M ( )S 307 92 :M ( )S 310 92 :M ( )S 313 92 :M ( )S 316 92 :M ( )S 319 92 :M ( )S 322 92 :M ( )S 325 92 :M ( )S 328 92 :M ( )S 331 92 :M ( )S 334 92 :M (X)S 343 95 :M f0_7 sf (6)S 347 92 :M f0_12 sf ( )S 350 92 :M ( )S 353 92 :M ( )S 356 92 :M ( )S 359 92 :M ( )S 362 92 :M ( )S 365 92 :M ( )S 368 92 :M ( )S 371 92 :M ( )S 374 92 :M ( )S 377 92 :M ( )S 380 92 :M ( )S 383 92 :M ( )S 386 92 :M ( )S 389 92 :M ( )S 392 92 :M ( )S 395 92 :M (X)S 404 95 :M f0_7 sf (2)S 126 140 :M f0_12 sf ( )S 129 140 :M ( )S 132 140 :M ( )S 135 140 :M ( )S 138 140 :M ( )S 141 140 :M ( )S 144 140 :M ( )S 147 140 :M ( )S 150 140 :M ( )S 153 140 :M ( )S 156 140 :M ( )S 159 140 :M ( )S 162 140 :M ( )S 165 140 :M ( )S 168 140 :M ( )S 171 140 :M ( )S 174 140 :M ( )S 177 140 :M ( )S 180 140 :M ( )S 183 140 :M ( )S 186 140 :M ( )S 189 140 :M ( )S 192 140 :M ( )S 195 140 :M ( )S 198 140 :M ( )S 201 140 :M ( )S 204 140 :M ( )S 207 140 :M ( )S 210 140 :M ( )S 213 140 :M ( )S 216 140 :M ( )S 219 140 :M ( )S 222 140 :M ( )S 225 140 :M ( )S 228 140 :M ( )S 231 140 :M ( )S 234 140 :M ( )S 237 140 :M ( )S 240 140 :M ( )S 243 140 :M ( )S 246 140 :M ( )S 249 140 :M ( )S 252 140 :M ( )S 255 140 :M ( )S 258 140 :M ( )S 261 140 :M ( )S 264 140 :M ( )S 267 140 :M ( )S 270 140 :M f4_12 sf (G)S gR gS 125 41 299 102 rC 146 54.75 -.75 .75 168.75 54 .75 146 54 @a np 166 50 :M 166 58 :L 174 54 :L 166 50 :L .75 lw eofill -.75 -.75 166.75 58.75 .75 .75 166 50 @b -.75 -.75 166.75 58.75 .75 .75 174 54 @b 166 50.75 -.75 .75 174.75 54 .75 166 50 @a np 148 58 :M 148 50 :L 140 54 :L 148 58 :L eofill -.75 -.75 148.75 58.75 .75 .75 148 50 @b -.75 -.75 140.75 54.75 .75 .75 148 50 @b 140 54.75 -.75 .75 148.75 58 .75 140 54 @a 201 54.75 -.75 .75 223.75 54 .75 201 54 @a np 221 50 :M 221 58 :L 229 54 :L 221 50 :L eofill -.75 -.75 221.75 58.75 .75 .75 221 50 @b -.75 -.75 221.75 58.75 .75 .75 229 54 @b 221 50.75 -.75 .75 229.75 54 .75 221 50 @a np 203 58 :M 203 50 :L 195 54 :L 203 58 :L eofill -.75 -.75 203.75 58.75 .75 .75 203 50 @b -.75 -.75 195.75 54.75 .75 .75 203 50 @b 195 54.75 -.75 .75 203.75 58 .75 195 54 @a 250 54.75 -.75 .75 272.75 54 .75 250 54 @a np 270 50 :M 270 58 :L 278 54 :L 270 50 :L eofill -.75 -.75 270.75 58.75 .75 .75 270 50 @b -.75 -.75 270.75 58.75 .75 .75 278 54 @b 270 50.75 -.75 .75 278.75 54 .75 270 50 @a np 252 58 :M 252 50 :L 244 54 :L 252 58 :L eofill -.75 -.75 252.75 58.75 .75 .75 252 50 @b -.75 -.75 244.75 54.75 .75 .75 252 50 @b 244 54.75 -.75 .75 252.75 58 .75 244 54 @a 305 54.75 -.75 .75 327.75 54 .75 305 54 @a np 325 50 :M 325 58 :L 333 54 :L 325 50 :L eofill -.75 -.75 325.75 58.75 .75 .75 325 50 @b -.75 -.75 325.75 58.75 .75 .75 333 54 @b 325 50.75 -.75 .75 333.75 54 .75 325 50 @a np 307 58 :M 307 50 :L 299 54 :L 307 58 :L eofill -.75 -.75 307.75 58.75 .75 .75 307 50 @b -.75 -.75 299.75 54.75 .75 .75 307 50 @b 299 54.75 -.75 .75 307.75 58 .75 299 54 @a 361 54.75 -.75 .75 383.75 54 .75 361 54 @a np 381 50 :M 381 58 :L 389 54 :L 381 50 :L eofill -.75 -.75 381.75 58.75 .75 .75 381 50 @b -.75 -.75 381.75 58.75 .75 .75 389 54 @b 381 50.75 -.75 .75 389.75 54 .75 381 50 @a np 363 58 :M 363 50 :L 355 54 :L 363 58 :L eofill -.75 -.75 363.75 58.75 .75 .75 363 50 @b -.75 -.75 355.75 54.75 .75 .75 363 50 @b 355 54.75 -.75 .75 363.75 58 .75 355 54 @a -.75 -.75 131.75 75.75 .75 .75 131 64 @b np 135 73 :M 127 73 :L 131 81 :L 135 73 :L eofill 127 73.75 -.75 .75 135.75 73 .75 127 73 @a 127 73.75 -.75 .75 131.75 81 .75 127 73 @a -.75 -.75 131.75 81.75 .75 .75 135 73 @b -.75 -.75 181.75 74.75 .75 .75 181 63 @b np 185 72 :M 177 72 :L 181 80 :L 185 72 :L eofill 177 72.75 -.75 .75 185.75 72 .75 177 72 @a 177 72.75 -.75 .75 181.75 80 .75 177 72 @a -.75 -.75 181.75 80.75 .75 .75 185 72 @b -.75 -.75 235.75 73.75 .75 .75 235 62 @b np 239 71 :M 231 71 :L 235 79 :L 239 71 :L eofill 231 71.75 -.75 .75 239.75 71 .75 231 71 @a 231 71.75 -.75 .75 235.75 79 .75 231 71 @a -.75 -.75 235.75 79.75 .75 .75 239 71 @b -.75 -.75 285.75 73.75 .75 .75 285 62 @b np 289 71 :M 281 71 :L 285 79 :L 289 71 :L eofill 281 71.75 -.75 .75 289.75 71 .75 281 71 @a 281 71.75 -.75 .75 285.75 79 .75 281 71 @a -.75 -.75 285.75 79.75 .75 .75 289 71 @b -.75 -.75 341.75 74.75 .75 .75 341 63 @b np 345 72 :M 337 72 :L 341 80 :L 345 72 :L eofill 337 72.75 -.75 .75 345.75 72 .75 337 72 @a 337 72.75 -.75 .75 341.75 80 .75 337 72 @a -.75 -.75 341.75 80.75 .75 .75 345 72 @b -.75 -.75 395.75 73.75 .75 .75 395 62 @b np 399 71 :M 391 71 :L 395 79 :L 399 71 :L eofill 391 71.75 -.75 .75 399.75 71 .75 391 71 @a 391 71.75 -.75 .75 395.75 79 .75 391 71 @a -.75 -.75 395.75 79.75 .75 .75 399 71 @b 148 89.75 -.75 .75 171.75 89 .75 148 89 @a np 150 93 :M 150 85 :L 142 89 :L 150 93 :L eofill -.75 -.75 150.75 93.75 .75 .75 150 85 @b -.75 -.75 142.75 89.75 .75 .75 150 85 @b 142 89.75 -.75 .75 150.75 93 .75 142 89 @a 352 89.75 -.75 .75 380.75 89 .75 352 89 @a np 378 85 :M 378 93 :L 386 89 :L 378 85 :L eofill -.75 -.75 378.75 93.75 .75 .75 378 85 @b -.75 -.75 378.75 93.75 .75 .75 386 89 @b 378 85.75 -.75 .75 386.75 89 .75 378 85 @a -.75 -.75 129.75 106.75 .75 .75 129 104 @b np 125 106 :M 133 106 :L 129 98 :L 125 106 :L eofill 125 106.75 -.75 .75 133.75 106 .75 125 106 @a 129 98.75 -.75 .75 133.75 106 .75 129 98 @a -.75 -.75 125.75 106.75 .75 .75 129 98 @b -.75 -.75 141.75 106.75 .75 .75 141 104 @b np 137 106 :M 145 106 :L 141 98 :L 137 106 :L eofill 137 106.75 -.75 .75 145.75 106 .75 137 106 @a 141 98.75 -.75 .75 145.75 106 .75 141 98 @a -.75 -.75 137.75 106.75 .75 .75 141 98 @b 0 90 100 48 180.5 99.5 @n 90 180 94 36 188.5 105.5 @n 90 180 160 56 210.5 105.5 @n 0 90 148 70 209.5 98.5 @n gR gS 126 156 297 103 rC 126 169 :M f1_12 sf ( )S 129 169 :M ( )S 132 169 :M ( )S 135 169 :M ( )S 138 169 :M ( )S 141 169 :M ( )S 144 169 :M ( )S 147 169 :M ( )S 150 169 :M ( )S 153 169 :M ( )S 156 169 :M (T)S 164 172 :M f1_7 sf (1)S 168 172 :M (3)S 172 169 :M f1_12 sf ( )S 175 169 :M ( )S 178 169 :M ( )S 181 169 :M ( )S 184 169 :M ( )S 187 169 :M ( )S 190 169 :M ( )S 193 169 :M ( )S 196 169 :M ( )S 199 169 :M ( )S 202 169 :M ( )S 205 169 :M ( )S 208 169 :M (T)S 216 172 :M f1_7 sf (3)S 220 172 :M (5)S 224 169 :M f1_12 sf ( )S 227 169 :M ( )S 230 169 :M ( )S 233 169 :M ( )S 236 169 :M ( )S 239 169 :M ( )S 242 169 :M ( )S 245 169 :M ( )S 248 169 :M ( )S 251 169 :M (T)S 259 172 :M f1_7 sf (5)S 263 172 :M (4)S 267 169 :M f1_12 sf ( )S 270 169 :M ( )S 273 169 :M ( )S 276 169 :M ( )S 279 169 :M ( )S 282 169 :M ( )S 285 169 :M ( )S 288 169 :M ( )S 291 169 :M ( )S 294 169 :M ( )S 297 169 :M ( )S 300 169 :M ( )S 303 169 :M (T)S 311 172 :M f1_7 sf (4)S 315 172 :M (6)S 319 169 :M f1_12 sf ( )S 322 169 :M ( )S 325 169 :M ( )S 328 169 :M ( )S 331 169 :M ( )S 334 169 :M ( )S 337 169 :M ( )S 340 169 :M ( )S 343 169 :M ( )S 346 169 :M ( )S 349 169 :M ( )S 352 169 :M ( )S 355 169 :M (T)S 363 172 :M f1_7 sf (2)S 367 172 :M (6)S 126 206 :M f0_12 sf (X)S 135 209 :M f0_7 sf (1)S 139 206 :M f0_12 sf ( )S 142 206 :M ( )S 145 206 :M ( )S 148 206 :M ( )S 151 206 :M ( )S 154 206 :M ( )S 157 206 :M ( )S 160 206 :M ( )S 163 206 :M ( )S 166 206 :M ( )S 169 206 :M ( )S 172 206 :M ( )S 175 206 :M ( )S 178 206 :M ( )S 181 206 :M (X)S 190 209 :M f0_7 sf (3)S 194 206 :M f0_12 sf ( )S 197 206 :M ( )S 200 206 :M ( )S 203 206 :M ( )S 206 206 :M ( )S 209 206 :M ( )S 212 206 :M ( )S 215 206 :M ( )S 218 206 :M ( )S 221 206 :M ( )S 224 206 :M ( )S 227 206 :M (X)S 236 209 :M f0_7 sf (5)S 240 206 :M f0_12 sf ( )S 243 206 :M ( )S 246 206 :M ( )S 249 206 :M ( )S 252 206 :M ( )S 255 206 :M ( )S 258 206 :M ( )S 261 206 :M ( )S 264 206 :M ( )S 267 206 :M ( )S 270 206 :M ( )S 273 206 :M (X)S 282 209 :M f0_7 sf (4)S 286 206 :M f0_12 sf ( )S 289 206 :M ( )S 292 206 :M ( )S 295 206 :M ( )S 298 206 :M ( )S 301 206 :M ( )S 304 206 :M ( )S 307 206 :M ( )S 310 206 :M ( )S 313 206 :M ( )S 316 206 :M ( )S 319 206 :M ( )S 322 206 :M ( )S 325 206 :M ( )S 328 206 :M ( )S 331 206 :M ( )S 334 206 :M (X)S 343 209 :M f0_7 sf (6)S 347 206 :M f0_12 sf ( )S 350 206 :M ( )S 353 206 :M ( )S 356 206 :M ( )S 359 206 :M ( )S 362 206 :M ( )S 365 206 :M ( )S 368 206 :M ( )S 371 206 :M ( )S 374 206 :M ( )S 377 206 :M ( )S 380 206 :M ( )S 383 206 :M ( )S 386 206 :M ( )S 389 206 :M (X)S 398 209 :M f0_7 sf (2)S 126 254 :M f0_12 sf ( )S 129 254 :M ( )S 132 254 :M ( )S 135 254 :M ( )S 138 254 :M ( )S 141 254 :M ( )S 144 254 :M ( )S 147 254 :M ( )S 150 254 :M ( )S 153 254 :M ( )S 156 254 :M ( )S 159 254 :M ( )S 162 254 :M ( )S 165 254 :M ( )S 168 254 :M ( )S 171 254 :M ( )S 174 254 :M ( )S 177 254 :M ( )S 180 254 :M ( )S 183 254 :M ( )S 186 254 :M ( )S 189 254 :M ( )S 192 254 :M ( )S 195 254 :M ( )S 198 254 :M ( )S 201 254 :M ( )S 204 254 :M ( )S 207 254 :M ( )S 210 254 :M ( )S 213 254 :M ( )S 216 254 :M ( )S 219 254 :M ( )S 222 254 :M ( )S 225 254 :M ( )S 228 254 :M ( )S 231 254 :M ( )S 234 254 :M ( )S 237 254 :M ( )S 240 254 :M ( )S 243 254 :M ( )S 246 254 :M ( )S 249 254 :M ( )S 252 254 :M ( )S 255 254 :M ( )S 258 254 :M ( )S 261 254 :M (T)S 268 254 :M (r)S 272 254 :M (a)S 277 254 :M (n)S 283 254 :M (s)S 288 254 :M (f)S 292 254 :M (o)S 298 254 :M (r)S 302 254 :M (m)S 311 254 :M <28>S 315 254 :M f4_12 sf (G)S 324 254 :M f0_12 sf <29>S gR gS 125 155 299 105 rC 148 203.75 -.75 .75 171.75 203 .75 148 203 @a np 150 207 :M 150 199 :L 142 203 :L 150 207 :L .75 lw eofill -.75 -.75 150.75 207.75 .75 .75 150 199 @b -.75 -.75 142.75 203.75 .75 .75 150 199 @b 142 203.75 -.75 .75 150.75 207 .75 142 203 @a 352 203.75 -.75 .75 380.75 203 .75 352 203 @a np 378 199 :M 378 207 :L 386 203 :L 378 199 :L eofill -.75 -.75 378.75 207.75 .75 .75 378 199 @b -.75 -.75 378.75 207.75 .75 .75 386 203 @b 378 199.75 -.75 .75 386.75 203 .75 378 199 @a -.75 -.75 129.75 220.75 .75 .75 129 218 @b np 125 220 :M 133 220 :L 129 212 :L 125 220 :L eofill 125 220.75 -.75 .75 133.75 220 .75 125 220 @a 129 212.75 -.75 .75 133.75 220 .75 129 212 @a -.75 -.75 125.75 220.75 .75 .75 129 212 @b -.75 -.75 141.75 220.75 .75 .75 141 218 @b np 137 220 :M 145 220 :L 141 212 :L 137 220 :L eofill 137 220.75 -.75 .75 145.75 220 .75 137 220 @a 141 212.75 -.75 .75 145.75 220 .75 141 212 @a -.75 -.75 137.75 220.75 .75 .75 141 212 @b 0 90 100 48 180.5 213.5 @n 90 180 94 36 188.5 219.5 @n 90 180 160 56 210.5 219.5 @n 0 90 148 70 209.5 212.5 @n -.75 -.75 140.75 194.75 .75 .75 159 175 @b np 144 195 :M 139 190 :L 136 198 :L 144 195 :L eofill 139 190.75 -.75 .75 144.75 195 .75 139 190 @a -.75 -.75 136.75 198.75 .75 .75 139 190 @b -.75 -.75 136.75 198.75 .75 .75 144 195 @b 160 174.75 -.75 .75 178.75 191 .75 160 174 @a np 179 187 :M 174 192 :L 182 195 :L 179 187 :L eofill -.75 -.75 174.75 192.75 .75 .75 179 187 @b 174 192.75 -.75 .75 182.75 195 .75 174 192 @a 179 187.75 -.75 .75 182.75 195 .75 179 187 @a -.75 -.75 196.75 190.75 .75 .75 214 176 @b np 199 192 :M 195 186 :L 191 194 :L 199 192 :L eofill 195 186.75 -.75 .75 199.75 192 .75 195 186 @a -.75 -.75 191.75 194.75 .75 .75 195 186 @b -.75 -.75 191.75 194.75 .75 .75 199 192 @b 214 175.75 -.75 .75 228.75 190 .75 214 175 @a np 229 186 :M 224 191 :L 232 194 :L 229 186 :L eofill -.75 -.75 224.75 191.75 .75 .75 229 186 @b 224 191.75 -.75 .75 232.75 194 .75 224 191 @a 229 186.75 -.75 .75 232.75 194 .75 229 186 @a -.75 -.75 241.75 191.75 .75 .75 258 174 @b np 245 192 :M 240 187 :L 237 195 :L 245 192 :L eofill 240 187.75 -.75 .75 245.75 192 .75 240 187 @a -.75 -.75 237.75 195.75 .75 .75 240 187 @b -.75 -.75 237.75 195.75 .75 .75 245 192 @b 259 174.75 -.75 .75 276.75 192 .75 259 174 @a np 277 188 :M 272 193 :L 280 196 :L 277 188 :L eofill -.75 -.75 272.75 193.75 .75 .75 277 188 @b 272 193.75 -.75 .75 280.75 196 .75 272 193 @a 277 188.75 -.75 .75 280.75 196 .75 277 188 @a -.75 -.75 289.75 192.75 .75 .75 307 174 @b np 293 193 :M 288 188 :L 285 196 :L 293 193 :L eofill 288 188.75 -.75 .75 293.75 193 .75 288 188 @a -.75 -.75 285.75 196.75 .75 .75 288 188 @b -.75 -.75 285.75 196.75 .75 .75 293 193 @b 307 174.75 -.75 .75 326.75 193 .75 307 174 @a np 327 189 :M 322 194 :L 330 197 :L 327 189 :L eofill -.75 -.75 322.75 194.75 .75 .75 327 189 @b 322 194.75 -.75 .75 330.75 197 .75 322 194 @a 327 189.75 -.75 .75 330.75 197 .75 327 189 @a -.75 -.75 345.75 191.75 .75 .75 361 174 @b np 349 192 :M 344 187 :L 341 195 :L 349 192 :L eofill 344 187.75 -.75 .75 349.75 192 .75 344 187 @a -.75 -.75 341.75 195.75 .75 .75 344 187 @b -.75 -.75 341.75 195.75 .75 .75 349 192 @b 362 174.75 -.75 .75 381.75 193 .75 362 174 @a np 382 189 :M 377 194 :L 385 197 :L 382 189 :L eofill -.75 -.75 377.75 194.75 .75 .75 382 189 @b 377 194.75 -.75 .75 385.75 197 .75 377 194 @a 382 189.75 -.75 .75 385.75 197 .75 382 189 @a gR gS 0 0 552 730 rC 251 281 :M f2_12 sf 2.949 .295(Figure 3)J 59 316 :M f0_12 sf .197 .02(Suppose )J 104 316 :M -.249(that )A 125 316 :M -.22(the )A 143 316 :M -.179(correlation )A 197 316 :M -.22(matrix )A 231 316 :M -.132(among )A 267 316 :M -.22(the )A 285 316 :M -.061(error )A 312 316 :M -.131(terms )A 342 316 :M (of )S 357 316 :M -.33(L )A 369 316 :M (is )S 382 316 :M f1_12 sf 1.029(S)A f0_12 sf .79 .079(. )J 399 316 :M -.326(We )A 420 316 :M -.166(will )A 443 316 :M .479 .048(show )J 474 316 :M -.331(that)A 59 334 :M -.113(there is a latent variable model of )A f1_12 sf -.177(S)A f0_12 sf -.129( of the form)A 232 350 86 27 rC 318 377 :M psb currentpoint pse 232 350 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2752 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) -16 346 sh (e) 2181 346 sh 224 /Times-Italic f1 (i) 172 442 sh (ij) 1354 442 sh (j) 1706 442 sh (j) 784 719 sh (i) 1014 719 sh (i) 2588 442 sh 384 ns (a) 1171 346 sh (T) 1500 346 sh 384 /Symbol f1 (=) 386 346 sh (+) 1895 346 sh 224 ns (\243) 874 719 sh 576 ns (\345) 712 433 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 2356 346 sh (\251) 2473 346 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 403 :M f0_12 sf -.099(where each of the T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.092( and )A f1_12 sf -.104(e)A f0_12 sf -.079A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.097( are uncorrelated.)A 59 433 :M (By )S 77 433 :M .32 .032(hypothesis, )J 136 433 :M f1_12 sf .632(S)A f0_12 sf .267 .027( )J 148 433 :M (is )S 160 433 :M -.326(a )A 170 433 :M -.124(positive )A 212 433 :M -.206(definite )A 252 433 :M -.046(matrix. )A 291 433 :M (By )S 310 433 :M -.33(Lemma )A 350 433 :M .833 .083(1, )J 365 433 :M -.196(there )A 393 433 :M (is )S 406 433 :M -.326(a )A 416 433 :M -.109(set )A 434 433 :M (of )S 449 433 :M -.163(variables)A 59 451 :M f1_12 sf .089(e)A f0_12 sf .067A f0_7 sf 0 3 rm .059(1)A 0 -3 rm f0_12 sf .051(,...,)A f1_12 sf .089(e)A f0_12 sf .067A f0_7 sf 0 3 rm .059(n)A 0 -3 rm f0_12 sf .289 .029( with positive definite )J 211 451 :M -.22(matrix )A 245 451 :M f1_12 sf .456(S)A f0_12 sf .408 .041J 261 451 :M .211 .021(= )J 272 451 :M f1_12 sf .632(S)A f0_12 sf .267 .027( )J 284 451 :M (- )S 292 451 :M f1_12 sf .402(d)A f0_12 sf .565 .056(I. )J 310 451 :M -.165(As )A 327 451 :M -.326(a )A 336 451 :M -.064(first )A 359 451 :M -.082(step )A 382 451 :M -.167(to )A 395 451 :M -.109(constructing )A 457 451 :M -.326(a )A 466 451 :M -.331(latent)A 59 469 :M -.205(variable )A 101 469 :M -.199(model )A 135 469 :M (of )S 150 469 :M f1_12 sf .342(e)A f0_12 sf .354 .035(, )J 164 469 :M (we )S 183 469 :M -.166(will )A 206 469 :M -.109(construct )A 254 469 :M -.326(a )A 264 469 :M -.276(latent )A 295 469 :M -.205(variable )A 338 469 :M -.199(model )A 373 469 :M (of )S 389 469 :M f1_12 sf .255(e)A f0_12 sf .403 .04J 408 469 :M -.117(represented )A 468 469 :M (by )S 486 469 :M (a)S 59 487 :M -.069(directed graph )A f4_12 sf -.124(H)A f0_12 sf -.063(. Note that )A f4_12 sf -.124(H)A f0_12 sf -.067( does not contain any of the X variables or )A f1_12 sf -.075(e)A f0_12 sf -.07( variables.)A 59 517 :M (By )S 77 517 :M -.33(Lemma )A 116 517 :M .833 .083(2, )J 130 517 :M -.196(there )A 157 517 :M (is )S 169 517 :M -.326(a )A 178 517 :M -.109(set )A 195 517 :M (of )S 209 517 :M -.145(variables )A 255 517 :M .51(T)A f0_7 sf 0 3 rm .243(1)A 0 -3 rm f0_12 sf .379 .038(, )J 274 517 :M 2.857 .286(..., )J 294 517 :M .09(T)A f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf ( )S 309 517 :M (such )S 335 517 :M -.249(that )A 356 517 :M f1_12 sf .348(e)A f0_12 sf .264A f0_7 sf 0 3 rm .231(1)A 0 -3 rm f0_12 sf .36 .036(, )J 377 517 :M 2.857 .286(..., )J 397 517 :M f1_12 sf .08(e)A f0_12 sf .061A f0_7 sf 0 3 rm .053(n)A 0 -3 rm f0_12 sf ( )S 415 517 :M -.083(with )A 441 517 :M -.197(correlation)A 59 535 :M -.22(matrix )A 93 535 :M f1_12 sf .149(S)A f0_12 sf .084A f1_12 sf .111(e)A f0_12 sf .192 .019J 126 535 :M -.215(are )A 144 535 :M -.326(a )A 153 535 :M -.063(lower )A 184 535 :M -.164(triangular )A 233 535 :M -.219(linear )A 263 535 :M -.117(transformation )A 336 535 :M (of )S 350 535 :M 1.314(T)A f0_7 sf 0 3 rm .628(1)A 0 -3 rm f0_12 sf 2.152 .215(,..., )J 385 535 :M .09(T)A f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf ( )S 400 535 :M -.109(and )A 421 535 :M (for )S 439 535 :M -.245(each )A 465 535 :M .555 .055(i, )J 477 535 :M -.33(the)A 59 553 :M -.109(coefficient of T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.105( in the equation for )A f1_12 sf -.123(e)A f0_12 sf -.093A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.107( is not equal to zero. That is)A 244 569 61 27 rC 305 596 :M psb currentpoint pse 244 569 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1952 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) -16 346 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 159 346 sh 224 /Times-Italic f1 (i) 274 442 sh (ij) 1392 442 sh (j) 1744 442 sh (j) 822 719 sh (i) 1053 719 sh 384 ns (a) 1209 346 sh (T) 1538 346 sh 384 /Symbol f1 (=) 424 346 sh 224 ns (<) 913 719 sh 576 ns (\345) 750 433 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 622 :M f0_12 sf -.062(where )A 92 622 :M -.094(a)A f0_7 sf 0 3 rm -.034(ii)A 0 -3 rm f0_12 sf ( )S 105 622 :M f1_12 sf .285A f0_12 sf .13 .013( )J 116 622 :M .833 .083(0. )J 130 622 :M -.219(The )A 152 622 :M -.117(transformation )A 225 622 :M -.217(can )A 245 622 :M -.163(be )A 260 622 :M -.117(represented )A 318 622 :M (by )S 334 622 :M -.326(a )A 343 622 :M -.205(directed )A 384 622 :M -.064(graph )A 415 622 :M f4_12 sf .25(H)A f0_12 sf .087 .009( )J 429 622 :M -.167(in )A 443 622 :M -.065(which )A 477 622 :M (for)S 59 640 :M -.039(each i, there are edges from T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.032( to )A f1_12 sf (e)S f0_12 sf S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.033(, j )A cF f1_12 sf -.033A sf -.033( i.)A endp %%Page: 7 7 %%BeginPageSetup initializepage (peter; page: 7 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (7)S gR gS 0 0 552 730 rC 59 55 :M f0_12 sf -.009(From the construction of )A f4_12 sf (H)S f0_12 sf -.009(, there are no edges from T)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( to )S f1_12 sf (e)S f0_12 sf (')S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.009( unless j = 1. Hence, for every)A 59 73 :M (j )S f1_12 sf S f0_12 sf .022 .002(\3121, in )J f4_12 sf (H)S f0_12 sf .037 .004( every every trek between )J f1_12 sf (e)S f0_12 sf S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .02 .002( and )J f1_12 sf (e)S f0_12 sf S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .038 .004( contains T)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .029 .003(. It follows )J 396 73 :M -.249(that )A 417 73 :M -.196(there )A 444 73 :M (is )S 456 73 :M -.33(at )A 468 73 :M -.112(most)A 59 91 :M -.055(one trek between )A f1_12 sf -.058(e)A f0_12 sf S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.056( and)A f1_12 sf -.046( e)A f0_12 sf S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.057(. The edge from T)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.042( to )A f1_12 sf -.058(e)A f0_12 sf S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.051( is not )A 340 91 :M .236 .024(zero. )J 368 91 :M -.128(Hence )A 402 91 :M -.164(if )A 413 91 :M f1_12 sf .104(e)A f0_7 sf 0 3 rm .069(1)A 0 -3 rm f0_12 sf .059 .006( )J 426 91 :M -.093(and)A f1_12 sf ( )S 447 91 :M -.11(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( )S 458 91 :M -.215(are )A 476 91 :M -.167(not)A 59 110 :M -.068(correlated in L \(i.e. X)A f0_7 sf 0 3 rm -.053(1)A 0 -3 rm f0_12 sf -.077( and)A f1_12 sf ( )S f0_12 sf -.13(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.074( are not d-adjacent in Transform\(G\)\) then the edge from )A 467 110 :M .09(T)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( )S 482 110 :M -.334(to)A 59 127 :M f1_12 sf (e)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .006 .001( is zero. In the example from Figure 3, )J f4_12 sf (a)S f0_7 sf 0 3 rm (12)S 0 -3 rm f0_12 sf ( =)S f4_12 sf ( a)S f0_7 sf 0 3 rm (14)S 0 -3 rm f0_12 sf ( = )S f4_12 sf (a)S f0_7 sf 0 3 rm (15)S 0 -3 rm f0_12 sf ( = )S f4_12 sf (a)S f0_7 sf 0 3 rm (16)S 0 -3 rm f0_12 sf ( = 0.)S 59 158 :M -.122(Applying this strategy to each of the T)A f0_7 sf 0 3 rm -.05(i)A 0 -3 rm f0_12 sf ( )S 245 158 :M -.145(variables )A 291 158 :M -.167(in )A 304 158 :M .448 .045(turn, )J 331 158 :M (we )S 349 158 :M -.217(can )A 369 158 :M .259 .026(now )J 394 158 :M .479 .048(show )J 424 158 :M -.249(that )A 445 158 :M (for )S 463 158 :M -.245(each )A 488 158 :M (i)S 59 175 :M -.097(and r > i, if there is no trek between )A f1_12 sf -.114(e)A f0_12 sf -.087A f0_7 sf 0 3 rm -.051(r)A 0 -3 rm f0_12 sf -.101( and )A f1_12 sf -.114(e)A f0_12 sf -.087A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.105( containing a variable )A 380 175 :M .348(T)A f0_7 sf 0 3 rm .092(j)A 0 -3 rm f0_12 sf .259 .026(, )J 397 175 :M -.062(where )A 430 175 :M -.334(j )A 437 175 :M .211 .021(< )J 448 175 :M .555 .055(i, )J 459 175 :M -.109(and )A 480 175 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm 59 193 :M f0_12 sf -.04(is not d-adjacent to X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.042( in Transform\()A f4_12 sf -.073(G)A f0_12 sf -.03(\) \(i.e. )A f1_12 sf (e)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.049( and )A 301 193 :M f1_12 sf -.11(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( )S 312 193 :M -.215(are )A 330 193 :M -.163(uncorrelated )A 392 193 :M -.167(in )A 405 193 :M .52 .052(L\), )J 424 193 :M -.165(then )A 448 193 :M -.22(the )A 466 193 :M -.164(T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 479 193 :M f1_12 sf S 59 211 :M -.071(e)A f0_12 sf -.054A f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf -.065( edge can be removed from the graph \(i.e. a)A f0_7 sf 0 3 rm -.029(ir)A 0 -3 rm f0_12 sf -.065( can be set to zero.\) Suppose on the )A 452 211 :M -.139(contrary)A 59 229 :M -.087(that there is no trek between )A f1_12 sf -.099(e)A f0_12 sf -.075A f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf -.087( and )A f1_12 sf -.099(e)A f0_12 sf -.075A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.089( containing a variable T)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.087(, where j < i, and X)A f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf -.088( is )A 462 229 :M -.111(not )A 481 229 :M (d-)S 59 247 :M -.067(adjacent to X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.069( in Transform\()A f4_12 sf -.119(G)A f0_12 sf -.049(\) \(i.e. )A f1_12 sf -.072(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.064( and )A f1_12 sf -.072(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.063( are uncorrelated in L\), but the T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.162A f0_12 sf ( )S 442 247 :M f1_12 sf -.213(e)A f0_12 sf -.162A f0_7 sf 0 3 rm -.094(r)A 0 -3 rm f0_12 sf ( )S 457 247 :M -.163(edge )A 483 247 :M (is)S 59 266 :M -.111(not )A 80 266 :M -.14(removed )A 127 266 :M -.08(from )A 156 266 :M -.22(the )A 176 266 :M -.064(graph )A 209 266 :M (by )S 227 266 :M -.084(this )A 251 266 :M -.107(procedure )A 305 266 :M .898 .09(\(i.e. )J 332 266 :M f4_12 sf -.576(a)A f0_7 sf 0 3 rm -.205(ir)A 0 -3 rm f0_12 sf ( )S 348 266 :M (is )S 363 266 :M -.111(not )A 385 266 :M -.109(set )A 405 266 :M -.167(to )A 421 266 :M .226 .023(zero.\) )J 456 266 :M (By )S 477 266 :M -.33(the)A 59 283 :M .032 .003(construction of )J f4_12 sf (H)S f0_12 sf .018 .002(, if k > i, then there is no edge from T)J f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf ( to )S f1_12 sf (e)S f0_12 sf (')S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (. )S 359 283 :M -.164(It )A 370 283 :M (follows )S 410 283 :M -.249(that )A 431 283 :M -.164(if )A 442 283 :M -.167(in )A 455 283 :M f4_12 sf .25(H)A f0_12 sf .087 .009( )J 468 283 :M -.245(there)A 59 301 :M -.125(is no trek between )A f1_12 sf -.138(e)A f0_12 sf -.105A f0_7 sf 0 3 rm -.061(r)A 0 -3 rm f0_12 sf -.122( and )A f1_12 sf -.138(e)A f0_12 sf -.105A f0_7 sf 0 3 rm -.051(i)A 0 -3 rm f0_12 sf -.127( containing a variable )A 297 301 :M .348(T)A f0_7 sf 0 3 rm .092(j)A 0 -3 rm f0_12 sf .259 .026(, )J 314 301 :M -.062(where )A 347 301 :M -.334(j )A 354 301 :M .211 .021(< )J 365 301 :M .555 .055(i, )J 376 301 :M -.165(then )A 400 301 :M -.129(every )A 430 301 :M -.163(trek )A 452 301 :M -.163(between)A 59 319 :M f1_12 sf -.097(e)A f0_12 sf -.073A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.089( and any other variable contains the edge from T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.07( to )A f1_12 sf -.097(e)A f0_12 sf -.073A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.089(, which is not )A 394 319 :M -.197(equal )A 423 319 :M -.167(to )A 436 319 :M -.038(zero.The )A 482 319 :M -.217(T)A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 337 :M f1_12 sf .126A f0_12 sf ( )S 75 337 :M f1_12 sf -.213(e)A f0_12 sf -.162A f0_7 sf 0 3 rm -.094(r)A 0 -3 rm f0_12 sf ( )S 90 337 :M -.163(edge )A 117 337 :M -.055(exists )A 149 337 :M (by )S 166 337 :M .32 .032(hypothesis, )J 226 337 :M .277 .028(so )J 242 337 :M -.196(there )A 270 337 :M (is )S 283 337 :M -.235(exactly )A 321 337 :M -.109(one )A 343 337 :M -.163(trek )A 366 337 :M -.139(between )A 410 337 :M f1_12 sf -.076(e)A f0_12 sf -.058A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 426 337 :M -.109(and )A 448 337 :M f1_12 sf -.213(e)A f0_12 sf -.162A f0_7 sf 0 3 rm -.094(r)A 0 -3 rm f0_12 sf ( )S 464 337 :M -.167(in )A 478 337 :M f4_12 sf .993(H)A f0_12 sf (.)S 59 355 :M -.173(Hence )A f1_12 sf -.163(e)A f0_12 sf -.123A f0_7 sf 0 3 rm -.06(i)A 0 -3 rm f0_12 sf -.144( and )A f1_12 sf -.163(e)A f0_12 sf -.123A f0_7 sf 0 3 rm -.072(r)A 0 -3 rm f0_12 sf -.148( are correlated in every parameterization of )A 341 355 :M f4_12 sf .79(H)A f0_12 sf .497 .05(. )J 358 355 :M -.063(\(Note )A 389 355 :M -.249(that )A 410 355 :M -.084(this )A 431 355 :M -.132(could )A 461 355 :M -.111(not )A 480 355 :M -.326(be)A 59 373 :M -.283(claimed )A 99 373 :M -.164(if )A 110 373 :M -.196(there )A 137 373 :M -.077(were )A 164 373 :M -.163(more )A 192 373 :M -.165(than )A 216 373 :M -.109(one )A 238 373 :M -.163(trek )A 261 373 :M -.139(between )A 305 373 :M f1_12 sf -.076(e)A f0_12 sf -.058A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 321 373 :M -.109(and )A 343 373 :M f1_12 sf -.213(e)A f0_12 sf -.162A f0_7 sf 0 3 rm -.094(r)A 0 -3 rm f0_12 sf ( )S 359 373 :M -.131(since )A 388 373 :M -.167(in )A 402 373 :M -.249(that )A 424 373 :M -.161(case )A 449 373 :M -.22(the )A 468 373 :M -.08(treks)A 59 391 :M -.137(might cancel each other.\) Since the covariances between distinct )A f1_12 sf -.15(e)A f0_12 sf -.134(\325 variables are equal to the)A 59 409 :M -.094(correlations between the corresponding )A f1_12 sf -.101(e)A f0_12 sf -.083( variables, it follows that )A f1_12 sf -.101(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.111( and )A 403 409 :M f1_12 sf -.297(e)A f0_7 sf 0 3 rm -.131(r)A 0 -3 rm f0_12 sf ( )S 414 409 :M -.215(are )A 432 409 :M -.196(correlated )A 482 409 :M -.334(in)A 59 428 :M .52 .052(L\325, )J 78 428 :M -.109(and )A 99 428 :M -.196(hence )A 130 428 :M -.197(d-adjacent )A 182 428 :M -.167(in )A 195 428 :M .053(Transform\()A f4_12 sf .084(G)A f0_12 sf .081 .008(\). )J 270 428 :M -.083(This )A 295 428 :M (is )S 307 428 :M -.326(a )A 316 428 :M -.093(contradiction. )A 385 428 :M -.219(The )A 407 428 :M -.109(end )A 428 428 :M -.109(result )A 459 428 :M (of )S 474 428 :M -.112(this)A 59 446 :M -.036(process of edge removal for the graph in Figure 3 is shown in Figure 4.)A 92 462 366 93 rC 92 475 :M f1_12 sf (T)S 100 478 :M f1_7 sf (1)S 104 475 :M f1_12 sf ( )S 107 475 :M ( )S 110 475 :M ( )S 113 475 :M ( )S 116 475 :M ( )S 119 475 :M ( )S 122 475 :M ( )S 125 475 :M ( )S 128 475 :M ( )S 131 475 :M ( )S 134 475 :M ( )S 137 475 :M ( )S 140 475 :M ( )S 143 475 :M ( )S 146 475 :M ( )S 149 475 :M ( )S 152 475 :M ( )S 155 475 :M (T)S 163 478 :M f1_7 sf (3)S 167 475 :M f1_12 sf ( )S 170 475 :M ( )S 173 475 :M ( )S 176 475 :M ( )S 179 475 :M ( )S 182 475 :M ( )S 185 475 :M ( )S 188 475 :M ( )S 191 475 :M ( )S 194 475 :M ( )S 197 475 :M ( )S 200 475 :M ( )S 203 475 :M ( )S 206 475 :M ( )S 209 475 :M ( )S 212 475 :M ( )S 215 475 :M (T)S 223 478 :M f1_7 sf (5)S 227 475 :M f1_12 sf ( )S 230 475 :M ( )S 233 475 :M ( )S 236 475 :M ( )S 239 475 :M ( )S 242 475 :M ( )S 245 475 :M ( )S 248 475 :M ( )S 251 475 :M ( )S 254 475 :M ( )S 257 475 :M ( )S 260 475 :M ( )S 263 475 :M ( )S 266 475 :M ( )S 269 475 :M ( )S 272 475 :M (T)S 280 478 :M f1_7 sf (4)S 284 475 :M f1_12 sf ( )S 287 475 :M ( )S 290 475 :M ( )S 293 475 :M ( )S 296 475 :M ( )S 299 475 :M ( )S 302 475 :M ( )S 305 475 :M ( )S 308 475 :M ( )S 311 475 :M ( )S 314 475 :M ( )S 317 475 :M ( )S 320 475 :M ( )S 323 475 :M ( )S 326 475 :M ( )S 329 475 :M ( )S 332 475 :M ( )S 335 475 :M ( )S 338 475 :M (T)S 346 478 :M f1_7 sf (6)S 350 475 :M f1_12 sf ( )S 353 475 :M ( )S 356 475 :M ( )S 359 475 :M ( )S 362 475 :M ( )S 365 475 :M ( )S 368 475 :M ( )S 371 475 :M ( )S 374 475 :M ( )S 377 475 :M ( )S 380 475 :M ( )S 383 475 :M ( )S 386 475 :M ( )S 389 475 :M ( )S 392 475 :M ( )S 395 475 :M ( )S 398 475 :M ( )S 401 475 :M (T)S 409 478 :M f1_7 sf (2)S 92 543 :M f1_12 sf (e)S 97 543 :M f0_12 sf (')S 101 546 :M f0_7 sf (X)S 106 546 :M (1)S 110 543 :M f0_12 sf ( )S 113 543 :M ( )S 116 543 :M ( )S 119 543 :M ( )S 122 543 :M ( )S 125 543 :M ( )S 128 543 :M ( )S 131 543 :M ( )S 134 543 :M ( )S 137 543 :M ( )S 140 543 :M ( )S 143 543 :M ( )S 146 543 :M ( )S 149 543 :M ( )S 152 543 :M ( )S 155 543 :M f1_12 sf (e)S 160 543 :M f0_12 sf (')S 164 546 :M f0_7 sf (X)S 169 546 :M (3)S 173 543 :M f0_12 sf ( )S 176 543 :M ( )S 179 543 :M ( )S 182 543 :M ( )S 185 543 :M ( )S 188 543 :M ( )S 191 543 :M ( )S 194 543 :M ( )S 197 543 :M ( )S 200 543 :M ( )S 203 543 :M ( )S 206 543 :M ( )S 209 543 :M ( )S 212 543 :M f1_12 sf (e)S 217 543 :M f0_12 sf (')S 221 546 :M f0_7 sf (X)S 226 546 :M (5)S 230 543 :M f0_12 sf ( )S 233 543 :M ( )S 236 543 :M ( )S 239 543 :M ( )S 242 543 :M ( )S 245 543 :M ( )S 248 543 :M ( )S 251 543 :M ( )S 254 543 :M ( )S 257 543 :M ( )S 260 543 :M ( )S 263 543 :M ( )S 266 543 :M ( )S 269 543 :M ( )S 272 543 :M ( )S 275 543 :M f1_12 sf (e)S 280 543 :M f0_12 sf (')S 284 546 :M f0_7 sf (X)S 289 546 :M (4)S 293 543 :M f0_12 sf ( )S 296 543 :M ( )S 299 543 :M ( )S 302 543 :M ( )S 305 543 :M ( )S 308 543 :M ( )S 311 543 :M ( )S 314 543 :M ( )S 317 543 :M ( )S 320 543 :M ( )S 323 543 :M ( )S 326 543 :M ( )S 329 543 :M ( )S 332 543 :M ( )S 335 543 :M ( )S 338 543 :M f1_12 sf (e)S 343 543 :M f0_12 sf (')S 347 546 :M f0_7 sf (X)S 352 546 :M (6)S 356 543 :M f0_12 sf ( )S 359 543 :M ( )S 362 543 :M ( )S 365 543 :M ( )S 368 543 :M ( )S 371 543 :M ( )S 374 543 :M ( )S 377 543 :M ( )S 380 543 :M ( )S 383 543 :M ( )S 386 543 :M ( )S 389 543 :M ( )S 392 543 :M ( )S 395 543 :M ( )S 398 543 :M ( )S 401 543 :M f1_12 sf (e)S 406 543 :M f0_12 sf (')S 410 546 :M f0_7 sf (X)S 415 546 :M (2)S gR gS 91 461 368 94 rC -.75 -.75 98.75 526.75 .75 .75 98 482 @b np 102 524 :M 94 524 :L 98 532 :L 102 524 :L .75 lw eofill 94 524.75 -.75 .75 102.75 524 .75 94 524 @a 94 524.75 -.75 .75 98.75 532 .75 94 524 @a -.75 -.75 98.75 532.75 .75 .75 102 524 @b -.75 -.75 161.75 525.75 .75 .75 161 481 @b np 165 523 :M 157 523 :L 161 531 :L 165 523 :L eofill 157 523.75 -.75 .75 165.75 523 .75 157 523 @a 157 523.75 -.75 .75 161.75 531 .75 157 523 @a -.75 -.75 161.75 531.75 .75 .75 165 523 @b -.75 -.75 220.75 526.75 .75 .75 220 482 @b np 224 524 :M 216 524 :L 220 532 :L 224 524 :L eofill 216 524.75 -.75 .75 224.75 524 .75 216 524 @a 216 524.75 -.75 .75 220.75 532 .75 216 524 @a -.75 -.75 220.75 532.75 .75 .75 224 524 @b -.75 -.75 280.75 526.75 .75 .75 280 482 @b np 284 524 :M 276 524 :L 280 532 :L 284 524 :L eofill 276 524.75 -.75 .75 284.75 524 .75 276 524 @a 276 524.75 -.75 .75 280.75 532 .75 276 524 @a -.75 -.75 280.75 532.75 .75 .75 284 524 @b -.75 -.75 343.75 525.75 .75 .75 343 481 @b np 347 523 :M 339 523 :L 343 531 :L 347 523 :L eofill 339 523.75 -.75 .75 347.75 523 .75 339 523 @a 339 523.75 -.75 .75 343.75 531 .75 339 523 @a -.75 -.75 343.75 531.75 .75 .75 347 523 @b -.75 -.75 405.75 527.75 .75 .75 405 483 @b np 409 525 :M 401 525 :L 405 533 :L 409 525 :L eofill 401 525.75 -.75 .75 409.75 525 .75 401 525 @a 401 525.75 -.75 .75 405.75 533 .75 401 525 @a -.75 -.75 405.75 533.75 .75 .75 409 525 @b 98 482.75 -.75 .75 150.75 525 .75 98 482 @a np 151 521 :M 147 527 :L 155 529 :L 151 521 :L eofill -.75 -.75 147.75 527.75 .75 .75 151 521 @b 147 527.75 -.75 .75 155.75 529 .75 147 527 @a 151 521.75 -.75 .75 155.75 529 .75 151 521 @a -.75 -.75 355.75 528.75 .75 .75 406 482 @b np 359 529 :M 355 524 :L 351 532 :L 359 529 :L eofill 355 524.75 -.75 .75 359.75 529 .75 355 524 @a -.75 -.75 351.75 532.75 .75 .75 355 524 @b -.75 -.75 351.75 532.75 .75 .75 359 529 @b 161 480.75 -.75 .75 212.75 525 .75 161 480 @a np 212 521 :M 208 526 :L 216 529 :L 212 521 :L eofill -.75 -.75 208.75 526.75 .75 .75 212 521 @b 208 526.75 -.75 .75 216.75 529 .75 208 526 @a 212 521.75 -.75 .75 216.75 529 .75 212 521 @a -.75 -.75 232.75 525.75 .75 .75 281 480 @b np 236 526 :M 231 521 :L 228 529 :L 236 526 :L eofill 231 521.75 -.75 .75 236.75 526 .75 231 521 @a -.75 -.75 228.75 529.75 .75 .75 231 521 @b -.75 -.75 228.75 529.75 .75 .75 236 526 @b 282 479.75 -.75 .75 332.75 525 .75 282 479 @a np 333 521 :M 328 526 :L 336 529 :L 333 521 :L eofill -.75 -.75 328.75 526.75 .75 .75 333 521 @b 328 526.75 -.75 .75 336.75 529 .75 328 526 @a 333 521.75 -.75 .75 336.75 529 .75 333 521 @a 221 481.75 -.75 .75 327.75 530 .75 221 481 @a np 326 525 :M 323 532 :L 332 532 :L 326 525 :L eofill -.75 -.75 323.75 532.75 .75 .75 326 525 @b 323 532.75 -.75 .75 332.75 532 .75 323 532 @a 326 525.75 -.75 .75 332.75 532 .75 326 525 @a gR gS 0 0 552 730 rC 156 576 :M f2_12 sf 1.715 .171(Figure 4: )J f6_12 sf .961(H)A f2_12 sf 2.027 .203( after extra edges are removed)J 59 611 :M f0_12 sf (From )S 89 611 :M -.22(the )A 107 611 :M -.276(latent )A 136 611 :M -.205(variable )A 177 611 :M -.199(model )A 210 611 :M -.095(without )A 250 611 :M -.196(correlated )A 300 611 :M (errors )S 332 611 :M (of )S 346 611 :M -.22(the )A 365 611 :M f1_12 sf .795(e)A f0_12 sf .708 .071(' )J 379 611 :M -.031(variables, )A 430 611 :M (we )S 449 611 :M -.217(can )A 470 611 :M .169(now)A 59 629 :M -.139(form a latent variable model without correlated errors )A 313 629 :M (of )S 327 629 :M -.22(the )A 345 629 :M f1_12 sf -.17(e)A f0_12 sf ( )S 354 629 :M -.031(variables. )A 404 629 :M .258 .026(For )J 425 629 :M -.245(each )A 450 629 :M .555 .055(i, )J 461 629 :M -.331(let )A 476 629 :M f1_12 sf -.069(e)A f0_12 sf -.052A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 647 :M f0_12 sf -.121(be a normally distributed variable with variance )A f1_12 sf -.15(d)A f0_12 sf -.115( that is independent of all of the T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.152(, )A 459 647 :M -.109(and )A 480 647 :M -.497(all)A 59 665 :M -.08(of the other )A f1_12 sf -.094(e)A f0_12 sf -.082(\325\325 variables. It follows then that)A endp %%Page: 8 8 %%BeginPageSetup initializepage (peter; page: 8 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (8)S gR gS 232 41 86 27 rC 318 68 :M psb currentpoint pse 232 41 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2752 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) -16 346 sh (e) 2181 346 sh 224 /Times-Italic f1 (i) 172 442 sh (ij) 1354 442 sh (j) 1706 442 sh (j) 784 719 sh (i) 1014 719 sh (i) 2588 442 sh 384 ns (a) 1171 346 sh (T) 1500 346 sh 384 /Symbol f1 (=) 386 346 sh (+) 1895 346 sh 224 ns (\243) 874 719 sh 576 ns (\345) 712 433 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 2356 346 sh (\251) 2473 346 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 94 :M f0_12 sf -.112(because the addition of the )A f1_12 sf -.126(e)A f0_12 sf -.096A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.123( term does not change )A 309 94 :M -.109(any )A 330 94 :M (of )S 344 94 :M -.22(the )A 362 94 :M -.049(correlations, )A 425 94 :M -.109(and )A 446 94 :M (adds )S 472 94 :M f1_12 sf (d)S f0_12 sf ( )S 482 94 :M -.334(to)A 59 112 :M -.078(the variance of )A f1_12 sf -.089(e)A f0_12 sf -.067A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (.)S 59 142 :M (From )S 89 142 :M -.22(the )A 107 142 :M -.276(latent )A 136 142 :M -.205(variable )A 177 142 :M -.199(model )A 210 142 :M -.095(without )A 251 142 :M -.196(correlated )A 302 142 :M (errors )S 335 142 :M (of )S 350 142 :M -.22(the )A 369 142 :M f1_12 sf -.17(e)A f0_12 sf ( )S 379 142 :M -.031(variables, )A 430 142 :M (we )S 449 142 :M -.217(can )A 470 142 :M .169(now)A 59 160 :M -.08(form )A 87 160 :M -.326(a )A 98 160 :M -.276(latent )A 129 160 :M -.205(variable )A 172 160 :M -.199(model )A 207 160 :M -.095(without )A 249 160 :M -.196(correlated )A 301 160 :M (errors )S 335 160 :M (of )S 351 160 :M f1_12 sf .776(S)A f0_12 sf .436<28>A f2_12 sf .946(X)A f0_12 sf .909 .091(\). )J 387 160 :M (If )S 401 160 :M (we )S 421 160 :M (use )S 443 160 :M -.22(the )A 463 160 :M -.163(above)A 59 178 :M -.165(equation )A 103 178 :M -.167(to )A 117 178 :M -.233(replace )A 155 178 :M -.245(each )A 181 178 :M f1_12 sf -.11(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 193 178 :M -.167(in )A 207 178 :M -.22(the )A 226 178 :M -.223(SEM )A 255 178 :M .558 .056(L, )J 271 178 :M (we )S 290 178 :M -.08(form )A 318 178 :M -.326(a )A 328 178 :M -.223(SEM )A 357 178 :M -.162<4CD520>A 373 178 :M -.065(which )A 407 178 :M (has )S 428 178 :M (no )S 445 178 :M -.218(correlated)A 59 197 :M -.098(errors, but has the same marginal covariance matrix as L. If the equations in L are:)A 233 212 84 27 rC 317 239 :M psb currentpoint pse 233 212 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2688 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (X) 17 346 sh (b) 1245 346 sh (X) 1591 346 sh 224 ns (i) 250 442 sh (ij) 1416 442 sh (j) 1857 442 sh (j) 859 719 sh (i) 1095 719 sh (i) 2520 442 sh 384 /Symbol f1 (=) 464 346 sh (+) 2046 346 sh 224 ns (\271) 952 719 sh 576 ns (\345) 790 433 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 2332 346 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 266 :M f0_12 sf -.141(then the equations in L\325 are:)A 206 281 137 27 rC 343 308 :M psb currentpoint pse 206 281 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4384 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (X) 17 346 sh (b) 1245 346 sh (X) 1591 346 sh (a) 2799 346 sh (T) 3128 346 sh 224 ns (i) 250 442 sh (ij) 1416 442 sh (j) 1857 442 sh (j) 859 719 sh (i) 1095 719 sh (ij) 2982 442 sh (j) 3334 442 sh (j) 2412 719 sh (i) 2643 719 sh (i) 4216 442 sh 384 /Symbol f1 (=) 464 346 sh (+) 2046 346 sh (+) 3523 346 sh 224 ns (\271) 952 719 sh (<) 2503 719 sh 576 ns (\345) 790 433 sh (\345) 2340 433 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 3809 346 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 3984 346 sh (\251) 4101 346 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 335 :M f0_12 sf -.162<4CD520>A 75 335 :M (has )S 96 335 :M -.326(a )A 106 335 :M -.064(graph )A 138 335 :M f4_12 sf .19(G)A f0_12 sf .139 .014J 156 335 :M -.165(obtained )A 201 335 :M -.08(from )A 229 335 :M f4_12 sf .25(G)A f0_12 sf .087 .009( )J 243 335 :M -.109(and )A 265 335 :M f4_12 sf .25(H)A f0_12 sf .087 .009( )J 280 335 :M -.167(in )A 295 335 :M -.22(the )A 315 335 :M -.073(following )A 367 335 :M -.081(way: )A 396 335 :M -.165(Remove )A 441 335 :M -.245(each )A 468 335 :M -.076(error)A 59 353 :M -.151(variable from )A 126 353 :M f4_12 sf .25(G)A f0_12 sf .087 .009( )J 139 353 :M -.121(\(because )A 184 353 :M -.331(all )A 199 353 :M (of )S 213 353 :M -.22(the )A 231 353 :M -.061(error )A 258 353 :M -.131(terms )A 288 353 :M -.215(are )A 306 353 :M -.163(uncorrelated )A 368 353 :M -.167(in )A 381 353 :M .488 .049(L\325\), )J 404 353 :M -.109(add )A 425 353 :M -.245(each )A 450 353 :M (of )S 464 353 :M -.22(the )A 482 353 :M -.217(T)A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 370 :M f0_12 sf -.056(variables to )A f4_12 sf -.108(G)A f0_12 sf -.064(, and add an edge from )A 238 370 :M -.164(T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 251 370 :M -.167(to )A 264 370 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 279 370 :M -.164(if )A 290 370 :M -.167(in )A 303 370 :M f4_12 sf .25(H)A f0_12 sf .087 .009( )J 316 370 :M -.196(there )A 343 370 :M (is )S 355 370 :M -.163(an )A 370 370 :M -.163(edge )A 396 370 :M -.08(from )A 423 370 :M -.164(T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 436 370 :M -.167(to )A 449 370 :M f1_12 sf .243(e)A f0_12 sf .184A f0_7 sf 0 3 rm .09(j)A 0 -3 rm f0_12 sf .252 .025(. )J 468 370 :M -.108(Note)A 59 389 :M -.094(that the ancestor relations among the )A f2_12 sf -.173(X)A f0_12 sf -.088( variables in )A f4_12 sf -.173(G)A f0_12 sf -.089(\325 is the )A 348 389 :M -.163(same )A 376 389 :M (as )S 390 389 :M -.22(the )A 408 389 :M -.122(ancestor )A 451 389 :M -.164(relations)A 59 407 :M -.043(among the )A f2_12 sf -.069(X)A f0_12 sf -.039( variables in Transform\()A f4_12 sf -.069(G)A f0_12 sf -.038(\). Given the graph )A f4_12 sf -.069(G)A f0_12 sf -.039( from Figure 3 and the graph )A f4_12 sf (H)S 59 425 :M f0_12 sf -.011(from Figure 4, the end result is shown in Figure 5. As in Transform\()A f4_12 sf (G)S f0_12 sf -.011(\), we will )A 444 425 :M -.33(call )A 464 425 :M -.326(a )A 473 425 :M -.218(trek)A 59 442 :M .069(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( )S f1_12 sf .094A f0_12 sf .069 .007( T)J f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf ( )S f1_12 sf .094A f0_12 sf .078 .008( X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .108 .011( that contains a T variable a )J f2_12 sf .255 .025(correlated error)J f2_7 sf 0 3 rm ( )S 0 -3 rm f2_12 sf .043(trek)A f0_12 sf .051 .005( in )J f4_12 sf .069(G)A f0_12 sf .056A 92 489 366 112 rC 92 502 :M f1_12 sf (T)S 100 505 :M f1_7 sf (1)S 104 502 :M f1_12 sf ( )S 107 502 :M ( )S 110 502 :M ( )S 113 502 :M ( )S 116 502 :M ( )S 119 502 :M ( )S 122 502 :M ( )S 125 502 :M ( )S 128 502 :M ( )S 131 502 :M ( )S 134 502 :M ( )S 137 502 :M ( )S 140 502 :M ( )S 143 502 :M ( )S 146 502 :M ( )S 149 502 :M ( )S 152 502 :M ( )S 155 502 :M (T)S 163 505 :M f1_7 sf (3)S 167 502 :M f1_12 sf ( )S 170 502 :M ( )S 173 502 :M ( )S 176 502 :M ( )S 179 502 :M ( )S 182 502 :M ( )S 185 502 :M ( )S 188 502 :M ( )S 191 502 :M ( )S 194 502 :M ( )S 197 502 :M ( )S 200 502 :M ( )S 203 502 :M ( )S 206 502 :M ( )S 209 502 :M ( )S 212 502 :M ( )S 215 502 :M (T)S 223 505 :M f1_7 sf (5)S 227 502 :M f1_12 sf ( )S 230 502 :M ( )S 233 502 :M ( )S 236 502 :M ( )S 239 502 :M ( )S 242 502 :M ( )S 245 502 :M ( )S 248 502 :M ( )S 251 502 :M ( )S 254 502 :M ( )S 257 502 :M ( )S 260 502 :M ( )S 263 502 :M ( )S 266 502 :M ( )S 269 502 :M ( )S 272 502 :M (T)S 280 505 :M f1_7 sf (4)S 284 502 :M f1_12 sf ( )S 287 502 :M ( )S 290 502 :M ( )S 293 502 :M ( )S 296 502 :M ( )S 299 502 :M ( )S 302 502 :M ( )S 305 502 :M ( )S 308 502 :M ( )S 311 502 :M ( )S 314 502 :M ( )S 317 502 :M ( )S 320 502 :M ( )S 323 502 :M ( )S 326 502 :M ( )S 329 502 :M ( )S 332 502 :M ( )S 335 502 :M ( )S 338 502 :M (T)S 346 505 :M f1_7 sf (6)S 350 502 :M f1_12 sf ( )S 353 502 :M ( )S 356 502 :M ( )S 359 502 :M ( )S 362 502 :M ( )S 365 502 :M ( )S 368 502 :M ( )S 371 502 :M ( )S 374 502 :M ( )S 377 502 :M ( )S 380 502 :M ( )S 383 502 :M ( )S 386 502 :M ( )S 389 502 :M ( )S 392 502 :M ( )S 395 502 :M ( )S 398 502 :M ( )S 401 502 :M (T)S 409 505 :M f1_7 sf (2)S 92 578 :M f0_12 sf (X)S 101 581 :M f0_7 sf (1)S 105 578 :M f0_12 sf ( )S 108 578 :M ( )S 111 578 :M ( )S 114 578 :M ( )S 117 578 :M ( )S 120 578 :M ( )S 123 578 :M ( )S 126 578 :M ( )S 129 578 :M ( )S 132 578 :M ( )S 135 578 :M ( )S 138 578 :M ( )S 141 578 :M ( )S 144 578 :M ( )S 147 578 :M ( )S 150 578 :M ( )S 153 578 :M (X)S 162 581 :M f0_7 sf (3)S 166 578 :M f0_12 sf ( )S 169 578 :M ( )S 172 578 :M ( )S 175 578 :M ( )S 178 578 :M ( )S 181 578 :M ( )S 184 578 :M ( )S 187 578 :M ( )S 190 578 :M ( )S 193 578 :M ( )S 196 578 :M ( )S 199 578 :M ( )S 202 578 :M ( )S 205 578 :M ( )S 208 578 :M ( )S 211 578 :M ( )S 214 578 :M ( )S 217 578 :M (X)S 226 581 :M f0_7 sf (5)S 230 578 :M f0_12 sf ( )S 233 578 :M ( )S 236 578 :M ( )S 239 578 :M ( )S 242 578 :M ( )S 245 578 :M ( )S 248 578 :M ( )S 251 578 :M ( )S 254 578 :M ( )S 257 578 :M ( )S 260 578 :M ( )S 263 578 :M ( )S 266 578 :M ( )S 269 578 :M ( )S 272 578 :M ( )S 275 578 :M (X)S 284 581 :M f0_7 sf (4)S 288 578 :M f0_12 sf ( )S 291 578 :M ( )S 294 578 :M ( )S 297 578 :M ( )S 300 578 :M ( )S 303 578 :M ( )S 306 578 :M ( )S 309 578 :M ( )S 312 578 :M ( )S 315 578 :M ( )S 318 578 :M ( )S 321 578 :M ( )S 324 578 :M ( )S 327 578 :M ( )S 330 578 :M ( )S 333 578 :M ( )S 336 578 :M ( )S 339 578 :M (X)S 348 581 :M f0_7 sf (6)S 352 578 :M f0_12 sf ( )S 355 578 :M ( )S 358 578 :M ( )S 361 578 :M ( )S 364 578 :M ( )S 367 578 :M ( )S 370 578 :M ( )S 373 578 :M ( )S 376 578 :M ( )S 379 578 :M ( )S 382 578 :M ( )S 385 578 :M ( )S 388 578 :M ( )S 391 578 :M ( )S 394 578 :M ( )S 397 578 :M ( )S 400 578 :M (X)S 409 581 :M f0_7 sf (2)S gR gS 91 488 368 136 rC -.75 -.75 98.75 553.75 .75 .75 98 509 @b np 102 551 :M 94 551 :L 98 559 :L 102 551 :L .75 lw eofill 94 551.75 -.75 .75 102.75 551 .75 94 551 @a 94 551.75 -.75 .75 98.75 559 .75 94 551 @a -.75 -.75 98.75 559.75 .75 .75 102 551 @b -.75 -.75 161.75 552.75 .75 .75 161 508 @b np 165 550 :M 157 550 :L 161 558 :L 165 550 :L eofill 157 550.75 -.75 .75 165.75 550 .75 157 550 @a 157 550.75 -.75 .75 161.75 558 .75 157 550 @a -.75 -.75 161.75 558.75 .75 .75 165 550 @b -.75 -.75 220.75 553.75 .75 .75 220 509 @b np 224 551 :M 216 551 :L 220 559 :L 224 551 :L eofill 216 551.75 -.75 .75 224.75 551 .75 216 551 @a 216 551.75 -.75 .75 220.75 559 .75 216 551 @a -.75 -.75 220.75 559.75 .75 .75 224 551 @b -.75 -.75 280.75 553.75 .75 .75 280 509 @b np 284 551 :M 276 551 :L 280 559 :L 284 551 :L eofill 276 551.75 -.75 .75 284.75 551 .75 276 551 @a 276 551.75 -.75 .75 280.75 559 .75 276 551 @a -.75 -.75 280.75 559.75 .75 .75 284 551 @b -.75 -.75 343.75 552.75 .75 .75 343 508 @b np 347 550 :M 339 550 :L 343 558 :L 347 550 :L eofill 339 550.75 -.75 .75 347.75 550 .75 339 550 @a 339 550.75 -.75 .75 343.75 558 .75 339 550 @a -.75 -.75 343.75 558.75 .75 .75 347 550 @b -.75 -.75 405.75 554.75 .75 .75 405 510 @b np 409 552 :M 401 552 :L 405 560 :L 409 552 :L eofill 401 552.75 -.75 .75 409.75 552 .75 401 552 @a 401 552.75 -.75 .75 405.75 560 .75 401 552 @a -.75 -.75 405.75 560.75 .75 .75 409 552 @b 98 509.75 -.75 .75 150.75 552 .75 98 509 @a np 151 548 :M 147 554 :L 155 556 :L 151 548 :L eofill -.75 -.75 147.75 554.75 .75 .75 151 548 @b 147 554.75 -.75 .75 155.75 556 .75 147 554 @a 151 548.75 -.75 .75 155.75 556 .75 151 548 @a -.75 -.75 355.75 555.75 .75 .75 406 509 @b np 359 556 :M 355 551 :L 351 559 :L 359 556 :L eofill 355 551.75 -.75 .75 359.75 556 .75 355 551 @a -.75 -.75 351.75 559.75 .75 .75 355 551 @b -.75 -.75 351.75 559.75 .75 .75 359 556 @b 161 507.75 -.75 .75 212.75 552 .75 161 507 @a np 212 548 :M 208 553 :L 216 556 :L 212 548 :L eofill -.75 -.75 208.75 553.75 .75 .75 212 548 @b 208 553.75 -.75 .75 216.75 556 .75 208 553 @a 212 548.75 -.75 .75 216.75 556 .75 212 548 @a -.75 -.75 232.75 552.75 .75 .75 281 507 @b np 236 553 :M 231 548 :L 228 556 :L 236 553 :L eofill 231 548.75 -.75 .75 236.75 553 .75 231 548 @a -.75 -.75 228.75 556.75 .75 .75 231 548 @b -.75 -.75 228.75 556.75 .75 .75 236 553 @b 282 506.75 -.75 .75 332.75 552 .75 282 506 @a np 333 548 :M 328 553 :L 336 556 :L 333 548 :L eofill -.75 -.75 328.75 553.75 .75 .75 333 548 @b 328 553.75 -.75 .75 336.75 556 .75 328 553 @a 333 548.75 -.75 .75 336.75 556 .75 333 548 @a 221 508.75 -.75 .75 327.75 557 .75 221 508 @a np 326 552 :M 323 559 :L 332 559 :L 326 552 :L eofill -.75 -.75 323.75 559.75 .75 .75 326 552 @b 323 559.75 -.75 .75 332.75 559 .75 323 559 @a 326 552.75 -.75 .75 332.75 559 .75 326 552 @a 114 574.75 -.75 .75 146.75 574 .75 114 574 @a np 116 578 :M 116 570 :L 108 574 :L 116 578 :L eofill -.75 -.75 116.75 578.75 .75 .75 116 570 @b -.75 -.75 108.75 574.75 .75 .75 116 570 @b 108 574.75 -.75 .75 116.75 578 .75 108 574 @a 357 574.75 -.75 .75 389.75 574 .75 357 574 @a np 387 570 :M 387 578 :L 395 574 :L 387 570 :L eofill -.75 -.75 387.75 578.75 .75 .75 387 570 @b -.75 -.75 387.75 578.75 .75 .75 395 574 @b 387 570.75 -.75 .75 395.75 574 .75 387 570 @a -.75 -.75 95.75 589.75 .75 .75 95 589 @b np 91 591 :M 99 591 :L 95 583 :L 91 591 :L eofill 91 591.75 -.75 .75 99.75 591 .75 91 591 @a 95 583.75 -.75 .75 99.75 591 .75 95 583 @a -.75 -.75 91.75 591.75 .75 .75 95 583 @b -.75 -.75 105.75 589.75 .75 .75 105 589 @b np 101 591 :M 109 591 :L 105 583 :L 101 591 :L eofill 101 591.75 -.75 .75 109.75 591 .75 101 591 @a 105 583.75 -.75 .75 109.75 591 .75 105 583 @a -.75 -.75 101.75 591.75 .75 .75 105 583 @b 90 180 160 62 175.5 591.5 @n 0 90 208 84 174.5 581.5 @n 90 180 90 30 150.5 591.5 @n 0 90 146 46 151.5 583.5 @n gR gS 0 0 552 730 rC 239 645 :M f2_12 sf 2.224 .222(Figure 5: )J f6_12 sf 1.157(G)A f2_12 sf S endp %%Page: 9 9 %%BeginPageSetup initializepage (peter; page: 9 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (9)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.326(We )A 79 56 :M -.166(will )A 101 56 :M .259 .026(now )J 126 56 :M .479 .048(show )J 157 56 :M -.249(that )A 179 56 :M -.164(if )A 191 56 :M -.196(there )A 219 56 :M (is )S 232 56 :M -.326(a )A 242 56 :M -.196(correlated )A 293 56 :M -.061(error )A 321 56 :M -.163(trek )A 344 56 :M -.139(between )A 388 56 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 404 56 :M -.109(and )A 426 56 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 442 56 :M -.167(in )A 456 56 :M f4_12 sf .19(G)A f0_12 sf .139 .014J 474 56 :M -.331(that)A 59 74 :M -.123(contains )A 103 74 :M -.326(a )A 113 74 :M -.205(variable )A 156 74 :M .159(T)A f0_7 sf 0 3 rm .051(r)A 0 -3 rm f0_12 sf .119 .012(, )J 175 74 :M -.165(then )A 201 74 :M -.167(in )A 216 74 :M -.022(Transform\()A f4_12 sf (G)S f0_12 sf (\) )S 289 74 :M -.196(there )A 318 74 :M (is )S 332 74 :M -.326(a )A 343 74 :M -.196(correlated )A 395 74 :M -.061(error )A 424 74 :M -.163(trek )A 448 74 :M -.139(sequence)A 59 92 :M .112 .011(between X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .051 .005( and X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .049 .005(, such )J 179 92 :M -.249(that )A 200 92 :M -.129(every )A 230 92 :M -.205(variable )A 271 92 :M -.167(in )A 284 92 :M -.22(the )A 302 92 :M -.196(correlated )A 352 92 :M -.061(error )A 379 92 :M -.163(trek )A 401 92 :M (sequence, )S 452 92 :M -.083(with )A 477 92 :M -.33(the)A 59 110 :M -.041(possible )A 103 110 :M -.183(exception )A 153 110 :M (of )S 168 110 :M -.22(the )A 187 110 :M .17 .017(endpoints, )J 242 110 :M (has )S 264 110 :M -.132(index )A 296 110 :M .898 .09(\(i.e. )J 322 110 :M -.031(subscript\) )A 375 110 :M (less )S 399 110 :M -.165(than )A 425 110 :M (or )S 441 110 :M -.197(equal )A 472 110 :M -.167(to )A 487 110 :M (r)S 59 128 :M -.099(\(henceforth referred to as the correlated error trek sequence in Transform\()A f4_12 sf -.181(G)A f0_12 sf -.146(\) )A 423 128 :M -.053(corresponding)A 59 146 :M -.167(to )A 73 146 :M -.22(the )A 92 146 :M -.196(correlated )A 143 146 :M -.061(error )A 171 146 :M -.163(trek )A 194 146 :M -.139(between )A 238 146 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 254 146 :M -.109(and )A 276 146 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 292 146 :M -.167(in )A 306 146 :M f4_12 sf .517(G)A f0_12 sf .641 .064J 332 146 :M -.219(The )A 355 146 :M (proof )S 386 146 :M (is )S 399 146 :M (by )S 416 146 :M -.148(induction )A 465 146 :M (on )S 483 146 :M 1.007(r.)A 59 164 :M .197 .02(Suppose )J 104 164 :M -.064(first )A 127 164 :M -.249(that )A 148 164 :M (r )S 156 164 :M .211 .021(= )J 167 164 :M .833 .083(1. )J 181 164 :M (If )S 193 164 :M -.196(there )A 220 164 :M (is )S 232 164 :M -.326(a )A 242 164 :M -.196(correlated )A 293 164 :M -.061(error )A 321 164 :M -.163(trek )A 344 164 :M -.139(between )A 388 164 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 404 164 :M -.109(and )A 426 164 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 442 164 :M -.167(in )A 456 164 :M f4_12 sf .19(G)A f0_12 sf .139 .014J 474 164 :M -.331(that)A 59 182 :M -.123(contains )A 102 182 :M .09(T)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( )S 117 182 :M -.165(then )A 141 182 :M -.196(there )A 168 182 :M -.215(are )A 186 182 :M -.196(correlated )A 236 182 :M -.061(error )A 263 182 :M -.064(treks )A 290 182 :M -.139(between )A 333 182 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 348 182 :M -.109(and )A 369 182 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 386 182 :M -.167(in )A 399 182 :M .053(Transform\()A f4_12 sf .084(G)A f0_12 sf .081 .008(\), )J 474 182 :M -.163(and)A 59 200 :M -.087(between X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.081( and X)A f0_7 sf 0 3 rm -.053(1)A 0 -3 rm f0_12 sf -.072(. The concatenation of these two correlated error treks forms a )A 445 200 :M -.218(correlated)A 59 218 :M -.061(error )A 88 218 :M -.163(trek )A 112 218 :M -.121(sequence )A 161 218 :M -.167(in )A 176 218 :M -.065(which )A 211 218 :M -.18(\(trivially\) )A 261 218 :M -.129(every )A 293 218 :M -.205(variable )A 336 218 :M -.167(in )A 351 218 :M -.22(the )A 371 218 :M -.121(sequence )A 420 218 :M -.219(except )A 456 218 :M (for )S 477 218 :M -.33(the)A 59 236 :M -.058(endpoints has an index less than or equal to 1. The induction hypothesis is that for all r )A cF f1_12 sf -.058A sf -.058( n,)A 59 254 :M -.164(if )A 72 254 :M -.196(there )A 101 254 :M (is )S 115 254 :M -.326(a )A 126 254 :M -.196(correlated )A 178 254 :M -.061(error )A 207 254 :M -.163(trek )A 231 254 :M -.139(between )A 276 254 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 293 254 :M -.109(and )A 316 254 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 333 254 :M -.167(in )A 348 254 :M f4_12 sf .19(G)A f0_12 sf .139 .014J 367 254 :M -.249(that )A 390 254 :M -.123(contains )A 435 254 :M .159(T)A f0_7 sf 0 3 rm .051(r)A 0 -3 rm f0_12 sf .119 .012(, )J 455 254 :M -.165(then )A 482 254 :M -.334(in)A 59 272 :M -.116(Transform\()A f4_12 sf -.184(G)A f0_12 sf -.096(\) there is a correlated )A 224 272 :M -.061(error )A 251 272 :M -.163(trek )A 273 272 :M -.121(sequence )A 320 272 :M -.139(between )A 363 272 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 378 272 :M -.109(and )A 399 272 :M .726(X)A f0_7 sf 0 3 rm .163(j)A 0 -3 rm f0_12 sf .457 .046(, )J 418 272 :M (such )S 444 272 :M -.249(that )A 465 272 :M -.161(every)A 59 290 :M -.093(variable in the sequence, with the possible exception of the endpoints has an index less than)A 59 308 :M .116 .012(r. Suppose now that )J 159 308 :M -.167(in )A 172 308 :M f4_12 sf 1.365(G)A f0_12 sf .739 .074(' )J 189 308 :M -.196(there )A 216 308 :M (is )S 228 308 :M -.326(a )A 237 308 :M -.196(correlated )A 287 308 :M -.061(error )A 314 308 :M -.163(trek )A 336 308 :M -.139(between )A 379 308 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 394 308 :M -.109(and )A 415 308 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 430 308 :M (such )S 456 308 :M -.249(that )A 477 308 :M -.33(the)A 59 326 :M -.163(trek )A 81 326 :M -.123(contains )A 124 326 :M .52(T)A f0_7 sf 0 3 rm .259(n+1)A 0 -3 rm f0_12 sf .387 .039(, )J 151 326 :M -.062(where )A 184 326 :M .555 .055(i, )J 195 326 :M -.334(j )A 202 326 :M cF f1_12 sf .038A sf .376 .038( )J 213 326 :M .88 .088(n+1. )J 240 326 :M -.131(Since )A 270 326 :M -.22(the )A 288 326 :M -.163(edge )A 314 326 :M -.139(between )A 357 326 :M .249(T)A f0_7 sf 0 3 rm .124(n+1)A 0 -3 rm f0_12 sf .102 .01( )J 380 326 :M -.109(and )A 401 326 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 417 326 :M -.055(exists )A 449 326 :M -.167(in )A 463 326 :M 1.033 .103(G\325, )J 485 326 :M -.668(it)A 59 344 :M (follows )S 99 344 :M -.08(from )A 126 344 :M -.22(the )A 144 344 :M -.166(method )A 183 344 :M (of )S 197 344 :M -.109(construction )A 259 344 :M (of )S 273 344 :M .286 .029<47D520>J 290 344 :M -.249(that )A 312 344 :M -.219(either )A 343 344 :M -.196(there )A 371 344 :M (is )S 384 344 :M -.326(a )A 394 344 :M -.196(correlated )A 445 344 :M -.061(error )A 473 344 :M -.218(trek)A 59 362 :M .18 .018(between X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .082 .008( and X)J f0_7 sf 0 3 rm .024(n+1)A 0 -3 rm f0_12 sf .042 .004( in )J f4_12 sf .057(G)A f0_12 sf .112 .011(' that contains some T)J f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf .063 .006(, r < n+1, )J 337 362 :M (or )S 351 362 :M .532(X)A f0_7 sf 0 3 rm .224(n+1)A 0 -3 rm f0_12 sf .184 .018( )J 376 362 :M -.109(and )A 397 362 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 412 362 :M -.215(are )A 430 362 :M -.197(d-adjacent )A 482 362 :M -.334(in)A 59 380 :M .053(Transform\()A f4_12 sf .084(G)A f0_12 sf .081 .008(\). )J 134 380 :M (In )S 148 380 :M -.22(the )A 166 380 :M -.107(former )A 202 380 :M .236 .024(case, )J 230 380 :M (by )S 246 380 :M -.22(the )A 264 380 :M -.148(induction )A 312 380 :M -.033(hypothesis )A 367 380 :M -.196(there )A 394 380 :M (is )S 407 380 :M -.326(a )A 417 380 :M -.196(correlated )A 468 380 :M -.076(error)A 59 398 :M -.163(trek )A 81 398 :M -.121(sequence )A 128 398 :M -.139(between )A 171 398 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 186 398 :M -.109(and )A 207 398 :M .532(X)A f0_7 sf 0 3 rm .224(n+1)A 0 -3 rm f0_12 sf .184 .018( )J 232 398 :M (that, )S 257 398 :M -.219(except )A 292 398 :M (for )S 311 398 :M -.22(the )A 330 398 :M .17 .017(endpoints, )J 385 398 :M -.123(contains )A 429 398 :M -.083(only )A 455 398 :M -.187(vertices)A 59 416 :M .229 .023(whose )J 94 416 :M -.141(indices )A 131 416 :M -.215(are )A 149 416 :M (less )S 171 416 :M -.165(than )A 195 416 :M (or )S 209 416 :M -.197(equal )A 238 416 :M -.167(to )A 251 416 :M .88 .088(n+1. )J 278 416 :M (In )S 292 416 :M -.22(the )A 310 416 :M -.275(latter )A 338 416 :M .236 .024(case, )J 367 416 :M .503( )J 422 416 :M (is )S 435 416 :M -.326(a )A 445 416 :M -.218(correlated)A 59 434 :M -.069(error trek sequence between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.072( and X)A f0_7 sf 0 3 rm -.049(n+1)A 0 -3 rm f0_12 sf -.061(. Similarly, there is a correlated error trek )A 448 434 :M -.139(sequence)A 59 452 :M -.139(between )A 102 452 :M .532(X)A f0_7 sf 0 3 rm .224(n+1)A 0 -3 rm f0_12 sf .184 .018( )J 127 452 :M -.109(and )A 148 452 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 163 452 :M (that, )S 188 452 :M -.219(except )A 222 452 :M (for )S 240 452 :M -.22(the )A 258 452 :M .17 .017(endpoints, )J 312 452 :M -.123(contains )A 355 452 :M -.083(only )A 381 452 :M -.163(vertices )A 422 452 :M .229 .023(whose )J 458 452 :M -.165(indices)A 59 470 :M -.101(are less than or equal to n+1. These two correlated error trek sequences can be concatenated)A 59 488 :M -.109(to form a )A 106 488 :M -.196(correlated )A 156 488 :M -.061(error )A 183 488 :M -.163(trek )A 205 488 :M -.121(sequence )A 252 488 :M -.139(between )A 295 488 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 310 488 :M -.109(and )A 331 488 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 346 488 :M (that, )S 371 488 :M -.219(except )A 405 488 :M (for )S 423 488 :M -.22(the )A 441 488 :M .038(endpoints,)A 59 506 :M -.123(contains )A 102 506 :M -.083(only )A 127 506 :M -.163(vertices )A 167 506 :M .229 .023(whose )J 202 506 :M -.141(indices )A 239 506 :M -.215(are )A 257 506 :M (less )S 279 506 :M -.165(than )A 303 506 :M (or )S 317 506 :M -.197(equal )A 346 506 :M -.167(to )A 359 506 :M .88 .088(n+1. )J 386 506 :M .258 .026(For )J 408 506 :M -.22(the )A 427 506 :M f4_12 sf 1.365(G)A f0_12 sf .739 .074(' )J 445 506 :M .447 .045(shown )J 482 506 :M -.334(in)A 59 524 :M -.042(Figure 5, there is a correlated error trek between X)A f0_7 sf 0 3 rm (5)S 0 -3 rm f0_12 sf -.048( and X)A f0_7 sf 0 3 rm (6)S 0 -3 rm f0_12 sf -.045(, and a corresponding )A 445 524 :M -.218(correlated)A 59 542 :M .111 .011(error trek sequence in the graph Transform\()J f4_12 sf (G)S f0_12 sf .076 .008(\) in Figure 3.)J 59 572 :M -.035(We will now show that if X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.036( and X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.033( are d-connected given )A f2_12 sf -.054(Z)A f0_12 sf -.026( in )A f4_12 sf -.059(G)A f0_12 sf -.034A 384 572 :M -.165(then )A 408 572 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 425 572 :M -.109(and )A 446 572 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 463 572 :M -.215(are )A 481 572 :M (d-)S 59 590 :M -.153(connected given )A 139 590 :M f2_12 sf (Z)S f0_12 sf ( )S 151 590 :M -.167(in )A 164 590 :M -.022(Transform\()A f4_12 sf (G)S f0_12 sf (\) )S 235 590 :M (using )S 265 590 :M -.33(Lemma )A 304 590 :M 1.395 .14(3.3.1+ )J 341 590 :M -.058(\(Richardson )A 403 590 :M .667 .067(1994, )J 435 590 :M -.065(which )A 468 590 :M (is )S 480 590 :M -.326(an)A 59 608 :M -.124(extension to the )A 137 608 :M -.274(cyclic )A 168 608 :M -.161(case )A 192 608 :M (of )S 206 608 :M -.33(Lemma )A 245 608 :M 1.333 .133(3.3.1 )J 275 608 :M -.167(in )A 288 608 :M -.046(Spirtes )A 325 608 :M -.33(et )A 337 608 :M .261 .026(al. )J 353 608 :M .629 .063(1993\). )J 389 608 :M -.33(Lemma )A 428 608 :M 1.395 .14(3.3.1+ )J 465 608 :M -.131(states)A 59 626 :M -.249(that )A 80 626 :M -.196(there )A 107 626 :M (is )S 119 626 :M -.326(a )A 128 626 :M -.165(path )A 152 626 :M -.167(in )A 165 626 :M -.326(a )A 174 626 :M -.205(directed )A 215 626 :M -.064(graph )A 246 626 :M .306 .031(G )J 259 626 :M -.249(that )A 280 626 :M -.097(d-connects )A 335 626 :M .306 .031(X )J 348 626 :M -.109(and )A 369 626 :M -.663(Y )A 381 626 :M -.132(given )A 411 626 :M f2_12 sf (Z)S f0_12 sf ( )S 424 626 :M -.164(if )A 436 626 :M -.109(and )A 458 626 :M -.083(only )A 484 626 :M -.327(if)A 59 644 :M -.196(there )A 87 644 :M (is )S 100 644 :M -.326(a )A 110 644 :M -.121(sequence )A 158 644 :M (of )S 173 644 :M -.163(vertices )A 214 644 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 228 644 :M -.109(and )A 250 644 :M -.326(a )A 260 644 :M -.109(set )A 278 644 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 291 644 :M (of )S 306 644 :M -.065(paths )A 336 644 :M -.167(in )A 350 644 :M f4_12 sf .25(G)A f0_12 sf .087 .009( )J 364 644 :M -.139(between )A 408 644 :M -.064(pairs )A 437 644 :M (of )S 453 644 :M -.282(adjacent)A 59 662 :M -.116(vertices in )A f2_12 sf -.248(Q)A f0_12 sf -.125( that have the following properties: \(i\) For each occurrence of a pair of adjacent)A 59 679 :M .32 .032(variables X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .15 .015( and X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .078 .008( in )J f2_12 sf .156 .016(Q, )J 183 679 :M f0_12 sf -.334(i )A 190 679 :M f1_12 sf 1.42A f0_12 sf 2.046 .205(\312j, )J 214 679 :M -.109(and )A 235 679 :M -.196(there )A 262 679 :M (is )S 274 679 :M -.326(a )A 283 679 :M -.11(unique )A 319 679 :M -.165(path )A 343 679 :M -.167(in )A 356 679 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 368 679 :M -.249(that )A 389 679 :M -.097(d-connects )A 444 679 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 459 679 :M -.109(and )A 480 679 :M .32(X)A f0_7 sf 0 3 rm (j)S 0 -3 rm endp %%Page: 10 10 %%BeginPageSetup initializepage (peter; page: 10 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (10)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf .035 .003(given )J f2_12 sf (Z)S f0_12 sf (\\{X)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (,X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .024 .002(}; \(ii\) if is a subsequence of )J f2_12 sf (Q)S f0_12 sf .042 .004(, the corresponding path between)J 59 74 :M (X)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.013( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( in )S f2_12 sf (P)S f0_12 sf -.011( is into X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.012(, and the corresponding path between X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.013( and X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf ( )S 404 74 :M -.167(in )A 417 74 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 429 74 :M (is )S 441 74 :M -.167(into )A 463 74 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 478 74 :M -.164(\(in)A 59 92 :M -.014(which case we say that the occurrence of X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.011( is a )A f2_12 sf -.014(collider)A f0_12 sf ( in )S f2_12 sf (Q)S f0_12 sf -.014(\) then X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.014( has a )A 426 92 :M -.131(descendant )A 482 92 :M -.334(in)A 59 110 :M f2_12 sf -.238(Z)A f0_12 sf -.129(; and \(iii\) )A 114 110 :M -.164(if )A 125 110 :M -.196(there )A 152 110 :M (is )S 164 110 :M -.163(an )A 179 110 :M -.162(occurrence )A 234 110 :M (of )S 248 110 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 263 110 :M -.249(that )A 284 110 :M (is )S 296 110 :M -.326(a )A 305 110 :M -.137(non-collider )A 366 110 :M -.167(in )A 379 110 :M f2_12 sf .405(Q)A f0_12 sf .237 .024(, )J 396 110 :M -.165(then )A 420 110 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 435 110 :M (is )S 447 110 :M -.111(not )A 466 110 :M -.167(in )A 479 110 :M f2_12 sf .724(Z)A f0_12 sf (.)S 59 128 :M -.118(Note that we do not require that a vertex occur )A 281 128 :M -.083(only )A 306 128 :M -.163(once )A 332 128 :M -.167(in )A 345 128 :M f2_12 sf .405(Q)A f0_12 sf .237 .024(. )J 362 128 :M -.128(Hence )A 396 128 :M -.109(one )A 417 128 :M -.162(occurrence )A 472 128 :M (of )S 486 128 :M (a)S 59 146 :M -.163(vertex )A 92 146 :M -.167(in )A 105 146 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 118 146 :M -.22(may )A 142 146 :M -.163(be )A 157 146 :M -.326(a )A 166 146 :M -.072(collider, )A 209 146 :M -.109(and )A 230 146 :M -.14(another )A 269 146 :M -.162(occurrence )A 324 146 :M (of )S 338 146 :M -.22(the )A 356 146 :M -.163(same )A 384 146 :M -.163(vertex )A 417 146 :M -.167(in )A 431 146 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 445 146 :M -.22(may )A 470 146 :M -.163(be )A 486 146 :M (a)S 59 164 :M -.053(non-collider.)A 59 194 :M -.084(Suppose now that there is an undirected )A 251 194 :M -.165(path )A 275 194 :M .306 .031(U )J 288 194 :M -.249(that )A 309 194 :M -.097(d-connects )A 364 194 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 381 194 :M -.109(and )A 402 194 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 419 194 :M -.132(given )A 449 194 :M f2_12 sf (Z)S f0_12 sf ( )S 461 194 :M -.167(in )A 474 194 :M f4_12 sf .744(G)A f0_12 sf .6A 59 212 :M -.027(Intuitively, in )A f4_12 sf -.055(G)A f0_12 sf -.03(' we would like to form )A f2_12 sf -.059(Q)A f0_12 sf -.03( and )A f2_12 sf (P)S f0_12 sf -.032( by breaking U into )A 386 212 :M (pieces, )S 423 212 :M (such )S 449 212 :M -.249(that )A 470 212 :M -.326(each)A 59 230 :M -.196(correlated )A 110 230 :M -.061(error )A 138 230 :M -.163(trek )A 161 230 :M -.052(occurs )A 197 230 :M (as )S 212 230 :M -.326(a )A 222 230 :M -.162(separate )A 265 230 :M -.052(piece. )A 298 230 :M -.247(More )A 328 230 :M -.035(formally, )A 377 230 :M -.08(form )A 406 230 :M -.326(a )A 417 230 :M -.121(sequence )A 466 230 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 481 230 :M (of)S 59 248 :M -.163(vertices )A 100 248 :M -.109(and )A 122 248 :M -.163(an )A 138 248 :M -.131(associated )A 191 248 :M -.121(sequence )A 239 248 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 252 248 :M (of )S 267 248 :M -.065(paths )A 297 248 :M -.167(in )A 311 248 :M f4_12 sf 1.365(G)A f0_12 sf .739 .074(' )J 329 248 :M -.083(with )A 355 248 :M -.22(the )A 374 248 :M -.073(following )A 425 248 :M -.119(properties: )A 480 248 :M -.16(\(i\))A 59 266 :M -.087(every vertex in )A f2_12 sf -.173(Q)A f0_12 sf -.07( is in )A f2_12 sf -.16(X)A f0_12 sf -.086( and occurs on U; \(ii\) no vertex occurs in )A f2_12 sf -.173(Q)A f0_12 sf -.086( more than once; \(iii\) if)A 59 284 :M -.079(A occurs before B in )A f2_12 sf -.15(Q)A f0_12 sf -.08(, then A occurs before )A 278 284 :M (B )S 290 284 :M (on )S 306 284 :M (U; )S 322 284 :M -.08(\(iv\) )A 343 284 :M -.164(if )A 354 284 :M -.22(the )A 372 284 :M -.047(subpath )A 413 284 :M (of )S 427 284 :M .306 .031(U )J 440 284 :M -.139(between )A 483 284 :M (A)S 59 302 :M -.096(and B is a correlated error trek, then A and )A 264 302 :M (B )S 276 302 :M -.083(both )A 301 302 :M -.129(occur )A 331 302 :M -.167(in )A 344 302 :M -.249(that )A 365 302 :M -.062(order )A 394 302 :M -.167(in )A 407 302 :M f2_12 sf .405(Q)A f0_12 sf .237 .024(. )J 424 302 :M -.219(The )A 446 302 :M -.165(path )A 470 302 :M -.167(in )A 483 302 :M f2_12 sf (P)S 59 320 :M f0_12 sf -.129(associated with a pair A and B of adjacent vertices in )A f2_12 sf -.256(Q)A f0_12 sf -.124( is the )A 351 320 :M -.047(subpath )A 392 320 :M (of )S 406 320 :M .306 .031(U )J 419 320 :M -.139(between )A 462 320 :M -.663(A )A 474 320 :M -.163(and)A 59 337 :M -.022(B. In the example in Figure 5, in )A f4_12 sf (G)S f0_12 sf -.023(' the d-connecting path between X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.025( and X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.022( given )A f2_12 sf (Z)S f0_12 sf -.02( = )A f1_12 sf S 59 355 :M f0_12 sf .327 .033(is X)J f0_7 sf 0 3 rm .081(1)A 0 -3 rm f0_12 sf .063 .006( )J f1_12 sf .275A f0_12 sf .226 .023( X)J f0_7 sf 0 3 rm .081(5)A 0 -3 rm f0_12 sf .063 .006( )J f1_12 sf .275A f0_12 sf .2 .02( T)J f0_7 sf 0 3 rm .081(4)A 0 -3 rm f0_12 sf .063 .006( )J f1_12 sf .275A f0_12 sf .07 .007( )J 162 355 :M .478(X)A f0_7 sf 0 3 rm .193(6)A 0 -3 rm f0_12 sf .166 .017( )J 179 355 :M f1_12 sf .126A f0_12 sf ( )S 195 355 :M .876(X)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(, )J 216 355 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 229 355 :M .211 .021(= )J 240 355 :M .781(, )J 326 355 :M -.109(and )A 347 355 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 359 355 :M .211 .021(= )J 370 355 :M .376(A f0_12 sf ( )S 410 355 :M .876(X)A f0_7 sf 0 3 rm .354(5)A 0 -3 rm f0_12 sf .552 .055(, )J 431 355 :M .478(X)A f0_7 sf 0 3 rm .193(5)A 0 -3 rm f0_12 sf .166 .017( )J 448 355 :M f1_12 sf .126A f0_12 sf ( )S 464 355 :M .09(T)A f0_7 sf 0 3 rm (4)S 0 -3 rm f0_12 sf ( )S 479 355 :M f1_12 sf S 59 373 :M f0_12 sf (X)S f0_7 sf 0 3 rm (6)S 0 -3 rm f0_12 sf .053 .005(, X)J f0_7 sf 0 3 rm (6)S 0 -3 rm f0_12 sf ( )S f1_12 sf .055A f0_12 sf .045 .005( X)J f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf .072 .007(>. In this example, there are no colliders in )J f2_12 sf (Q)S f0_12 sf (.)S 59 404 :M -.057(Because U is a path that d-connects X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.062( and X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.054( given )A f2_12 sf -.092(Z)A f0_12 sf -.05( in G\325, it is easy to see that )A 448 404 :M -.22(the )A 466 404 :M -.082(paths)A 59 422 :M -.076(in )A f2_12 sf -.136(P)A f0_12 sf -.087( have the following properties in )A f4_12 sf -.16(G)A f0_12 sf -.081(\325: \(i\) Each path in )A f2_12 sf -.173(Q)A f0_12 sf -.09( d-connects its endpoints )A 459 422 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 474 422 :M -.163(and)A 59 440 :M (X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.025( given )A f2_12 sf (Z)S f0_12 sf -.032(\\{X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.032(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.023(}; \(ii\) if there is an )A 236 440 :M -.162(occurrence )A 291 440 :M (of )S 305 440 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 320 440 :M -.167(in )A 333 440 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 346 440 :M -.249(that )A 367 440 :M (is )S 379 440 :M -.326(a )A 388 440 :M -.206(collider )A 427 440 :M -.165(then )A 451 440 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 466 440 :M (has )S 486 440 :M (a)S 59 458 :M -.112(descendant in )A f2_12 sf -.183(Z)A f0_12 sf -.102(; and \(iii\) if there is an occurrence of X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 323 458 :M -.167(in )A 336 458 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 349 458 :M -.249(that )A 370 458 :M (is )S 382 458 :M -.326(a )A 391 458 :M -.049(non-collider, )A 456 458 :M -.165(then )A 480 458 :M .32(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 476 :M f0_12 sf .065 .006(is not in )J f2_12 sf (Z)S f0_12 sf (.)S 59 506 :M -.326(We )A 79 506 :M -.166(will )A 101 506 :M .259 .026(now )J 126 506 :M .479 .048(show )J 156 506 :M .259 .026(how )J 181 506 :M -.167(to )A 194 506 :M -.109(construct )A 241 506 :M -.326(a )A 250 506 :M -.121(sequence )A 297 506 :M (of )S 311 506 :M -.163(vertices )A 351 506 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 369 506 :M -.109(and )A 391 506 :M -.326(a )A 401 506 :M -.109(set )A 419 506 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 437 506 :M (of )S 452 506 :M -.065(paths )A 482 506 :M -.334(in)A 59 524 :M -.061(Transform\()A f4_12 sf -.096(G)A f0_12 sf -.062(\) between )A 172 524 :M -.064(pairs )A 199 524 :M (of )S 213 524 :M -.247(adjacent )A 255 524 :M -.163(vertices )A 295 524 :M -.167(in )A 308 524 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 326 524 :M -.249(that )A 347 524 :M -.163(have )A 373 524 :M -.22(the )A 391 524 :M -.073(following )A 441 524 :M -.131(properties:)A 59 542 :M -.09(\(i\) For each occurrence of a pair of adjacent variables X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.102( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.074( in )A f2_12 sf -.128<51D5>A f0_12 sf -.086( there is a )A 435 542 :M -.11(unique )A 471 542 :M -.22(path)A 59 560 :M -.017(in )A f2_12 sf <50D5>S f0_12 sf -.02( that d-connects X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.024( and )A 196 560 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 211 560 :M -.132(given )A 241 560 :M f2_12 sf .239(Z)A f0_12 sf .176(\\{X)A f0_7 sf 0 3 rm .058(i)A 0 -3 rm f0_12 sf .174(,X)A f0_7 sf 0 3 rm .058(j)A 0 -3 rm f0_12 sf .3 .03(}; )J 297 560 :M -.164(\(ii\) )A 315 560 :M -.164(if )A 326 560 :M -.196(there )A 353 560 :M (is )S 365 560 :M -.163(an )A 380 560 :M -.162(occurrence )A 435 560 :M (of )S 449 560 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 464 560 :M -.167(in )A 477 560 :M f2_12 sf .673<51D5>A 59 578 :M f0_12 sf -.082(that is a collider, then X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 178 578 :M (has )S 198 578 :M -.326(a )A 207 578 :M -.131(descendant )A 263 578 :M -.167(in )A 276 578 :M f2_12 sf -.189(Z)A f0_12 sf -.149(; )A 291 578 :M -.109(and )A 312 578 :M -.198(\(iii\) )A 333 578 :M -.164(if )A 344 578 :M -.196(there )A 371 578 :M (is )S 383 578 :M -.163(an )A 398 578 :M -.162(occurrence )A 453 578 :M (of )S 467 578 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 482 578 :M -.334(in)A 59 596 :M f2_12 sf -.054<51D5>A f0_12 sf -.037( that is a non-collider, then X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.033( is not in )A f2_12 sf -.065(Z)A f0_12 sf -.04(. It will follow from Lemma 3.3.1+ that )A 461 596 :M .306 .031(X )J 474 596 :M -.163(and)A 59 614 :M -.076(Y are d-connected given )A f2_12 sf -.12(Z)A f0_12 sf -.075( in Transform\()A f4_12 sf -.13(G)A f0_12 sf -.105(\).)A 59 644 :M -.09(We will create )A f2_12 sf -.124<51D5>A f0_12 sf -.088( by several modifications of )A f2_12 sf -.174(Q)A f0_12 sf -.094(. Step )A 318 644 :M (\(1\) )S 336 644 :M -.167(in )A 349 644 :M -.205(creating )A 390 644 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 408 644 :M (is )S 420 644 :M -.167(to )A 433 644 :M -.233(replace )A 470 644 :M -.326(each)A 59 662 :M -.053(subsequence of )A f2_12 sf -.088(Q)A f0_12 sf -.045( such that X)A f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf -.05( and X)A f0_7 sf 0 3 rm (s)S 0 -3 rm f0_12 sf -.044( are on a correlated error trek )A 422 662 :M -.167(in )A 435 662 :M f2_12 sf .405(Q)A f0_12 sf .237 .024(, )J 452 662 :M -.083(with )A 477 662 :M -.33(the)A 59 679 :M -.038(corresponding correlated error trek sequence A f0_12 sf -.038(, X)A f0_7 sf 0 3 rm (s)S 0 -3 rm f0_12 sf -.039(> in Transform\()A f4_12 sf -.067(G)A f0_12 sf -.038(\). Note that each)A endp %%Page: 11 11 %%BeginPageSetup initializepage (peter; page: 11 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (11)S gR gS 0 0 552 730 rC 59 55 :M f0_12 sf -.162(occurrence )A 114 55 :M (of )S 128 55 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 145 55 :M -.139(between )A 188 55 :M .403(A f0_12 sf .298(,X)A f0_7 sf 0 3 rm .139(s)A 0 -3 rm f0_12 sf .453 .045(> )J 253 55 :M (is )S 265 55 :M -.326(a )A 274 55 :M -.206(collider )A 313 55 :M -.167(in )A 326 55 :M f2_12 sf .577<51D5>A f0_12 sf .472 .047(. )J 348 55 :M (In )S 362 55 :M -.22(the )A 380 55 :M -.081(example, )A 427 55 :M -.195(after )A 453 55 :M -.22(the )A 472 55 :M -.08(first)A 59 73 :M .825 .083(step )J f2_12 sf .369<51D5>A f0_12 sf .625 .063( = and )J f2_12 sf .296(P')A f0_12 sf .625 .063( = A f0_12 sf .539 .054( X)J f0_7 sf 0 3 rm .194(5)A 0 -3 rm f0_12 sf .625 .063(, X)J f0_7 sf 0 3 rm .194(5)A 0 -3 rm f0_12 sf .151 .015( )J f1_12 sf .656A f0_12 sf .477 .048( T)J f0_7 sf 0 3 rm .194(45)A 0 -3 rm f0_12 sf .151 .015( )J f1_12 sf .656A f0_12 sf .539 .054( X)J f0_7 sf 0 3 rm .194(4)A 0 -3 rm f0_12 sf .625 .063(, X)J f0_7 sf 0 3 rm .194(4)A 0 -3 rm f0_12 sf .151 .015( )J f1_12 sf .656A f0_12 sf .477 .048( T)J f0_7 sf 0 3 rm .194(46)A 0 -3 rm f0_12 sf .151 .015( )J f1_12 sf .656A f0_12 sf .539 .054( X)J f0_7 sf 0 3 rm .194(6)A 0 -3 rm f0_12 sf .625 .063(, X)J f0_7 sf 0 3 rm (6)S 0 -3 rm 59 91 :M f1_12 sf .312A f0_12 sf .256 .026( X)J f0_7 sf 0 3 rm .092(2)A 0 -3 rm f0_12 sf .461 .046(>, i.e. we replaced the subsequence in )J f2_12 sf .246(Q)A f0_12 sf .339 .034( by .)A 59 122 :M -.111(Recall that the ancestor relations among the )A f2_12 sf -.203(X)A f0_12 sf -.116( variables \(which includes )A 402 122 :M -.22(the )A 420 122 :M -.145(variables )A 466 122 :M -.167(in )A 479 122 :M f2_12 sf (Z)S f0_12 sf <29>S 59 140 :M (in )S f4_12 sf (G)S f0_12 sf S 88 140 :M (is )S 100 140 :M -.22(the )A 118 140 :M -.163(same )A 146 140 :M (as )S 160 140 :M -.22(the )A 178 140 :M -.122(ancestor )A 221 140 :M -.146(relations )A 265 140 :M -.132(among )A 301 140 :M -.22(the )A 319 140 :M f2_12 sf .993(X)A f0_12 sf .344 .034( )J 333 140 :M -.145(variables )A 379 140 :M -.167(in )A 392 140 :M .053(Transform\()A f4_12 sf .084(G)A f0_12 sf .081 .008(\). )J 467 140 :M -.327(After)A 59 158 :M -.02(stage \(1\) in creating )A f2_12 sf -.029<51D5>A f0_12 sf -.018(, if X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.02( is not )A 232 158 :M -.163(an )A 247 158 :M -.122(ancestor )A 290 158 :M (of )S 304 158 :M f2_12 sf (Z)S f0_12 sf ( )S 316 158 :M -.167(in )A 329 158 :M -.022(Transform\()A f4_12 sf (G)S f0_12 sf (\) )S 400 158 :M (\(or )S 418 158 :M -.167(in )A 431 158 :M f4_12 sf .517(G)A f0_12 sf .641 .064J 456 158 :M -.111(but )A 475 158 :M (has)S 59 176 :M -.163(an )A 74 176 :M -.162(occurrence )A 129 176 :M -.167(in )A 142 176 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 160 176 :M -.249(that )A 181 176 :M (is )S 193 176 :M -.326(a )A 202 176 :M -.072(collider, )A 245 176 :M -.334(it )A 255 176 :M (follows )S 295 176 :M -.249(that )A 316 176 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 333 176 :M .264 .026(was )J 356 176 :M -.13(added )A 388 176 :M -.167(to )A 402 176 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 421 176 :M (by )S 438 176 :M -.182(replacing )A 486 176 :M (a)S 59 193 :M -.039(subsequence of )A f2_12 sf -.065(Q)A f0_12 sf -.034( by a corresponding correlated error )A 361 193 :M -.163(trek )A 383 193 :M -.121(sequence )A 430 193 :M .403(A f0_12 sf .325(,X)A f0_7 sf 0 3 rm .152(s)A 0 -3 rm f0_12 sf (>)S 59 212 :M -.042(in Transform\()A f4_12 sf -.07(G)A f0_12 sf -.041(\). Hence any such X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.041( lies between some pair )A 350 212 :M (of )S 364 212 :M -.163(vertices )A 404 212 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 419 212 :M -.109(and )A 440 212 :M .37(X)A f0_7 sf 0 3 rm .116(s)A 0 -3 rm f0_12 sf .128 .013( )J 456 212 :M -.249(that )A 477 212 :M -.323(are)A 59 229 :M -.107(adjacent in )A f2_12 sf -.217(Q)A f0_12 sf -.139(. )A 129 229 :M -.139(Because )A 172 229 :M -.129(every )A 202 229 :M -.163(vertex )A 235 229 :M -.167(in )A 248 229 :M .403(A f0_12 sf .303 .03(, )J 294 229 :M .347(X)A f0_7 sf 0 3 rm .109(s)A 0 -3 rm f0_12 sf .356 .036(> )J 317 229 :M -.167(in )A 330 229 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 348 229 :M -.187(\(except )A 386 229 :M (for )S 404 229 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 419 229 :M -.109(and )A 440 229 :M .293(X)A f0_7 sf 0 3 rm .092(s)A 0 -3 rm f0_12 sf .215 .021(\) )J 460 229 :M (has )S 480 229 :M -.326(an)A 59 248 :M .018 .002(index less than r and s, and X)J f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf ( is )S 219 248 :M -.111(not )A 238 248 :M -.163(an )A 253 248 :M -.122(ancestor )A 296 248 :M (of )S 310 248 :M f2_12 sf (Z)S f0_12 sf ( )S 322 248 :M -.167(in )A 335 248 :M f4_12 sf .624(G)A f0_12 sf .6 .06J 356 248 :M -.334(it )A 366 248 :M (follows )S 406 248 :M -.08(from )A 433 248 :M -.22(the )A 451 248 :M -.092(ordering)A 59 266 :M -.068(of the variables that we chose, that X)A f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf -.078( and X)A f0_7 sf 0 3 rm (s)S 0 -3 rm f0_12 sf -.069( are not ancestors of )A 370 266 :M f2_12 sf (Z)S f0_12 sf ( )S 382 266 :M -.167(in )A 395 266 :M f4_12 sf .624(G)A f0_12 sf .6 .06J 416 266 :M -.139(Because )A 459 266 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 474 266 :M -.163(and)A 59 284 :M .37(X)A f0_7 sf 0 3 rm .116(s)A 0 -3 rm f0_12 sf .128 .013( )J 75 284 :M -.215(are )A 93 284 :M (on )S 109 284 :M .306 .031(U )J 122 284 :M -.111(but )A 141 284 :M -.111(not )A 160 284 :M -.071(ancestors )A 208 284 :M (of )S 222 284 :M f2_12 sf (Z)S f0_12 sf ( )S 234 284 :M -.167(in )A 247 284 :M f4_12 sf .624(G)A f0_12 sf .6 .06J 268 284 :M -.196(there )A 295 284 :M (is )S 307 284 :M -.326(a )A 316 284 :M -.047(subpath )A 357 284 :M (of )S 371 284 :M .306 .031(U )J 384 284 :M -.249(that )A 406 284 :M (is )S 419 284 :M -.326(a )A 429 284 :M -.205(directed )A 471 284 :M -.22(path)A 59 302 :M -.028(from X)A f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf -.023( to X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.025( and a )A 156 302 :M -.047(subpath )A 197 302 :M (of )S 211 302 :M .306 .031(U )J 224 302 :M -.249(that )A 245 302 :M (is )S 257 302 :M -.326(a )A 266 302 :M -.205(directed )A 307 302 :M -.165(path )A 331 302 :M .37(X)A f0_7 sf 0 3 rm .116(s)A 0 -3 rm f0_12 sf .128 .013( )J 347 302 :M -.167(to )A 360 302 :M .876(X)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(, )J 381 302 :M (or )S 395 302 :M -.247(vice )A 418 302 :M .43 .043(versa. )J 451 302 :M (In )S 465 302 :M -.263(either)A 59 320 :M (case, in )S f4_12 sf (G)S f0_12 sf (\325, X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( is an ancestor of X)S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( and X)S f0_7 sf 0 3 rm (s)S 0 -3 rm f0_12 sf ( is an ancestor of )S 344 320 :M .876(X)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(, )J 365 320 :M (or )S 379 320 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 394 320 :M (is )S 406 320 :M -.163(an )A 421 320 :M -.122(ancestor )A 464 320 :M (of )S 478 320 :M .596(X)A f0_7 sf 0 3 rm (2)S 0 -3 rm 59 338 :M f0_12 sf -.015(and X)A f0_7 sf 0 3 rm (s)S 0 -3 rm f0_12 sf -.012( is an ancestor of X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.013(. Because in )A 249 338 :M f4_12 sf .624(G)A f0_12 sf .6 .06J 270 338 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 285 338 :M (is )S 297 338 :M -.163(an )A 312 338 :M -.122(ancestor )A 355 338 :M (of )S 369 338 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 386 338 :M -.109(and )A 407 338 :M .37(X)A f0_7 sf 0 3 rm .116(s)A 0 -3 rm f0_12 sf .128 .013( )J 423 338 :M -.163(an )A 438 338 :M -.122(ancestor )A 481 338 :M (of)S 59 356 :M .057(X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf .105 .01( or vice-versa, )J 144 356 :M -.109(and )A 165 356 :M (k )S 175 356 :M .211 .021(< )J 186 356 :M (r )S 194 356 :M -.109(and )A 215 356 :M 1.111 .111(s, )J 228 356 :M -.334(it )A 238 356 :M (follows )S 278 356 :M -.08(from )A 305 356 :M -.22(the )A 323 356 :M -.081(ordering )A 367 356 :M (of )S 381 356 :M -.22(the )A 399 356 :M -.145(variables )A 445 356 :M -.249(that )A 466 356 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 483 356 :M (is)S 59 374 :M .065 .006(also an ancestor of X)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .038 .004( or X)J f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf .025 .002( in )J f4_12 sf (G)S f0_12 sf .068 .007(\325. Hence X)J f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .031 .003( is an )J 302 374 :M -.122(ancestor )A 345 374 :M (of )S 359 374 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 376 374 :M (or )S 390 374 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 407 374 :M -.167(in )A 420 374 :M .056(Transform\()A f4_12 sf .088(G)A f0_12 sf .071(\).)A 59 392 :M -.072(In the example, in Transform\()A f4_12 sf -.125(G)A f0_12 sf -.075(\) X)A f0_7 sf 0 3 rm -.05(4)A 0 -3 rm f0_12 sf -.067( is not an ancestor of the empty set but is an ancestor of)A 59 410 :M (X)S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.013(, and it is between two vertices X)A f0_7 sf 0 3 rm (5)S 0 -3 rm f0_12 sf -.015( and X)A f0_7 sf 0 3 rm (6)S 0 -3 rm f0_12 sf -.014( which also )A 328 410 :M -.215(are )A 346 410 :M -.111(not )A 365 410 :M -.071(ancestors )A 413 410 :M (of )S 427 410 :M -.22(the )A 445 410 :M -.199(empty )A 478 410 :M -.164(set)A 59 428 :M .13 .013(but are ancestors of X)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .075 .008( or X)J f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf (.)S 59 458 :M -.036(If there is some vertex X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.029( in )A f2_12 sf -.051<51D5>A f0_12 sf -.034( that is not an ancestor of )A f2_12 sf -.061(Z)A f0_12 sf (,)S f2_12 sf ( )S f0_12 sf -.038(but occurs in )A 411 458 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 429 458 :M (as )S 443 458 :M -.326(a )A 452 458 :M -.081(collider,)A 59 476 :M -.101(suppose without loss of generality that there is a vertex that )A 342 476 :M (is )S 354 476 :M -.163(an )A 369 476 :M -.122(ancestor )A 412 476 :M (of )S 426 476 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 443 476 :M -.111(but )A 462 476 :M -.111(not )A 481 476 :M (of)S 59 494 :M f2_12 sf -.154(Z)A f0_12 sf -.084(, that occurs as a collider in )A f2_12 sf -.129<51D5>A f0_12 sf -.093(. Let X)A f0_7 sf 0 3 rm -.06(a)A 0 -3 rm f0_12 sf -.089( be the last occurrence of a collider in )A 429 494 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 447 494 :M -.249(that )A 468 494 :M (is )S 480 494 :M -.326(an)A 59 512 :M -.01(ancestor of X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.009( but not of )A f2_12 sf (Z)S f0_12 sf -.009(, if there is one, )A 266 512 :M -.072(otherwise )A 316 512 :M -.331(let )A 331 512 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 347 512 :M .211 .021(= )J 358 512 :M .876(X)A f0_7 sf 0 3 rm .354(1)A 0 -3 rm f0_12 sf .552 .055(. )J 379 512 :M -.083(Step )A 404 512 :M (\(2\) )S 422 512 :M -.167(in )A 435 512 :M -.093(forming )A 477 512 :M f2_12 sf .673<51D5>A 59 529 :M f0_12 sf .566 .057(and )J f2_12 sf .221<50D5>A f0_12 sf .117 .012( )J 96 529 :M (is )S 108 529 :M -.167(to )A 121 529 :M -.233(replace )A 158 529 :M -.22(the )A 176 529 :M -.058(subsequence )A 240 529 :M .466(A f0_12 sf .352(,X)A f0_7 sf 0 3 rm .188(a)A 0 -3 rm f0_12 sf .536 .054(> )J 303 529 :M (by )S 319 529 :M .449( )J 366 529 :M -.164(if )A 377 529 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 393 529 :M cF f1_12 sf 1.151A sf 1.151A f0_7 sf 0 3 rm .662(1)A 0 -3 rm f0_12 sf 1.032 .103(, )J 427 529 :M -.109(and )A 448 529 :M -.205(replacing)A 59 548 :M -.22(the )A 78 548 :M -.049(corresponding )A 151 548 :M -.065(paths )A 181 548 :M -.167(in )A 195 548 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 213 548 :M (by )S 230 548 :M -.326(a )A 240 548 :M -.205(directed )A 282 548 :M -.165(path )A 307 548 :M -.08(from )A 335 548 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 352 548 :M -.167(to )A 366 548 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 385 548 :M -.164(if )A 398 548 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 416 548 :M cF f1_12 sf 1.151A sf 1.151A f0_7 sf 0 3 rm .662(1)A 0 -3 rm f0_12 sf 1.032 .103(. )J 452 548 :M (\(Such )S 486 548 :M (a)S 59 566 :M -.205(directed )A 102 566 :M -.165(path )A 128 566 :M -.055(exists )A 161 566 :M -.164(if )A 174 566 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 192 566 :M cF f1_12 sf 1.045A sf 1.045A f0_7 sf 0 3 rm .601(1)A 0 -3 rm f0_12 sf .515 .052( )J 224 566 :M -.139(because )A 267 566 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 285 566 :M (is )S 299 566 :M -.163(an )A 316 566 :M -.122(ancestor )A 361 566 :M (of )S 377 566 :M .721(X)A f0_7 sf 0 3 rm .291(1)A 0 -3 rm f0_12 sf .693 .069(.\) )J 405 566 :M -.083(This )A 433 566 :M -.092(removes )A 480 566 :M -.497(all)A 59 584 :M -.099(occurences of )A 128 584 :M -.163(vertices )A 168 584 :M -.139(between )A 211 584 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 228 584 :M -.109(and )A 249 584 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 265 584 :M -.249(that )A 286 584 :M -.215(are )A 304 584 :M -.111(not )A 323 584 :M -.071(ancestors )A 371 584 :M (of )S 385 584 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 401 584 :M -.111(but )A 420 584 :M -.215(are )A 438 584 :M -.146(colliders )A 482 584 :M -.334(in)A 59 601 :M f2_12 sf .215<51D5>A f0_12 sf .502 .05(. In the example, X)J f0_7 sf 0 3 rm .1(a)A 0 -3 rm f0_12 sf .287 .029( = X)J f0_7 sf 0 3 rm .113(4)A 0 -3 rm f0_12 sf .394 .039(, and after step 2, )J f2_12 sf .215<51D5>A f0_12 sf .363 .036( = and )J f2_12 sf .182<50D5>A f0_12 sf .363 .036( = A f0_12 sf ( )S 474 601 :M 1.049(X)A f0_7 sf 0 3 rm .424(4)A 0 -3 rm f0_12 sf (,)S 59 619 :M .44(X)A f0_7 sf 0 3 rm .178(4)A 0 -3 rm f0_12 sf .139 .014( )J f1_12 sf .602A f0_12 sf .438 .044( T)J f0_7 sf 0 3 rm .178(46)A 0 -3 rm f0_12 sf .139 .014( )J f1_12 sf .602A f0_12 sf .494 .049( X)J f0_7 sf 0 3 rm .178(6)A 0 -3 rm f0_12 sf .573 .057(, X)J f0_7 sf 0 3 rm .178(6)A 0 -3 rm f0_12 sf .139 .014( )J f1_12 sf .602A f0_12 sf .494 .049( X)J f0_7 sf 0 3 rm .178(2)A 0 -3 rm f0_12 sf .497(>.)A 59 650 :M -.07(By definition, every vertex that occurs as a collider between X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf -.079( and X)A f0_7 sf 0 3 rm -.052(2)A 0 -3 rm f0_12 sf -.056( in )A f2_12 sf -.098<51D5>A f0_12 sf ( )S 425 650 :M (is )S 437 650 :M -.163(an )A 452 650 :M -.139(ancestor)A 59 668 :M (of )S f2_12 sf (Z)S f0_12 sf -.015( or of X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.015(. Let X)A f0_7 sf 0 3 rm (b)S 0 -3 rm f0_12 sf -.014( be the first vertex )A 249 668 :M -.195(after )A 274 668 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 290 668 :M -.167(in )A 303 668 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 321 668 :M -.249(that )A 342 668 :M (is )S 354 668 :M -.163(an )A 369 668 :M -.122(ancestor )A 412 668 :M (of )S 426 668 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 443 668 :M -.111(but )A 462 668 :M -.111(not )A 481 668 :M (of)S 59 686 :M f2_12 sf (Z)S f0_12 sf -.021(, if there is )A 122 686 :M .481 .048(one, )J 147 686 :M -.072(otherwise )A 197 686 :M -.331(let )A 212 686 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 229 686 :M .211 .021(= )J 240 686 :M .876(X)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(. )J 261 686 :M -.083(Step )A 286 686 :M (\(3\) )S 304 686 :M -.167(in )A 317 686 :M -.093(forming )A 359 686 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 377 686 :M -.109(and )A 398 686 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 415 686 :M (is )S 427 686 :M -.167(to )A 440 686 :M -.233(replace )A 477 686 :M -.33(the)A endp %%Page: 12 12 %%BeginPageSetup initializepage (peter; page: 12 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (12)S gR gS 0 0 552 730 rC 59 55 :M f0_12 sf 1.136 .114(subsequence A f0_12 sf .201(,X)A f0_7 sf 0 3 rm .121(2)A 0 -3 rm f0_12 sf .513 .051(> by if X)J f0_7 sf 0 3 rm .121(b)A 0 -3 rm f0_12 sf .094 .009( )J f1_12 sf .227A f0_12 sf .103 .01( )J 287 55 :M .876(X)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(, )J 308 55 :M -.109(and )A 329 55 :M -.182(replacing )A 376 55 :M -.22(the )A 394 55 :M -.049(corresponding )A 466 55 :M -.082(paths)A 59 74 :M -.167(in )A 73 74 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 91 74 :M (by )S 108 74 :M -.326(a )A 118 74 :M -.205(directed )A 160 74 :M -.165(path )A 185 74 :M -.08(from )A 213 74 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 231 74 :M -.167(to )A 245 74 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 263 74 :M -.164(if )A 275 74 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 293 74 :M cF f1_12 sf 1.151A sf 1.151A f0_7 sf 0 3 rm .662(2)A 0 -3 rm f0_12 sf 1.032 .103(. )J 329 74 :M -.083(This )A 356 74 :M -.092(removes )A 402 74 :M -.331(all )A 419 74 :M -.117(occurrences )A 481 74 :M (of)S 59 92 :M -.146(colliders )A 104 92 :M -.139(between )A 148 92 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 166 92 :M -.109(and )A 188 92 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 206 92 :M -.249(that )A 228 92 :M -.215(are )A 247 92 :M -.111(not )A 267 92 :M -.071(ancestors )A 316 92 :M (of )S 332 92 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(. )J 350 92 :M -.081(Note )A 379 92 :M -.249(that )A 402 92 :M -.331(all )A 419 92 :M -.117(occurrences )A 481 92 :M (of)S 59 110 :M -.089(colliders that are left are between X)A f0_7 sf 0 3 rm -.059(a)A 0 -3 rm f0_12 sf -.102( and X)A f0_7 sf 0 3 rm -.067(b)A 0 -3 rm f0_12 sf -.095(, and every occurrence of )A 388 110 :M -.326(a )A 397 110 :M -.206(collider )A 436 110 :M -.139(between )A 479 110 :M .169(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm 59 128 :M f0_12 sf -.016(and X)A f0_7 sf 0 3 rm (b)S 0 -3 rm f0_12 sf -.013( is an ancestor of )A f2_12 sf (Z)S f0_12 sf -.014( by construction. In the example, X)A f0_7 sf 0 3 rm (b)S 0 -3 rm f0_12 sf -.015( = X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.013(, and after step )A 455 128 :M .724 .072(\(3\), )J 477 128 :M f2_12 sf .673<51D5>A 59 146 :M f0_12 sf .098 .01(and )J f2_12 sf .038<50D5>A f0_12 sf .151 .015( are unchanged.)J 59 176 :M -.091(We will now show that every path between a pair of )A 309 176 :M -.145(variables )A 355 176 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 372 176 :M -.109(and )A 393 176 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 410 176 :M -.167(in )A 423 176 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 440 176 :M -.108(d-connects)A 59 194 :M .116(X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .165 .016( and X)J f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .162 .016( given )J f2_12 sf .107(Z)A f0_12 sf .079(\\{X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .078(,X)A f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .21 .021(}. If the path between X)J f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .165 .016( and X)J f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .125 .013( is also in )J f2_12 sf .098(P)A f0_12 sf .073 .007(, )J 406 194 :M -.165(then )A 430 194 :M -.334(it )A 440 194 :M -.108(d-connects)A 59 212 :M (X)S f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .058 .006( and X)J f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .057 .006( given )J f2_12 sf (Z)S f0_12 sf .028(\\{X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .027(,X)A f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .073 .007(} because every path in )J f2_12 sf (P)S f0_12 sf .071 .007( has this property. )J 398 212 :M (If )S 410 212 :M -.22(the )A 428 212 :M -.165(path )A 452 212 :M -.163(between)A 59 230 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 76 230 :M -.109(and )A 97 230 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 114 230 :M (is )S 126 230 :M -.111(not )A 145 230 :M -.167(in )A 158 230 :M f2_12 sf .918(P)A f0_12 sf .683 .068(, )J 174 230 :M -.111(but )A 193 230 :M .264 .026(was )J 216 230 :M -.13(added )A 248 230 :M -.167(in )A 262 230 :M -.082(step )A 286 230 :M (\(1\) )S 305 230 :M (of )S 320 230 :M -.22(the )A 339 230 :M -.146(formation )A 390 230 :M (of )S 405 230 :M f2_12 sf .875<50D5>A f0_12 sf .843 .084(, )J 427 230 :M -.165(then )A 452 230 :M -.22(the )A 471 230 :M -.22(path)A 59 248 :M -.139(between )A 104 248 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 124 248 :M -.109(and )A 148 248 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 168 248 :M (is )S 183 248 :M -.326(a )A 195 248 :M -.196(correlated )A 248 248 :M -.061(error )A 278 248 :M .231 .023(trek, )J 307 248 :M -.065(which )A 343 248 :M -.097(d-connects )A 401 248 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 421 248 :M -.109(and )A 445 248 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 465 248 :M -.165(given)A 59 266 :M f2_12 sf .135(Z)A f0_12 sf .1(\\{X)A f0_7 sf 0 3 rm .059(u)A 0 -3 rm f0_12 sf .099(,X)A f0_7 sf 0 3 rm .059(v)A 0 -3 rm f0_12 sf .26 .026(} because no )J 171 266 :M -.33(T )A 182 266 :M -.205(variable )A 223 266 :M (is )S 235 266 :M -.167(in )A 248 266 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(. )J 264 266 :M (If )S 276 266 :M -.22(the )A 294 266 :M -.165(path )A 318 266 :M -.139(between )A 361 266 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 378 266 :M -.109(and )A 399 266 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 416 266 :M (is )S 428 266 :M -.111(not )A 447 266 :M -.167(in )A 460 266 :M f2_12 sf .918(P)A f0_12 sf .683 .068(, )J 476 266 :M -.167(but)A 59 284 :M .173 .017(was added in step \(2\) of the formation of )J f2_12 sf .065<50D5>A f0_12 sf .157 .016(, then X)J f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .103 .01( = X)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .13 .013(, X)J f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .103 .01( = X)J f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf .121 .012(, and )J 410 284 :M -.22(the )A 428 284 :M -.165(path )A 452 284 :M -.163(between)A 59 302 :M -.078(X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf -.048( and X)A f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf -.043( is a directed path from X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf -.046( to )A 247 302 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 264 302 :M -.249(that )A 285 302 :M (does )S 311 302 :M -.111(not )A 330 302 :M -.189(contain )A 368 302 :M -.109(any )A 389 302 :M -.219(member )A 431 302 :M (of )S 445 302 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(. )J 461 302 :M -.16(Hence)A 59 320 :M -.069(the path d-connects X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf -.073( and X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf -.064( given )A f2_12 sf -.11(Z)A f0_12 sf -.066(. Similarly, if the path between path between )A 457 320 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 474 320 :M -.163(and)A 59 338 :M (X)S f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .026 .003( is not in )J f2_12 sf (P)S f0_12 sf .041 .004(, but was added in step )J 237 338 :M (\(3\) )S 255 338 :M (of )S 269 338 :M -.22(the )A 287 338 :M -.146(formation )A 337 338 :M .387(of)A f2_12 sf .232 .023( )J 352 338 :M .875<50D5>A f0_12 sf .843 .084(, )J 373 338 :M -.165(then )A 397 338 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 414 338 :M .211 .021(= )J 425 338 :M .876(X)A f0_7 sf 0 3 rm .354(b)A 0 -3 rm f0_12 sf .552 .055(, )J 446 338 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 463 338 :M .211 .021(= )J 474 338 :M 1.049(X)A f0_7 sf 0 3 rm .424(2)A 0 -3 rm f0_12 sf (,)S 59 356 :M -.084(and the path between X)A f0_7 sf 0 3 rm -.057(u)A 0 -3 rm f0_12 sf -.087( and X)A f0_7 sf 0 3 rm -.057(v)A 0 -3 rm f0_12 sf -.075( is a directed path )A 296 356 :M -.08(from )A 323 356 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 340 356 :M -.167(to )A 353 356 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 370 356 :M -.249(that )A 391 356 :M (does )S 417 356 :M -.111(not )A 436 356 :M -.189(contain )A 474 356 :M -.163(any)A 59 374 :M -.03(member of )A f2_12 sf (Z)S f0_12 sf -.027(. Hence the path d-connects X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf -.029( and X)A f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf -.025( given )A f2_12 sf (Z)S f0_12 sf (.)S 59 404 :M -.092(We will now show that every vertex that occurs )A 288 404 :M (as )S 302 404 :M -.326(a )A 311 404 :M -.206(collider )A 350 404 :M -.167(in )A 363 404 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 381 404 :M (has )S 401 404 :M -.326(a )A 410 404 :M -.131(descendant )A 466 404 :M -.167(in )A 479 404 :M f2_12 sf .724(Z)A f0_12 sf (,)S 59 422 :M -.072(and every vertex that occurs as a non-collider in )A f2_12 sf -.103<51D5>A f0_12 sf -.063( is not in )A f2_12 sf -.123(Z)A f0_12 sf -.083(. Every )A 392 422 :M -.163(vertex )A 425 422 :M -.249(that )A 446 422 :M -.052(occurs )A 481 422 :M (as)S 59 440 :M -.326(a )A 69 440 :M -.206(collider )A 109 440 :M -.167(in )A 123 440 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 142 440 :M (is )S 155 440 :M -.163(an )A 171 440 :M -.122(ancestor )A 215 440 :M (of )S 230 440 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 247 440 :M -.139(because )A 289 440 :M (steps )S 318 440 :M (\(2\) )S 337 440 :M -.109(and )A 359 440 :M (\(3\) )S 378 440 :M -.167(in )A 392 440 :M -.22(the )A 411 440 :M -.146(formation )A 462 440 :M (of )S 477 440 :M f2_12 sf .673<51D5>A 59 458 :M f0_12 sf -.114(removed all occurrences of colliders that were not ancestors )A 344 458 :M (of )S 358 458 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(. )J 374 458 :M -.13(Every )A 406 458 :M -.163(vertex )A 439 458 :M -.249(that )A 460 458 :M -.062(occurs)A 59 476 :M (as )S 73 476 :M -.326(a )A 82 476 :M -.137(non-collider )A 143 476 :M -.167(in )A 156 476 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 169 476 :M -.109(and )A 190 476 :M (as )S 204 476 :M -.326(a )A 213 476 :M -.137(non-collider )A 274 476 :M -.167(in )A 287 476 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 305 476 :M (is )S 317 476 :M -.111(not )A 336 476 :M -.167(in )A 350 476 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 367 476 :M -.139(because )A 409 476 :M -.129(every )A 440 476 :M -.163(vertex )A 474 476 :M -.331(that)A 59 494 :M -.068(occurs as a non-collider in )A f2_12 sf -.138(Q)A f0_12 sf -.06( is not in )A f2_12 sf -.118(Z)A f0_12 sf -.077(. The only )A 300 494 :M -.163(vertices )A 340 494 :M -.249(that )A 361 494 :M -.22(may )A 385 494 :M -.129(occur )A 415 494 :M (as )S 429 494 :M -.109(non-colliders)A 59 512 :M -.167(in )A 72 512 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 90 512 :M -.111(but )A 109 512 :M -.111(not )A 128 512 :M -.167(in )A 141 512 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 154 512 :M -.215(are )A 172 512 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 188 512 :M -.109(and )A 209 512 :M .876(X)A f0_7 sf 0 3 rm .354(b)A 0 -3 rm f0_12 sf .552 .055(. )J 230 512 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 246 512 :M (is )S 258 512 :M -.111(not )A 277 512 :M -.167(in )A 290 512 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 306 512 :M -.139(because )A 347 512 :M -.219(either )A 377 512 :M -.334(it )A 387 512 :M (is )S 399 512 :M -.197(equal )A 428 512 :M -.167(to )A 441 512 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 459 512 :M (or )S 474 512 :M 1.049(X)A f0_7 sf 0 3 rm .424(2)A 0 -3 rm f0_12 sf (,)S 59 530 :M -.049(neither of which is in )A f2_12 sf -.085(Z)A f0_12 sf -.042(, or it is )A 211 530 :M -.111(not )A 230 530 :M -.163(an )A 245 530 :M -.122(ancestor )A 288 530 :M (of )S 302 530 :M f2_12 sf (Z)S f0_12 sf ( )S 314 530 :M (by )S 330 530 :M -.024(construction. )A 396 530 :M -.066(Similarly, )A 447 530 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 464 530 :M (is )S 476 530 :M -.167(not)A 59 548 :M .269 .027(in )J f2_12 sf .227(Z)A f0_12 sf (.)S 59 578 :M -.128(Hence )A 93 578 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 111 578 :M (is )S 123 578 :M -.326(a )A 132 578 :M -.121(sequence )A 179 578 :M (of )S 193 578 :M -.065(paths )A 222 578 :M -.249(that )A 243 578 :M -.046(satisfy )A 278 578 :M -.097(properties )A 329 578 :M .486 .049(\(i\), )J 348 578 :M .23 .023(\(ii\), )J 370 578 :M -.109(and )A 391 578 :M (\(iii\). )S 416 578 :M -.164(It )A 427 578 :M (follows )S 468 578 :M -.107(from)A 59 596 :M -.025(Lemma 3.3.1+ that X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.025( and X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.023( are d-connected given )A f2_12 sf (Z)S f0_12 sf -.023( in Transform\()A f4_12 sf (G)S f0_12 sf (\).)S 59 626 :M -.025(By contraposition, since X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.03( and )A 214 626 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 231 626 :M -.215(are )A 249 626 :M -.117(d-separated )A 307 626 :M -.167(in )A 320 626 :M .053(Transform\()A f4_12 sf .084(G)A f0_12 sf .081 .008(\), )J 395 626 :M -.165(they )A 419 626 :M -.215(are )A 437 626 :M -.129(d-separated)A 59 644 :M -.132(given )A 91 644 :M f2_12 sf (Z)S f0_12 sf ( )S 105 644 :M -.167(in )A 120 644 :M f4_12 sf .624(G)A f0_12 sf .6 .06J 143 644 :M -.139(Because )A 188 644 :M f4_12 sf .19(G)A f0_12 sf .139 .014J 207 644 :M (is )S 221 644 :M -.22(the )A 241 644 :M -.205(directed )A 284 644 :M -.064(graph )A 317 644 :M (of )S 333 644 :M -.326(a )A 344 644 :M -.276(latent )A 375 644 :M -.205(variable )A 418 644 :M -.199(model )A 453 644 :M -.162<4CD520>A 470 644 :M -.11(with)A 59 661 :M -.179(correlation )A 115 661 :M -.22(matrix )A 151 661 :M -.249(that )A 174 661 :M (has )S 196 661 :M -.206(marginal )A 243 661 :M f1_12 sf .776(S)A f0_12 sf .436<28>A f2_12 sf .946(X)A f0_12 sf .909 .091(\), )J 279 661 :M (no )S 297 661 :M -.196(correlated )A 350 661 :M .604 .06(errors, )J 389 661 :M -.109(and )A 413 661 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 433 661 :M -.109(and )A 457 661 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 477 661 :M -.323(are)A endp %%Page: 13 13 %%BeginPageSetup initializepage (peter; page: 13 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (13)S gR gS 0 0 552 730 rC 59 55 :M f0_12 sf -.117(d-separated )A 117 55 :M -.132(given )A 147 55 :M f2_12 sf (Z)S f0_12 sf ( )S 159 55 :M -.167(in )A 173 55 :M f4_12 sf .624(G)A f0_12 sf .6 .06J 195 55 :M -.334(it )A 206 55 :M (follows )S 247 55 :M -.08(from )A 275 55 :M -.187(Theorem )A 322 55 :M (3 )S 333 55 :M -.04(\(Spirtes )A 375 55 :M (1995\) )S 408 55 :M -.249(that )A 430 55 :M f1_12 sf .224(r)A f0_7 sf 0 3 rm .134(XY.)A 0 -3 rm f2_7 sf 0 3 rm .159(Z)A 0 -3 rm f0_12 sf .102 .01( )J 459 55 :M .211 .021(= )J 471 55 :M (0 )S 482 55 :M -.334(in)A 59 73 :M f1_12 sf .513(S)A f0_12 sf .288<28>A f2_12 sf .626(X)A f0_12 sf .555 .055(\). )J f1_12 sf <5C>S 59 104 :M f2_12 sf 1.973 .197(Theorem )J 113 104 :M .387(3:)A f0_12 sf .232 .023( )J 128 104 :M (If )S 140 104 :M f4_12 sf .25(G)A f0_12 sf .087 .009( )J 153 104 :M (is )S 165 104 :M -.22(the )A 183 104 :M -.064(graph )A 214 104 :M (of )S 228 104 :M -.223(SEM )A 256 104 :M -.33(L )A 267 104 :M -.083(with )A 292 104 :M -.165(normally )A 338 104 :M -.12(distributed )A 392 104 :M -.196(correlated )A 442 104 :M (errors )S 474 104 :M -.163(and)A 59 121 :M -.088(marginal correlation matrix )A f1_12 sf -.129(S)A f0_12 sf -.089(\(X\), and X is d-connected to Y given )A f2_12 sf -.146(Z)A f0_12 sf -.093( in )A 400 121 :M -.022(Transform\()A f4_12 sf (G)S f0_12 sf (\) )S 471 121 :M -.22(then)A 59 139 :M -.067(L does not linearly entail that )A f1_12 sf -.099(r)A f0_7 sf 0 3 rm -.059(XY.)A 0 -3 rm f2_7 sf 0 3 rm -.07(Z)A 0 -3 rm f0_12 sf -.082( = 0.)A 59 170 :M .839 .084(Proof. )J 95 170 :M (If )S 108 170 :M .306 .031(X )J 122 170 :M (is )S 135 170 :M -.111(not )A 155 170 :M -.117(d-separated )A 214 170 :M -.08(from )A 242 170 :M -.663(Y )A 255 170 :M -.132(given )A 287 170 :M f2_12 sf (Z)S f0_12 sf ( )S 301 170 :M -.167(in )A 316 170 :M .053(Transform\()A f4_12 sf .084(G)A f0_12 sf .081 .008(\), )J 393 170 :M -.165(then )A 419 170 :M (by )S 437 170 :M -.187(Theorem )A 485 170 :M (3)S 59 187 :M -.04(\(Spirtes )A 100 187 :M (1995\) )S 132 187 :M -.196(there )A 159 187 :M (is )S 171 187 :M -.326(a )A 181 187 :M -.226(parameterization )A 263 187 :M (of )S 278 187 :M -.022(Transform\()A f4_12 sf (G)S f0_12 sf (\) )S 350 187 :M -.083(with )A 376 187 :M -.179(correlation )A 431 187 :M -.22(matrix )A 466 187 :M f1_12 sf .373(S)A f0_12 sf .438(\(X\))A 59 205 :M (such )S 87 205 :M -.249(that )A 110 205 :M f1_12 sf .224(r)A f0_7 sf 0 3 rm .134(XY.)A 0 -3 rm f2_7 sf 0 3 rm .159(Z)A 0 -3 rm f0_12 sf .102 .01( )J 140 205 :M f1_12 sf .285A f0_12 sf .13 .013( )J 154 205 :M .833 .083(0. )J 171 205 :M (By )S 192 205 :M -.22(the )A 213 205 :M -.132(convention )A 272 205 :M -.141(adopted )A 316 205 :M (for )S 337 205 :M (new )S 364 205 :M -.276(latent )A 396 205 :M -.145(variables )A 445 205 :M -.131(names )A 482 205 :M -.334(in)A 59 224 :M -.077(Transform\()A f4_12 sf -.122(G)A f0_12 sf -.067(\), no new latent variable )A 240 224 :M .264 .026(was )J 263 224 :M -.274(called )A 294 224 :M -.113(T)A f0_7 sf 0 3 rm -.03(ji)A 0 -3 rm f0_12 sf ( )S 309 224 :M -.062(where )A 342 224 :M -.334(j )A 349 224 :M .691 .069(>i. )J 367 224 :M .258 .026(For )J 388 224 :M -.22(the )A 406 224 :M -.08(sake )A 431 224 :M (of )S 445 224 :M -.221(notational)A 59 242 :M -.101(convenience, we will also use the name T)A f0_7 sf 0 3 rm -.039(ji)A 0 -3 rm f0_12 sf -.088( to refer to T)A f0_7 sf 0 3 rm -.039(ij)A 0 -3 rm f0_12 sf -.096(. In that parameterization,)A 203 257 144 27 rC 347 284 :M psb currentpoint pse 203 257 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4608 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (X) 17 346 sh (b) 1285 346 sh (X) 1673 346 sh (a) 2985 346 sh (T) 3356 346 sh 224 ns (i) 271 442 sh (ij) 1477 442 sh (j) 1960 442 sh (j) 839 719 sh (i) 1112 719 sh (ij) 3189 442 sh (ij) 3550 442 sh (j) 2532 719 sh (i) 2810 719 sh (i) 4439 442 sh 384 /Symbol f1 (=) 484 346 sh (+) 2180 346 sh (+) 3832 346 sh 224 ns (<) 951 719 sh (\271) 2646 719 sh 576 ns (\345) 788 433 sh (\345) 2484 433 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 4128 346 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 4303 346 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 311 :M f0_12 sf -.034(Now define)A 233 326 83 27 rC 316 353 :M psb currentpoint pse 233 326 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2656 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) -16 346 sh (e) 2210 346 sh 224 /Times-Italic f1 (i) 172 442 sh (ij) 1354 442 sh (ij) 1673 442 sh (j) 781 719 sh (i) 1017 719 sh (i) 2500 442 sh 384 ns (a) 1171 346 sh (T) 1500 346 sh 384 /Symbol f1 (=) 386 346 sh (+) 1924 346 sh 224 ns (\271) 874 719 sh 576 ns (\345) 712 433 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 2385 346 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 380 :M f0_12 sf -.104(It follows then that)A 233 395 84 27 rC 317 422 :M psb currentpoint pse 233 395 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2688 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (X) 17 346 sh (b) 1245 346 sh (X) 1591 346 sh 224 ns (i) 250 442 sh (ij) 1416 442 sh (j) 1857 442 sh (j) 862 719 sh (i) 1093 719 sh (i) 2520 442 sh 384 /Symbol f1 (=) 464 346 sh (+) 2046 346 sh 224 ns (<) 953 719 sh 576 ns (\345) 790 433 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 2332 346 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 448 :M f0_12 sf -.061(which is a parameterization of L in which )A f1_12 sf -.083(r)A f0_7 sf 0 3 rm -.05(XY.)A 0 -3 rm f2_7 sf 0 3 rm -.059(Z)A 0 -3 rm f0_12 sf ( )S f1_12 sf -.083A f0_12 sf -.047( 0. )A f1_12 sf <5C>S 59 479 :M f0_12 sf -.17(Because the covariance matrix )A 205 479 :M (of )S 219 479 :M -.22(the )A 237 479 :M -.033(non-error )A 286 479 :M -.145(variables )A 332 479 :M -.167(in )A 345 479 :M -.326(a )A 354 479 :M -.219(linear )A 384 479 :M -.223(SEM )A 412 479 :M (does )S 438 479 :M -.111(not )A 457 479 :M -.13(depend)A 59 497 :M (upon )S 87 497 :M -.092(whether )A 129 497 :M -.22(the )A 147 497 :M -.061(error )A 174 497 :M -.131(terms )A 204 497 :M -.215(are )A 222 497 :M -.165(normally )A 268 497 :M -.027(distributed, )A 327 497 :M -.111(but )A 347 497 :M -.046(depends )A 391 497 :M -.083(only )A 417 497 :M (upon )S 446 497 :M -.22(the )A 465 497 :M -.263(linear)A 59 515 :M -.163(coefficients )A 117 515 :M -.109(and )A 138 515 :M -.22(the )A 156 515 :M -.196(covariance )A 210 515 :M -.22(matrix )A 244 515 :M -.132(among )A 281 515 :M -.22(the )A 300 515 :M .604 .06(errors, )J 337 515 :M -.187(Theorem )A 384 515 :M (2 )S 395 515 :M -.109(and )A 417 515 :M -.187(Theorem )A 464 515 :M (3 )S 475 515 :M -.326(can)A 59 533 :M -.097(obviously be extended to the case where the error terms are not normally distributed.)A endp %%Page: 14 14 %%BeginPageSetup initializepage (peter; page: 14 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (14)S gR gS 0 0 552 730 rC 248 68 :M f2_12 sf .57(Appendix)A 59 98 :M f0_12 sf -.093(Sets of variables and defined terms )A 228 98 :M -.215(are )A 246 98 :M -.167(in )A 259 98 :M -.034(boldface. )A 307 98 :M -.663(A )A 319 98 :M f2_12 sf 2.237 .224(directed )J 369 98 :M .243(graph)A f0_12 sf .119 .012( )J 405 98 :M (is )S 417 98 :M -.163(an )A 432 98 :M -.091(ordered )A 472 98 :M -.218(pair)A 59 116 :M -.068(of a finite set of vertices )A f2_12 sf -.138(V)A f0_12 sf -.079(, and a )A 220 116 :M -.109(set )A 237 116 :M (of )S 251 116 :M -.205(directed )A 292 116 :M -.064(edges )A 323 116 :M f2_12 sf .569(E)A f0_12 sf .388 .039(. )J 339 116 :M -.663(A )A 351 116 :M -.205(directed )A 392 116 :M -.163(edge )A 418 116 :M -.08(from )A 445 116 :M -.663(A )A 457 116 :M -.167(to )A 470 116 :M (B )S 482 116 :M (is)S 59 134 :M -.079(an ordered pair of distinct vertices in )A f2_12 sf -.146(V)A f0_12 sf -.091( in which A )A 339 134 :M (is )S 351 134 :M -.22(the )A 369 134 :M f2_12 sf .284(tail)A f0_12 sf .204 .02( )J 391 134 :M (of )S 405 134 :M -.22(the )A 423 134 :M -.163(edge )A 449 134 :M -.109(and )A 470 134 :M (B )S 482 134 :M (is)S 59 152 :M .211 .021(the )J f2_12 sf .103(head)A f0_12 sf .188 .019(; the edge is )J f2_12 sf .31 .031(out of)J f0_12 sf .158 .016( A and )J f2_12 sf .084(into)A f0_12 sf .159 .016( B, and A is )J f2_12 sf .095(parent)A f0_12 sf .149 .015( of B and B is a )J f2_12 sf .085(child)A f0_12 sf .178 .018( of A. A)J 59 169 :M -.121(sequence )A 106 169 :M (of )S 120 169 :M -.064(edges )A 151 169 :M .741( )J 213 169 :M -.167(in )A 226 169 :M .306 .031(G )J 239 169 :M (is )S 251 169 :M -.163(an )A 266 169 :M f2_12 sf 2.342 .234(undirected )J 330 169 :M 1.189 .119(path )J 359 169 :M f0_12 sf -.164(if )A 370 169 :M -.109(and )A 391 169 :M -.083(only )A 416 169 :M -.164(if )A 427 169 :M -.196(there )A 454 169 :M -.055(exists )A 485 169 :M (a)S 59 187 :M -.121(sequence )A 107 187 :M (of )S 122 187 :M -.163(vertices )A 163 187 :M .618( )J 235 187 :M (such )S 262 187 :M -.249(that )A 284 187 :M (for )S 303 187 :M (1 )S 315 187 :M cF f1_12 sf .038A sf .376 .038( )J 328 187 :M -.334(i )A 337 187 :M .211 .021(< )J 350 187 :M (n )S 362 187 :M -.219(either )A 394 187 :M .137( )J 451 187 :M .211 .021(= )J 464 187 :M -.061(E)A f0_10 sf 0 2 rm (i)S 0 -2 rm f0_12 sf ( )S 480 187 :M (or)S 59 205 :M .197( = E)J f0_10 sf 0 2 rm .071(i)A 0 -2 rm f0_12 sf .255 .026(. A path U is )J f2_12 sf .126(acyclic)A f0_12 sf .201 .02( if no )J 265 205 :M -.163(vertex )A 298 205 :M -.108(occurring )A 347 205 :M (on )S 363 205 :M -.163(an )A 378 205 :M -.163(edge )A 404 205 :M -.167(in )A 417 205 :M -.22(the )A 435 205 :M -.165(path )A 459 205 :M -.062(occurs)A 59 223 :M .221 .022(more than once. A sequence of edges )J 300 223 :M -.167(in )A 313 223 :M .306 .031(G )J 326 223 :M (is )S 338 223 :M -.326(a )A 347 223 :M f2_12 sf 2.237 .224(directed )J 397 223 :M 1.189 .119(path )J 426 223 :M f0_12 sf -.164(if )A 437 223 :M -.109(and )A 458 223 :M -.083(only )A 483 223 :M -.327(if)A 59 241 :M -.006(there exists a sequence of vertices such that for 1 )A 367 241 :M cF f1_12 sf .038A sf .376 .038( )J 378 241 :M -.334(i )A 385 241 :M .211 .021(< )J 396 241 :M (n )S 406 241 :M ( )S 410 241 :M .137( )J 465 241 :M .211 .021(= )J 476 241 :M .499(E)A f0_10 sf 0 2 rm .189(i)A 0 -2 rm f0_12 sf (.)S 59 260 :M -.051(If there is an acyclic directed path from A to B or B = A then A is an )A f2_12 sf -.058(ancestor)A f0_12 sf -.054( of B, and )A 482 260 :M (B)S 59 278 :M (is )S 71 278 :M -.326(a )A 80 278 :M f2_12 sf .445(descendant)A f0_12 sf .233 .023( )J 146 278 :M (of )S 160 278 :M .281 .028(A. )J 176 278 :M -.663(A )A 188 278 :M -.205(directed )A 229 278 :M -.064(graph )A 260 278 :M (is )S 272 278 :M f2_12 sf .572(acyclic)A f0_12 sf .347 .035( )J 315 278 :M -.164(if )A 326 278 :M -.109(and )A 347 278 :M -.083(only )A 372 278 :M -.164(if )A 383 278 :M -.334(it )A 393 278 :M -.123(contains )A 436 278 :M (no )S 453 278 :M -.234(directed)A 59 296 :M -.081(cyclic paths.)A 118 291 :M f0_7 sf (1)S 59 326 :M f0_12 sf .309 .031(Vertex X is a )J f2_12 sf .121(collider)A f0_12 sf .075 .008( )J 171 326 :M (on )S 187 326 :M -.163(an )A 202 326 :M -.282(acyclic )A 238 326 :M -.164(undirected )A 291 326 :M -.165(path )A 315 326 :M .306 .031(U )J 328 326 :M -.167(in )A 341 326 :M -.205(directed )A 382 326 :M -.064(graph )A 413 326 :M .306 .031(G )J 426 326 :M -.164(if )A 437 326 :M -.109(and )A 458 326 :M -.083(only )A 483 326 :M -.327(if)A 59 344 :M -.084(there are two edges on U that are directed into X. )A 295 344 :M -.195(Three )A 326 344 :M -.125(disjoint )A 365 344 :M (sets )S 387 344 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(, )J 405 344 :M f2_12 sf .79(Y)A f0_12 sf .497 .05(, )J 422 344 :M -.109(and )A 443 344 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 459 344 :M f2_12 sf .993(X)A f0_12 sf .344 .034( )J 473 344 :M -.163(and)A 59 362 :M f2_12 sf -.064(Y)A f0_12 sf -.03( are )A f2_12 sf -.041(d-separated)A f0_12 sf -.034( given )A f2_12 sf -.059(Z)A f0_12 sf -.035( in G if and only if there is no acyclic undirected path U from )A 485 362 :M (a)S 59 380 :M -.04(member of )A f2_12 sf -.062(X)A f0_12 sf -.034( to a )A 147 380 :M -.219(member )A 189 380 :M (of )S 203 380 :M f2_12 sf .25(Y)A f0_12 sf .087 .009( )J 216 380 :M (such )S 242 380 :M -.249(that )A 263 380 :M -.129(every )A 293 380 :M -.137(non-collider )A 354 380 :M (on )S 370 380 :M .306 .031(U )J 383 380 :M (is )S 395 380 :M -.111(not )A 414 380 :M -.167(in )A 427 380 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 443 380 :M -.109(and )A 464 380 :M -.161(every)A 59 398 :M -.004(collider on U has a descendant in )A f2_12 sf (Z)S f0_12 sf -.003(. For three disjoint sets )A f2_12 sf (X)S f0_12 sf (, )S f2_12 sf (Y)S f0_12 sf (, )S 375 398 :M -.109(and )A 396 398 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 412 398 :M f2_12 sf .993(X)A f0_12 sf .344 .034( )J 426 398 :M -.109(and )A 447 398 :M f2_12 sf .25(Y)A f0_12 sf .087 .009( )J 460 398 :M -.215(are )A 478 398 :M f2_12 sf 1.336(d-)A 59 416 :M .12(connected)A f0_12 sf .255 .026( given )J f2_12 sf .169(Z)A f0_12 sf .214 .021( in G if and only if )J f2_12 sf .183(X)A f0_12 sf .197 .02( and )J f2_12 sf .183(Y)A f0_12 sf .211 .021( are not )J f2_12 sf .116(d-separated)A f0_12 sf .255 .026( given )J f2_12 sf .169(Z)A f0_12 sf (.)S 59 446 :M -.148(The variables in a linear structural equation model )A 297 446 :M -.131(\(SEM\) )A 333 446 :M -.217(can )A 353 446 :M -.163(be )A 368 446 :M -.142(divided )A 407 446 :M -.167(into )A 429 446 :M (two )S 451 446 :M .67 .067(sets, )J 477 446 :M -.33(the)A 59 464 :M -.105(\322error )A 91 464 :M -.163(variables\323 )A 142 464 :M (or )S 157 464 :M -.105(\322error )A 190 464 :M (terms,\323 )S 230 464 :M -.109(and )A 252 464 :M -.22(the )A 271 464 :M -.09(substantive )A 329 464 :M -.031(variables. )A 380 464 :M -.024(Corresponding )A 456 464 :M -.167(to )A 470 464 :M -.326(each)A 59 482 :M -.097(substantive variable X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.092( is a linear equation with X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.093( on the left hand side of the equation, )A 474 482 :M -.163(and)A 59 499 :M -.084(the direct causes of X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.08( plus the error term )A f1_12 sf -.094(e)A f7_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.086( on the right hand side of the equation. Since we)A 59 518 :M -.089(have no interest in first moments, without loss of generality each )A 368 518 :M -.205(variable )A 409 518 :M -.217(can )A 429 518 :M -.163(be )A 444 518 :M -.038(expressed)A 59 536 :M -.093(as a deviation from its mean.)A 59 638 :M ( )S 59 635.48 -.48 .48 203.48 635 .48 59 635 @a 81 646 :M f8_6 sf (1)S 84 651 :M f0_10 sf -.101(An undirected path is often defined as a sequence of vertices )A 324 651 :M -.135(rather )A 350 651 :M .202 .02(than )J 371 651 :M .056 .006(a )J 379 651 :M -.268(sequence )A 417 651 :M .144 .014(of )J 429 651 :M -.128(edges. )A 457 651 :M (The )S 476 651 :M (two)S 59 663 :M -.047(definitions are essentially equivalent for )A 221 663 :M -.116(acyclic )A 252 663 :M -.337(directed )A 285 663 :M .203 .02(graphs, )J 318 663 :M -.163(because )A 352 663 :M .056 .006(a )J 360 663 :M (pair )S 379 663 :M .144 .014(of )J 391 663 :M -.074(vertices )A 425 663 :M -.126(can )A 442 663 :M .051 .005(be )J 455 663 :M -.258(identified)A 59 675 :M -.029(with a unique edge in the graph. However, a cyclic graph may contain more than one edge between a )A 464 675 :M (pair )S 483 675 :M -.328(of)A 59 687 :M .045 .005(vertices. In that case it is no longer possible to identify a pair of vertices with a unique edge.)J endp %%Page: 15 15 %%BeginPageSetup initializepage (peter; page: 15 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (15)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf .431 .043(Consider, for example, two SEMs S)J f0_7 sf 0 3 rm .072(1)A 0 -3 rm f0_12 sf .239 .024( and S)J f0_7 sf 0 3 rm .072(2)A 0 -3 rm f0_12 sf .217 .022( over )J f2_12 sf .179(X)A f0_12 sf .225 .022( = {X)J f0_7 sf 0 3 rm .072(1)A 0 -3 rm f0_12 sf .233 .023(, X)J f0_7 sf 0 3 rm .072(2)A 0 -3 rm f0_12 sf .233 .023(, X)J f0_7 sf 0 3 rm .072(3)A 0 -3 rm f0_12 sf .285 .028(}, where in )J 437 56 :M -.083(both )A 462 56 :M -.112(SEMs)A 59 74 :M -.079(X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.041( is a direct cause of X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.048( and X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.041( is a direct cause of X)A f0_7 sf 0 3 rm (3)S 0 -3 rm f0_12 sf -.044(. The structural equations)A 440 69 :M f0_7 sf (2)S 444 74 :M f0_12 sf -.111( in )A 460 74 :M -.065(Figure)A 59 92 :M (6 are common to both S)S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( and S)S f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf (.)S 230 109 :M .748(X)A f7_7 sf 0 3 rm .41 .041(1 )J 0 -3 rm f7_12 sf .764 .076(= )J f1_12 sf .455(e)A f7_7 sf 0 3 rm (1)S 0 -3 rm 230 127 :M f0_12 sf .861(X)A f7_7 sf 0 3 rm .378(2)A 0 -3 rm f7_12 sf .599 .06( = )J f1_12 sf .654(b)A f7_7 sf 0 3 rm .378(1)A 0 -3 rm f7_12 sf .294 .029( )J f0_12 sf .861(X)A f7_7 sf 0 3 rm .472 .047( 1)J 0 -3 rm f7_12 sf .599 .06( + )J f1_12 sf .523(e)A f7_7 sf 0 3 rm (2)S 0 -3 rm 230 145 :M f0_12 sf .861(X)A f7_7 sf 0 3 rm .378(3)A 0 -3 rm f7_12 sf .599 .06( = )J f1_12 sf .654(b)A f7_7 sf 0 3 rm .378(2)A 0 -3 rm f7_12 sf .294 .029( )J f0_12 sf .861(X)A f7_7 sf 0 3 rm .472 .047( 2)J 0 -3 rm f7_12 sf .599 .06( + )J f1_12 sf .523(e)A f7_7 sf 0 3 rm (3)S 0 -3 rm 132 170 :M f2_12 sf 2.273 .227(Figure 6: Structural Equations for SEMs S)J f2_7 sf 0 3 rm .365(1)A 0 -3 rm f2_12 sf 1.285 .128( and S)J f2_7 sf 0 3 rm (2)S 0 -3 rm 59 211 :M f0_12 sf -.062(where )A 92 211 :M f1_12 sf .648(b)A f0_7 sf 0 3 rm .469 .047(1 )J 0 -3 rm 106 211 :M f0_12 sf -.109(and )A 127 211 :M f1_12 sf .46(b)A f0_7 sf 0 3 rm .244(2)A 0 -3 rm f0_12 sf .21 .021( )J 142 211 :M -.215(are )A 160 211 :M -.16(free )A 182 211 :M -.163(parameters )A 237 211 :M -.093(ranging )A 277 211 :M -.08(over )A 302 211 :M -.245(real )A 323 211 :M .204 .02(values, )J 361 211 :M -.109(and )A 383 211 :M f1_12 sf .578(e)A f0_7 sf 0 3 rm .384(1)A 0 -3 rm f0_12 sf .329(,)A f0_7 sf 0 3 rm .192 .019( )J 0 -3 rm 400 211 :M f1_12 sf .242(e)A f0_7 sf 0 3 rm .219 .022(2 )J 0 -3 rm 413 211 :M f0_12 sf -.109(and )A 435 211 :M f1_12 sf .104(e)A f0_7 sf 0 3 rm .069(3)A 0 -3 rm f0_12 sf .059 .006( )J 449 211 :M -.215(are )A 468 211 :M -.076(error)A 59 229 :M -.056(terms. In addition suppose that )A f1_12 sf -.063(e)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf (,)S f0_7 sf 0 3 rm ( )S 0 -3 rm f1_12 sf -.063(e)A f0_7 sf 0 3 rm -.031(2 )A 0 -3 rm f0_12 sf -.061(and )A f1_12 sf -.063(e)A f0_7 sf 0 3 rm (3)S 0 -3 rm f0_12 sf -.056( are distributed as multivariate normal. )A 448 229 :M (In )S 462 229 :M .42(S)A f0_7 sf 0 3 rm .22(1)A 0 -3 rm f0_12 sf .189 .019( )J 477 229 :M (we)S 59 248 :M -.166(will )A 81 248 :M -.053(assume )A 120 248 :M -.249(that )A 141 248 :M -.22(the )A 159 248 :M -.179(correlation )A 213 248 :M -.139(between )A 256 248 :M -.245(each )A 281 248 :M -.163(pair )A 303 248 :M (of )S 317 248 :M -.166(distinct )A 355 248 :M -.061(error )A 382 248 :M -.131(terms )A 412 248 :M (is )S 425 248 :M -.131(fixed )A 454 248 :M -.33(at )A 467 248 :M .089(zero.)A 59 265 :M .072 .007(The free parameters of S)J f0_7 sf 0 3 rm (1 )S 0 -3 rm f0_12 sf .049 .005(are )J f3_12 sf (q )S f0_12 sf .05 .005(= <)J f3_12 sf (b)S f0_12 sf (, )S f2_12 sf (P)S f0_12 sf .061 .006(>, where )J f3_12 sf (b )S f0_12 sf .043 .004(is the set of )J 361 265 :M -.219(linear )A 391 265 :M -.163(coefficients )A 449 265 :M .568({)A f1_12 sf .649(b)A f0_7 sf 0 3 rm .345(1)A 0 -3 rm f0_12 sf .538 .054(, )J 474 265 :M f1_12 sf .48(b)A f0_7 sf 0 3 rm .255(2)A 0 -3 rm f0_12 sf (})S 59 283 :M -.109(and )A 82 283 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 96 283 :M (is )S 110 283 :M -.22(the )A 130 283 :M -.109(set )A 149 283 :M (of )S 165 283 :M -.144(variances )A 215 283 :M (of )S 231 283 :M -.22(the )A 251 283 :M -.061(error )A 280 283 :M .216 .022(terms. )J 316 283 :M -.326(We )A 338 283 :M -.166(will )A 362 283 :M (use )S 384 283 :M f3_12 sf .558(S)A f2_7 sf 0 3 rm .29(S1)A 0 -3 rm f0_12 sf .314<28>A f3_12 sf .491(q)A f3_7 sf 0 3 rm .275(1)A 0 -3 rm f0_12 sf .5 .05(\) )J 425 283 :M -.167(to )A 440 283 :M -.164(denote )A 477 283 :M -.33(the)A 59 301 :M -.183(covariance matrix parameterized by )A 229 301 :M -.22(the )A 247 301 :M -.163(vector )A 280 301 :M f3_12 sf .612(q)A f3_7 sf 0 3 rm .343(1)A 0 -3 rm f0_12 sf .294 .029( )J 295 301 :M (for )S 313 301 :M -.199(model )A 346 301 :M .755(S)A f0_7 sf 0 3 rm .396(1)A 0 -3 rm f0_12 sf .617 .062(, )J 365 301 :M -.109(and )A 386 301 :M -.164(occasionally )A 448 301 :M -.262(leave )A 476 301 :M -.167(out)A 59 320 :M -.135(the model subscript if the context makes it clear which model )A 350 320 :M (is )S 362 320 :M -.132(being )A 392 320 :M -.119(referred )A 433 320 :M .512 .051(to. )J 450 320 :M (If )S 462 320 :M -.331(all )A 477 320 :M -.33(the)A 59 338 :M -.07(pairs of error terms in )A 166 338 :M -.326(a )A 175 338 :M -.223(SEM )A 203 338 :M .299 .03(S )J 214 338 :M -.215(are )A 232 338 :M -.074(uncorrelated, )A 298 338 :M (we )S 316 338 :M (say )S 336 338 :M .299 .03(S )J 347 338 :M (is )S 359 338 :M -.326(a )A 368 338 :M -.223(SEM )A 396 338 :M -.083(with )A 421 338 :M f2_12 sf .366(uncorrelated)A 59 356 :M .461(errors)A f0_12 sf (.)S 59 386 :M (S)S f7_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf .021 .002( contains the same structural equations as S)J f7_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf (, but in S)S f7_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf ( we )S 350 386 :M -.166(will )A 372 386 :M -.131(allow )A 402 386 :M -.22(the )A 420 386 :M (errors )S 452 386 :M -.163(between)A 59 404 :M f7_12 sf -.131(X)A f7_7 sf 0 3 rm -.053(2)A 0 -3 rm f0_12 sf -.065( and )A f7_12 sf -.131(X)A f7_7 sf 0 3 rm -.053(3)A 0 -3 rm f0_12 sf -.064( to be correlated, i.e., we make the correlation between the errors of )A f7_12 sf -.131(X)A f7_7 sf 0 3 rm -.053(2)A 0 -3 rm f0_12 sf ( )S 448 404 :M -.109(and )A 469 404 :M f7_12 sf -.128(X)A f7_7 sf 0 3 rm -.052(3)A 0 -3 rm f0_12 sf ( )S 486 404 :M (a)S 59 421 :M -.117(free parameter, instead of fixing it at )A 234 421 :M .236 .024(zero, )J 262 421 :M (as )S 276 421 :M -.167(in )A 289 421 :M 1.022(S)A f7_7 sf 0 3 rm .583(1)A 0 -3 rm f0_12 sf .836 .084(. )J 309 421 :M (In )S 323 421 :M .968(S)A f7_7 sf 0 3 rm .552(2)A 0 -3 rm f0_7 sf 0 3 rm .254 .025( )J 0 -3 rm 338 421 :M f0_12 sf -.22(the )A 356 421 :M -.16(free )A 378 421 :M -.163(parameters )A 433 421 :M -.215(are )A 451 421 :M f3_12 sf .506(q)A f0_12 sf .242 .024( )J 462 421 :M .211 .021(= )J 473 421 :M .681(<)A f3_12 sf .663(b)A f0_12 sf (,)S 59 439 :M f2_12 sf .093<50D5>A f0_12 sf .255 .025(>, where )J f3_12 sf .131 .013(b )J f0_12 sf .179 .018(is the set of )J 186 439 :M -.219(linear )A 216 439 :M -.163(coefficients )A 274 439 :M .506({)A f1_12 sf .578(b)A f0_7 sf 0 3 rm .307(1)A 0 -3 rm f0_12 sf .263(,)A f1_12 sf .578(b)A f0_7 sf 0 3 rm .307(2)A 0 -3 rm f0_12 sf .699 .07(} )J 316 439 :M -.109(and )A 337 439 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 354 439 :M (is )S 366 439 :M -.22(the )A 384 439 :M -.109(set )A 401 439 :M (of )S 415 439 :M -.144(variances )A 463 439 :M (of )S 477 439 :M -.33(the)A 59 457 :M -.061(error )A 86 457 :M -.131(terms )A 116 457 :M -.109(and )A 137 457 :M -.22(the )A 155 457 :M -.179(correlation )A 209 457 :M -.139(between )A 252 457 :M f1_12 sf .242(e)A f0_7 sf 0 3 rm .219 .022(2 )J 0 -3 rm 264 457 :M f0_12 sf -.109(and )A 286 457 :M f1_12 sf .44(e)A f0_7 sf 0 3 rm .292(3)A 0 -3 rm f0_12 sf .455 .046(. )J 304 457 :M (If )S 317 457 :M -.22(the )A 336 457 :M -.136(correlations )A 396 457 :M -.139(between )A 440 457 :M -.109(any )A 462 457 :M (of )S 477 457 :M -.33(the)A 59 476 :M -.009(error terms in a SEM are not fixed at zero, we will call it a SEM with )A f2_12 sf -.01(correlated errors)A f0_12 sf (.)S 484 471 :M f0_7 sf (3)S 59 506 :M f0_12 sf -.164(It )A 70 506 :M (is )S 82 506 :M -.041(possible )A 126 506 :M -.167(to )A 140 506 :M -.145(associate )A 187 506 :M -.083(with )A 213 506 :M -.245(each )A 239 506 :M -.223(SEM )A 268 506 :M -.083(with )A 294 506 :M -.163(uncorrelated )A 357 506 :M (errors )S 390 506 :M -.326(a )A 400 506 :M -.205(directed )A 442 506 :M -.064(graph )A 474 506 :M -.331(that)A 59 524 :M -.063(represents )A 112 524 :M -.22(the )A 131 524 :M -.163(causal )A 165 524 :M -.108(structure )A 211 524 :M (of )S 226 524 :M -.22(the )A 245 524 :M -.199(model )A 279 524 :M -.109(and )A 302 524 :M -.22(the )A 322 524 :M -.08(form )A 351 524 :M (of )S 367 524 :M -.22(the )A 387 524 :M -.219(linear )A 419 524 :M (equations. )S 474 524 :M .168(For)A 59 541 :M -.109(example, the )A 123 541 :M -.205(directed )A 164 541 :M -.064(graph )A 195 541 :M -.131(associated )A 247 541 :M -.083(with )A 272 541 :M -.22(the )A 290 541 :M -.09(substantive )A 347 541 :M -.145(variables )A 393 541 :M -.167(in )A 406 541 :M .42(S)A f0_7 sf 0 3 rm .22(1)A 0 -3 rm f0_12 sf .189 .019( )J 421 541 :M (is )S 433 541 :M .319(X)A f0_7 sf 0 3 rm .129(1)A 0 -3 rm f1_12 sf .436A f0_12 sf .11 .011( )J 462 541 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 479 541 :M f1_12 sf S 59 560 :M f0_12 sf .876(X)A f0_7 sf 0 3 rm .354(3)A 0 -3 rm f0_12 sf .552 .055(, )J 80 560 :M -.139(because )A 121 560 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 138 560 :M (is )S 150 560 :M -.22(the )A 168 560 :M -.083(only )A 193 560 :M -.09(substantive )A 250 560 :M -.205(variable )A 291 560 :M -.249(that )A 312 560 :M -.052(occurs )A 347 560 :M (on )S 364 560 :M -.22(the )A 383 560 :M -.132(right )A 410 560 :M -.082(hand )A 438 560 :M -.082(side )A 462 560 :M (of )S 477 560 :M -.33(the)A 59 578 :M -.048(equation for X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.048(, and X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.045( is the only substantive variable )A 324 578 :M -.249(that )A 345 578 :M -.091(appears )A 385 578 :M (on )S 401 578 :M -.22(the )A 419 578 :M -.132(right )A 445 578 :M -.082(hand )A 472 578 :M -.109(side)A 59 596 :M (of )S 74 596 :M -.22(the )A 93 596 :M -.165(equation )A 138 596 :M (for )S 157 596 :M .876(X)A f0_7 sf 0 3 rm .354(3)A 0 -3 rm f0_12 sf .552 .055(. )J 179 596 :M -.326(We )A 200 596 :M -.182(generally )A 248 596 :M (do )S 265 596 :M -.111(not )A 285 596 :M -.189(include )A 325 596 :M -.061(error )A 354 596 :M -.131(terms )A 386 596 :M -.167(in )A 401 596 :M -.22(the )A 421 596 :M -.205(directed )A 464 596 :M -.08(graph)A 59 636 :M ( )S 59 633.48 -.48 .48 203.48 633 .48 59 633 @a 81 645 :M f0_6 sf (2)S 84 649 :M f0_10 sf .24 .024( We realize that it is slightly unconventional to )J 278 649 :M (write )S 302 649 :M .218 .022(the )J 318 649 :M .365 .036(trivial )J 346 649 :M -.117(equation )A 383 649 :M -.052(for )A 398 649 :M .218 .022(the )J 414 649 :M -.03(exogenous )A 460 649 :M -.171(variable)A 59 661 :M .138(X)A f0_9 sf 0 2 rm .086(1)A 0 -2 rm f0_10 sf .215 .022( in terms of its error, but this serves to give the error )J 287 661 :M .192 .019(terms )J 313 661 :M .056 .006(a )J 321 661 :M -.118(unified )A 352 661 :M -.313(and )A 369 661 :M -.037(special )A 400 661 :M .454 .045(status )J 427 661 :M .144 .014(as )J 439 661 :M -.043(providing )A 481 661 :M (all)S 59 675 :M -.009(the external sources of variation for the system)A 81 683 :M f0_6 sf (3)S 84 687 :M f0_10 sf .085 .009(We do not consider SEMs with other sorts of constraints on the parameters, e.g., equality constraints.)J endp %%Page: 16 16 %%BeginPageSetup initializepage (peter; page: 16 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (16)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.109(associated with )A 135 56 :M -.326(a )A 144 56 :M -.223(SEM )A 172 56 :M (unless )S 206 56 :M -.22(the )A 224 56 :M (errors )S 256 56 :M -.215(are )A 274 56 :M -.087(correlated. )A 328 56 :M -.326(We )A 348 56 :M -.14(enclose )A 387 56 :M -.122(measured )A 436 56 :M -.145(variables )A 482 56 :M -.334(in)A 59 74 :M -.081(boxes, latent variables in circles, and leave error variables unenclosed.)A 180 111 19 15 rC 180 123 :M f8_12 sf (X)S gR gS 189 117 14 12 rC 189 126 :M f8_10 sf (1)S gR gS 272 111 19 15 rC 272 123 :M f8_12 sf (X)S gR gS 281 117 14 12 rC 281 126 :M f8_10 sf (2)S gR gS 358 111 19 15 rC 358 123 :M f8_12 sf (X)S gR gS 367 117 14 12 rC 367 126 :M f8_10 sf (3)S gR gS 274 154 14 15 rC 274 167 :M f1_12 sf (e)S gR gS 280 161 15 12 rC 280 170 :M f8_10 sf (2)S gR gS 358 153 14 15 rC 358 166 :M f1_12 sf (e)S gR gS 364 160 14 12 rC 364 169 :M f8_10 sf (3)S gR gS 169 107 212 89 rC 201 120.75 -.75 .75 256.75 120 .75 201 120 @a np 254 116 :M 254 124 :L 262 120 :L 254 116 :L .75 lw eofill -.75 -.75 254.75 124.75 .75 .75 254 116 @b -.75 -.75 254.75 124.75 .75 .75 262 120 @b 254 116.75 -.75 .75 262.75 120 .75 254 116 @a 294 119.75 -.75 .75 343.75 119 .75 294 119 @a np 341 115 :M 341 122 :L 349 119 :L 341 115 :L eofill -.75 -.75 341.75 122.75 .75 .75 341 115 @b -.75 -.75 341.75 122.75 .75 .75 349 119 @b 341 115.75 -.75 .75 349.75 119 .75 341 115 @a -.75 -.75 280.75 151.75 .75 .75 280 140 @b np 277 142 :M 284 142 :L 280 134 :L 277 142 :L eofill 277 142.75 -.75 .75 284.75 142 .75 277 142 @a 280 134.75 -.75 .75 284.75 142 .75 280 134 @a -.75 -.75 277.75 142.75 .75 .75 280 134 @b -.75 -.75 366.75 149.75 .75 .75 366 136 @b np 362 138 :M 369 138 :L 366 130 :L 362 138 :L eofill 362 138.75 -.75 .75 369.75 138 .75 362 138 @a 366 130.75 -.75 .75 369.75 138 .75 366 130 @a -.75 -.75 362.75 138.75 .75 .75 366 130 @b 90 180 80 26 322.5 181.5 @n 0 90 78 26 319.5 181.5 @n 175.5 109.5 23 19 rS 268.5 111.5 23 19 rS 354.5 111.5 23 19 rS -.75 -.75 358.75 177.75 .75 .75 358 177 @b np 354 179 :M 362 179 :L 358 171 :L 354 179 :L eofill 354 179.75 -.75 .75 362.75 179 .75 354 179 @a 358 171.75 -.75 .75 362.75 179 .75 358 171 @a -.75 -.75 354.75 179.75 .75 .75 358 171 @b -.75 -.75 281.75 178.75 .75 .75 281 178 @b np 277 180 :M 285 180 :L 281 172 :L 277 180 :L eofill 277 180.75 -.75 .75 285.75 180 .75 277 180 @a 281 172.75 -.75 .75 285.75 180 .75 281 172 @a -.75 -.75 277.75 180.75 .75 .75 281 172 @b gR gS 0 0 552 730 rC 161 217 :M f2_12 sf 2.078 .208(Figure 7. SEM S)J f2_7 sf 0 3 rm .387(2)A 0 -3 rm f2_12 sf 2.469 .247( with correlated errors)J 59 259 :M f0_12 sf -.219(The )A 81 259 :M -.236(typical )A 116 259 :M -.165(path )A 140 259 :M -.188(diagram )A 183 259 :M -.249(that )A 205 259 :M (would )S 240 259 :M -.163(be )A 256 259 :M -.132(given )A 287 259 :M (for )S 306 259 :M .42(S)A f0_7 sf 0 3 rm .22(2)A 0 -3 rm f0_12 sf .189 .019( )J 322 259 :M (is )S 335 259 :M .447 .045(shown )J 372 259 :M -.167(in )A 386 259 :M -.054(Figure )A 422 259 :M .833 .083(7. )J 437 259 :M -.083(This )A 463 259 :M (is )S 476 259 :M -.167(not)A 59 276 :M -.115(strictly a directed graph because of the double-headed arrow between )A 388 276 :M -.061(error )A 415 276 :M -.131(terms )A 445 276 :M f1_12 sf .242(e)A f0_7 sf 0 3 rm .219 .022(2 )J 0 -3 rm 457 276 :M f0_12 sf -.109(and )A 478 276 :M f1_12 sf .552(e)A f0_7 sf 0 3 rm .367(3)A 0 -3 rm f0_12 sf (,)S 59 294 :M -.103(which indicates that )A f1_12 sf -.115(e)A f0_7 sf 0 3 rm -.057(2 )A 0 -3 rm f0_12 sf -.111(and )A f1_12 sf -.115(e)A f0_7 sf 0 3 rm -.076(3)A 0 -3 rm f0_12 sf -.098( are correlated. It is generally )A 336 294 :M -.246(accepted )A 380 294 :M -.249(that )A 401 294 :M -.179(correlation )A 455 294 :M (is )S 467 294 :M -.167(to )A 480 294 :M -.326(be)A 59 312 :M -.183(explained )A 108 312 :M (by )S 124 312 :M -.082(some )A 153 312 :M -.08(form )A 180 312 :M (of )S 194 312 :M -.163(causal )A 227 312 :M -.059(connection. )A 286 312 :M -.18(Accordingly )A 348 312 :M -.164(if )A 359 312 :M f1_12 sf .242(e)A f0_7 sf 0 3 rm .219 .022(2 )J 0 -3 rm 371 312 :M f0_12 sf -.109(and )A 393 312 :M f1_12 sf .104(e)A f0_7 sf 0 3 rm .069(3)A 0 -3 rm f0_12 sf .059 .006( )J 407 312 :M -.215(are )A 426 312 :M -.196(correlated )A 477 312 :M (we)S 59 330 :M -.056(will assume that either )A f1_12 sf -.064(e)A f0_7 sf 0 3 rm -.032(2 )A 0 -3 rm f0_12 sf -.059(causes )A f1_12 sf -.064(e)A f0_7 sf 0 3 rm (3)S 0 -3 rm f0_12 sf -.036(, )A f1_12 sf -.064(e)A f0_7 sf 0 3 rm -.032(3 )A 0 -3 rm f0_12 sf -.059(causes )A f1_12 sf -.064(e)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.059(, some latent variable causes )A 420 330 :M -.083(both )A 445 330 :M f1_12 sf .242(e)A f0_7 sf 0 3 rm .219 .022(2 )J 0 -3 rm 457 330 :M f0_12 sf -.109(and )A 478 330 :M f1_12 sf .552(e)A f0_7 sf 0 3 rm .367(3)A 0 -3 rm f0_12 sf (,)S 59 349 :M (or )S 73 349 :M -.082(some )A 102 349 :M -.181(combination )A 164 349 :M (of )S 178 349 :M .217 .022(these. )J 210 349 :M (In )S 224 349 :M -.131(other )A 253 349 :M 1.048 .105(words, )J 292 349 :M -.125(double-headed )A 366 349 :M .223 .022(arrows )J 404 349 :M -.215(are )A 423 349 :M -.163(an )A 439 349 :M -.083(ambiguous)A 59 367 :M -.106(representation of a causal connection.)A endp %%Page: 17 17 %%BeginPageSetup initializepage (peter; page: 17 of 17)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (17)S gR gS 0 0 552 730 rC 238 68 :M f2_12 sf .605(Bibliography)A 59 92 :M f0_12 sf -.054(\(T. Richardson\) Properties of Cyclic Graphical Models, Master's Thesis, Carnegie Mellon)A 59 104 :M .519 .052(University, 1994.)J 59 128 :M -.022(\(P. Spirtes, C. Glymour and R. Scheines\) )A f4_12 sf -.023(Causation, Prediction, and Search)A f0_12 sf -.023(, Springer-)A 59 140 :M -.005(Verlag Lecture Notes in Statistics 81, N.Y., 1993.)A 59 164 :M 1.028 .103(\(P. )J 83 164 :M -.04(Spirtes\) )A 129 164 :M -.244("Directed )A 182 164 :M -.221(Cyclic )A 221 164 :M -.145(Graphical )A 276 164 :M -.141(Representation )A 355 164 :M (of )S 374 164 :M -.122(Feedback )A 428 164 :M .061 .006(Models", )J 482 164 :M -.334(in)A 59 176 :M f4_12 sf -.18(Proceedings )A 121 176 :M .555 .055(of )J 135 176 :M -.22(the )A 154 176 :M -.122(Eleventh )A 200 176 :M -.131(Conference )A 259 176 :M (on )S 276 176 :M -.21(Uncertainty )A 336 176 :M -.167(in )A 350 176 :M -.233(Artificial )A 396 176 :M -.123(Intelligence)A f0_12 sf -.156(, )A 459 176 :M .518 .052(ed. )J 479 176 :M (by)S 59 188 :M .023 .002(Philippe Besnard and Steve Hanks, Morgan Kaufmann Publishers, Inc., San Mateo, 1995.)J endp %%Trailer end %%EOF