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1 Bayesianprobabilisticextensionsofadeterministicclassicationmodel K.U.Leuven,Belgium IwinLeenenandIvenVanMechelen AndrewGelman ColumbiaUniversity,NewYork binarypredictorvariablesx1;:::;xk,abooleanregressionmodelisaconjunctive(ordisjunctive)logicalcombinationconsistingofasubsetsofthe aspecicationofak-dimensionalbinaryindicatorvector(1;:::;k)with Xvariables,whichpredictsY.Formally,Booleanregressionmodelsinclude yitodier(foranyobservationi).withinbayesianestimation,aposterior distributionoftheparameters(1;:::;k;)islookedfor.theadvantages ofsuchabayesianapproachincludeaproperaccountfortheuncertainty ThispaperextendsdeterministicmodelsforBooleanregressionwithina Bayesianframework.ForagivenbinarycriterionvariableYandasetofk Summary j=1ixj2s.inaprobabilisticextension,aparameterisaddedwhich representstheprobabilityofthepredictedvalue^yiandtheobservedvalue Tiensestraat102,B-3000Leuven,Belgium. posteriorpredictivechecks).weillustrateinanexampleusingrealdata. inthemodelestimatesandvariouspossibilitiesformodelchecking(using draftofthispaper,andjohannesberkhofforhelpfuldiscussions. TheauthorsgratefullyacknowledgeBrianJunkerforhishelpfulcommentsonanearlier ThisworkwassupportedinpartbytheU.S.NationalScienceFoundationGrantSBR- AddresscorrespondencetoIwinLeenen,DepartmentofPsychology,K.U.Leuven,

2 Keywords:Bayesianestimation,Booleanregression,logicalruleanalysis, posteriorpredictivechecks 2 Inmanyresearchlines,predictionproblemsareconsideredwiththepredictors 1and/orcriteriabeingbinaryvariables.Asaresult,anumberofmodelsand associatedtechniqueshavebeendevelopedtoexaminetherelationsinthis Introduction though,oneaimsatndingthesucientand/ornecessaryconditionsfor theprobabilitythatthecriterionvariableassumeseitherofthetwopossible valuesisalinearfunctionofanumberofpredictors.inmanyrelevantcases, example,inalogisticregressionmodelwithbinaryvariables,thelogitof typeofdata,includinginstantiationsofthegeneralizedlinearmodel.for approach,whichassumesacompensatoryassociationrule,lessappropriate acriteriontooccur,which,asaresult,makesthegeneralizedlinearmodel fromatheoreticalpointofview.inmedicaldiagnoses,forexample,assigning andconceptsassumethatassignmenttoacategoryisbasedonthepresence ofasetofsinglynecessaryandjointlysucientattributes. adiseasetoagivenpatientisoftenbasedonconsideringalistofnecessary model(vanmechelen,1988;vanmechelen&deboeck,1990)maybehelpful andsucientconditions;asanotherexample,sometheoriesoncategories asitidentiesforagivenbinarycriterionandagivensetofbinarypredictors asubsetofthepredictorsthatareconjunctively(resp.disjunctively)combinedtopredictthevalueonthecriterionvariable.besidesapplicationsin Insearchofnecessaryand/orsucientconditions,aBooleanregression 1984;VanMechelen&DeBoeck,1990),techniquesrelatedtoBooleanregressionhavebeenstudiedindiscretemathematicsandinthecontextofthe thesocialsciences(mckenzie,clarke,&low,1992;ragin,mayer,&drass, designofswitchingcircuitsinelectronics(biswas,1975;halder,1978;mc- suchasdisjunctivecombinationsofconjunctions(orviceversa),arealso Cluskey,1965;Sen,1983).Inthelatterpublications,morecomplexrules, considered. ExistingalgorithmsforBooleanregressionaimatndingasubsetofthe 1988).However,atleastthreeshortcomingsgowiththeapproachofnding predictorswhichminimizesthenumberofpredictionerrors(vanmechelen, asinglebestsolution:first,inmanyempiricalapplications,severaldierent Booleanregressionhasinitiallybeenformulatedasadeterministicmodel. subsetsofthepredictorsmaytthedata(almost)equallywell,whereasfrom asubstantiveviewpointtheymaybequitedierent.second,itisnotobvious provideanytoolsformodelcheckingduetothefactthatthemodeldoes thenumberofpredictionerrors).third,thedeterministicmodeldoesnot howtodrawstatisticalinferencesaboutthesizeofthepredictionerroras notspecifyitsrelationtothedata.hence,amethodwhichgivesinsight estimatesthetruemodelerror(becausethealgorithmaimsatminimizing thepredictionerrorassociatedwiththesinglebestsolutionprobablyunder-

3 them,isofgreatinterest. inseveralconcurringmodelsandinthelevelofuncertaintyassociatedwith 3 followsthegeneralrecipeproposedbygelman,leenen,vanmechelen,and uralconceptualframeworkforexploringthelikelihoodofseveralpossible concurringmodelsforagivendataset.themodelextensionpresentedhere withinabayesianframework.bayesianstatisticscanbeconsideredanat- Therefore,thepresentpaperextendsthemodelforBooleanregression DeBoeck(inpreparation),whichbringsmostofthetoolsthatareavailable forstochasticmodelswithintherealmofdeterministicmodels(likethemodel ofbooleanregression). thedeterministicmodelofbooleanregression.insection3,thestochastic extensionispresentedandestimationandcheckingofthemodelwithina Bayesianframeworkisdiscussed.InSection4anexampleondenitionsof emotionsillustratestheapplicationofthenewmodeltorealdata.section5 Theremainderofthepaperisorganizedasfollows:Section2recapitulates dealswithpossibleextensionsandcontainssomeconcludingremarks. 2.1Modelformulation ConsiderannkbinarymatrixX,whichdenotestheobservationsforn 2 TheDeterministicBooleanRegressionModel y=(y1;:::;yi;:::;yn),whichcontainstheobservedvaluesforthenunits unitsonkexplanatoryvariablesx1;:::;xj;:::;xk,andabinaryvector onacriterionvariabley.booleanregression,then,speciesaparameter conjunctivemodel, existwhichdierinthewaythatandxarecombinedtoget^y.ina onthecriterion.bothadisjunctiveandaconjunctivevariantofthemodel combinedwithxtogetabinaryvector^y=(^y1;:::;^yn)ofpredictedvalues vector=(1;:::;k)withj2f0;1g(j=1;:::;k)whichissubsequently whereasinthedisjunctivevariant: ^yi^y(;x)i=y ^yi^y(;x)i=1 Y jjj=1xij; (1) Despitetheirsubstantivedierence,conjunctiveanddisjunctivemodelsare dualmodels,though:acomparisonofeq.(1)andeq.(2)showsthatif jjj=1(1 xij): (2) aconjunctivemodeltssomedatasetxandythensimultaneouslythe Eq.(1). theconjunctivemodeland,unlessotherwisestated,any^yiiscalculatedasin onlyoneofbothvariantsneedstobeconsidered;inthispaper,wefocuson wherexcij=1 xijandyci=1 yi(i=1;:::;n;j=1;:::;k).asaresult, disjunctivemodeltsthecomplementeddataxcandyc,andviceversa,

4 icationoftherelationbetweentheobservedyandthepredicted^y.even Booleanregressionbeingadeterministicmodeldoesnotincludeaspec- 4 themodelshouldberejected.inpracticalapplicationsofthemodel,though, oneallowsforpredictionerrorsandthemodelgoeswithalgorithmsthataim everanobservationiexistsforwhichyiand^yiarediscrepant(i.e.,yi6=^yi), more,strictlyspeaking,themodelrequiresthemtobeequal.hence,when- atndingwiththeminimalnumberofdiscrepancies: 2.2Modelestimation D(y;)=nXi=1[yi ^y(;x)i]2: Mechelen&DeBoeck,1990)useagreedyheuristicwhichinitializesthe eachtimeselectingthatjforwhichthechangeyieldsthelargestdecrease algorithms(mickey,mundle,&engelman,1983;vanmechelen,1988;van entriesinto1andsuccessivelychangesthevalueofsomeentryjinto0, TondathatminimizesD(y;),twostrategieshavebeenproposed.Most innumberofdiscrepancies,untilchanginganyoftheremainingj'sdoesnot boundalgorithmthatguaranteesthatasolutionwithminimalvalueon furtherimprovethesolution. D(y;)isfound.Thisalgorithmpassesthroughatree,makingextensively useofthepropertythatinaconjunctivemodelchanginganarbitraryentry jfrom1into0doesnotdecreasethenumberoffalsenegatives(afalse Recently,LeenenandVanMechelen(1998)haveproposedabranch-andertyallowsthealgorithmtoapplybranchingandboundingtoalargeextent, equals0andtheobservedvalueyiequals1).inmanycases,thelatterprop- negativebeingdenedasanobservationiforwhichthepredictedvalue^yi therebystronglyreducingtheprocessingtimecomparedtoanenumerative searchamongallpossiblesolutions. numberofdescriptivestatistics,includingproportionofdiscrepancies,jaccard'sgoodness-of-tstatistic(sneath&sokal,1973;tversky,1977),andictivegainbyknowingthemodeloverapredictionbasedonthemarginal criterionprobabilityonly.however,thesestatisticsarelimitedinthatthey arebasedonthetotalgoodness-of-tanddonotexaminethestructureof notasolutionis\sucientlygood." 2.3Modelchecking Thegoodnessoftofthedeterministicmodelcanbesummarizedintoa VanMechelenandDeBoeck's(1990)^p,whichindicatestheamountofpre- theerrors.also,onlyrulesofthumbareavailabletodecideonwhetheror

5 3 BayesianBooleanRegression 5 maythereforebeconsideredthatexplicitlyincludesthepossibilityofapredictionerror. modelunderlyingthedeterministicmodel.anaturalextensionofthemodel 3.1Modelformulation Allowingfordiscrepanciesrevealstheimplicitassumptionofastochastic themodelandwhichisassumedtobeidenticalacrossobservations.hence, variablepossiblychangingfrom0into1orviceversa.forthispurpose,a newparameterisaddedtothemodel,whichistheexpectederrorrateof tothedeterministicmodel,whichaccountsforthevaluesonthecriterion ThestochasticextensionimpliestheadditionofaBernoulli-likeprocess foranyobservationi,itholdsthat: itlyindicatedbecausethepredictorvaluesareconsideredxed.)underlocal (Inthelatterandallfollowingequations,thedependenceonXisnotexplic- stochasticindependence,itfurtherholdsthatthelikelihoodofyunderthis Pr(yi=^yij;)=1 : (3) modelis: work,whichprovidestoolsforexploringtheposteriordistribution: Forconvenience,D(y;)isabbreviatedtoDinformulas. Inanextstep,thestochasticmodelisconsideredwithinaBayesianframe- p(yj;)=d(1 )n D: p(;jy)=p(yj;)p(;) ingdeterministicmodel:for,inthiscasemaximizingthelikelihood(which Uniformpriordistributionsimplyaminimalextensionofthealreadyexist- Wewillassumeandtohaveindependentanduniformpriordistributions. p(y) : (4) impliesminimizingthenumberofdiscrepancies)correspondstondingthe modeoftheposteriordistribution(gelmanetal.,inpreparation). AsshownintheAppendix,workingouttheposterioryields: wherethesuminthedenominatorisoverall2kvaluesintheparameter p(;jy)=(n+1)d(1 )n D space.clearly,evaluatingthissumisfeasibleforsmallkonly. #21 P(n D#) ; (5) Eq.(5)resultsin: parameter.againintheappendix,itisshownthatintegratingoutin Often,onewillbeinterestedinthemarginalposteriordistributionofthe p(jy)= #21 P(nD) 1 D#): (6)

6 yhaveequalposteriorprobabilities.furthermore,itfollowsthatifhas Thelatterimpliesthattwoparameterswhichareequallydiscrepantwith 6 probabilitiesequals: onediscrepancyfewerthanthentheratiooftheirmarginalposterior 3.2Modelestimation p(jy)=n D p(jy) D: (7) Inthissectionweshowhowonecangaininsightintheposteriordistribution Step0Asaninitializationstep,mestimates(s;0)andmestimates(s;0), bydrawingsimulationswithagibbs-metropolisalgorithm: value: vectorwithpr((s;0) (s=1;:::;m),areconstructedasfollows:(s;0)isarandombinary j =1)=0.5(j=1;:::;k)and(s;0)isgiventhe estimatesoftobe0or1(gelmanetal.,inpreparation). Weadd1inthenominatorand2inthedenominatortoavoidinitial D(y;(s;0))+1 n+2 : Step1WerunmparallelsequencesofaMetropolisalgorithm,with((s;0);(s;0)) asthestartingpointforsequences(s=1;:::;m).ateachiterationt (t=1;2;:::),thefollowingsubstepsareexecutedforeachsequences: 1.Acandidatevalueisconstructedbasedonthevalue(s;t 1) inthepreviousiteration.therefore,rstanintegerw(s;t)from adiscretedensity(e.g.,poissonorbinomial)isdrawnwiththe or1into0)toobtain.assuch,w(s;t)representsthenumberof randomlyselectedandsubsequentlychanged(fromeither0into1 entriesinthatarechangedfrom(s;t 1). restriction1w(s;t)k.next,w(s;t)entriesin(s;t 1)are ingfromthefollowingjumpingdistribution: Thisprocedureforconstructingtechnicallycorrespondstodraw- J(j(s;t 1))=kXw=11 Thejumpingdistributionreturnstheprobabilityofconsideringthe wherep(w)isthe(truncated)discretedensitymentionedabove. kwp(w); algorithmisofthemetropolistype. Clearly,Jissymmetric:J(j)=J(j)suchthattheresulting candidate,giventhevalueof(s;t 1)ofthepreviousiteration.

7 2.Theratiooftheposteriordensities,orequivalently,theratioof thelikelihoods,iscalculated: 7 3.Valuesareassignedto(s;t)and(s;t): r=p(yj;(s;t 1)) p(yj(s;t 1);(s;t 1))=1 (s;t 1) (s;t 1)D(y;(s;t 1)) D(y;): Thevaluefor(s;t)isobtainedbyadrawfromaBeta(D(y;(s;t))+ (s;t 1)otherwise withprobabilitymin(1,r) Thesestepsarerepeateduntilthemsequencesappearmixed.Gelman andrubin's(1992)p^rstatisticmaybeusedasadiagnosticinstrument inmonitoringtheconvergence. 1;n D(y;(s;t))+1)distribution. Step2InordertoobtainLposteriorsimulationdraws,theproceduredescribedinstep1continues,afterconvergenceofthesequences,for anotherl=miterations.thelatterdrawsinthemsequencesare 3.3Modelchecking collectedandwilleventuallyconstitutethesetofsimulationdraws AnaturalwayformodelcheckinginBayesianstatisticsisusingposterior f((l);(l))j(l=1;:::;l)gfromtheposteriordistribution. Step3ForeachoftheLposteriorsimulationdrawsareplicateddataset predictivechecks.therefore,weproceedwiththenextsteps: Step4AtestvariableT(y;)isdenedwhichsummarizessomeaspectof simulatedfrom^y(l)basedoneq.(3)(with(l)substitutedfor). y(l)issimulatedasfollows:first,^y(l)=^y((l);x)iscomputedusing interestofthedataorthediscrepancybetweenmodelanddata. Eq.(1),and,subsequently,thencomponentsofy(l)areindependently Step5TherealizedvalueT(y;(l))fortheobserveddataandthereplicated Step6Therealizedvalueandthereplicatedvaluearecomparedtoestimate Lsimulationdraws. valuet(y(l);(l))forthereplicateddataarecomputedforeachofthe Themodelcheckingprocedurepresentedherewillbeillustratedintheexample. theposteriorpredictivep-valueastheproportionofthelsimulations forwhicht(y(l);(l))>t(y;(l)).

8 4 IllustrativeApplication 8 4.1ProblemandData standable(nontechnical),andwhichthemselvesarenotnamesofspecic semanticprimitives,whichare\termsofwordswhichareintuitivelyunder- conceptscanbedenedbyasetofsinglynecessaryandjointlysucient Inthissectionweillustratethenewapproachbyanexampleintheeldof emotionsoremotionalstates."table1listssomeofthesemanticprimitives deningemotionconcepts.accordingtowierzbicka(1992,p.541),emotion explicitdenitions(i.e.,byexperts),thepresentstudyconsidersimplicittheoriesinlaymenandevaluateswhethertheseimplicittheoriesareconjunctive PredictorSemanticprimitive X1Apersondidsomethingbad X3Iwouldwanttochangethis X2Idon'twantthis sheproposed.asherdenitionsofemotionsareconjunctivecombinations andanemotionconcept(asthecriterion).whereaswierzbickadealswith propriatelydescribetherelationbetweensemanticprimitives(aspredictors) ofsemanticprimitives,abooleanregressionmodelmaybeexpectedtoap- combinationsofsemanticprimitivesaswell. X7Iwouldwantthatsomethingdidn'thappen X6Somethingbadhappened X5Ifeelbad X4Iwouldwanttodosomethingbadtosomebody X11Ifeelgood X12Somebodydidsomethinggood X10Iwantsomethinglikethis X9Somethinggoodhappened X8Ican'tchangethesituation Table1:Listofthe(noncomplemented)predictorsfortheBooleanregression X13Idon'twanttochangethis analysesintheapplication X14Iwouldwanttodosomethinggoodforsomebody eachof14semanticprimitivesintable1wastrueforthegivensituationand askedtogeneratetwentydierentsituationsinwhichtheyhadrecentlybeen askedtospecifyforthetwentysituationstheygenerated:(1)whetherornot involvedandfelteitherangry,sad,grateful,orhappy.next,thesubjectswere Fiverst-yearpsychologystudentsoftheUniversityofLeuvenwereeach (2)whetherornottheyexperiencedeachofthe4forementionedemotions: anger,sadness,gratitude,andhappiness.intheanalyses,the520situa-

9 originalandthecomplementedsemanticprimitivesareincludedaspredictors,eventuallyresultingin28predictors(x1;:::;x14;:x1;:::;:x14)anativeemotions,angerandsadness,wereverysimilar,astheresultsforboth tionswereconcatenated,resultinginton=100observations,andboththe 9 positiveemotions,gratitudeandhappiness,were,onlyanalyseswithanger 4criteriaYangry,Ysad,Ygrateful,andYhappy.Becausetheresultsforbothneg- andhappinessarepresentedinthefollowingsections. (Leenen&VanMechelen,1998).ForYangry,thebestlogicalrulecombines cies)werefoundusingthepreviouslydiscussedbranch-and-boundalgorithm Optimalconjunctivelogicalrules(i.e.,withminimalnumberofdiscrepan- 4.2Deterministicanalysis thecomplementsofthepredictors9,10,and14:apersonreports(s)heexperiencesangerinagivensituationi\itisnotthecasethatsomethinggood happenedand(s)hedoesnotwantsomethinglikethisand(s)hedoesnot predictedbythesinglepredictor9:apersonreports(s)hefeelshappyi \somethinggoodhappened."table2presentssomegoodness-of-tindices wanttodoanythinggoodforsomebody."yhappyontheotherhandisbest forbothoptimalrules. Emotion Anger :X9^:X10^:X14 Optimalrule %discrepanciesjaccardindex^p Table2:OptimallogicalrulesforYangryandYhappyandassociatedgoodnessof-tstatisticsasfoundbyadeterministicanalysis Happiness Bayesiananalysis TheprocedurediscussedinSection3.2wasusedtosimulatetheposterior 4.3.1Modelestimation distributionof(;).foreachcriterion,weranm=5sequencesofthe describedgibbs-metropolisalgorithm.afterconvergence,namelywhengelmanandrubin's(1992)^r-statisticwassmallerthan1.1foreachofthe parametersj(j=1;:::;28)and,another2000runsineachsequence wereexecuted,endingupwithl=10000posteriordrawsforeachcriterion. lently,theconjunctivecombinations)forangerandhappiness,respectively. ministicbranch-and-boundalgorithmmaybeoneofseveral\best"solutions: TheresultsoftheBayesiananalysisshowthattherulefoundbythedeter- Foranger,(atleast)vedierentconjunctivecombinationshaveaminimal Table3givesthemarginaldistributionofthe-parameters(or,equiva-

10 Logicalrule 10 Angry :X10^:X11^:X12^:X14 Posteriorprobability%discrepancies :X9^:X10^:X11^:X12^:X14 :X9^:X10^:X14 :X9^:X10^:X11^:X other :X10^:X11^:X14 :X9^:X10^:X12^:X14 :X10^:X12^:X14 <: :X4^X9 :X1^:X4^X9 Happy X9 :X1^X9 :X4^X :X5^X9.170 :X1^:X5^X9.030 :X1^:X4^X :X4^:X5^X9 :X4^:X5^X11 :X1^:X4^:X5^X11 X :X1^:X5^X11 :X5^X :X1^:X4^:X5^X9 :X1^X other Table3:SimulatedposteriordistributionofforYangryandYhappy <: numberofdiscrepanciesandtwohaveonly1discrepancymore;forhappiness, isentirelyduetosimulationvariability.)thewiderangeofavailablemodels numberofdiscrepancieshavedierentcomputedposteriorprobabilities;this areequaltothenumberofdiscrepancies.inthetable,modelswiththesame crepancymore.(inourexample,n=100,sothe%discrepanciesintable3 fourconjunctivecombinationsdoequallywelland12logicalruleshave1dis- thattaboutequallywellindicatesthatthestochasticextensioncanadda therulefoundbythedeterministicalgorithm,whichmakethemlessimpor- remarkthatmostoftheotherrulesmerelyaddoneormorepredictorsto wiseonlyasinglerulemightbeconsidered.forthisparticularcase,onemay considerableamountofinformationtothedeterministicanalysis,asother-

11 tantastheaddedpredictorscannotbeconsideredsinglynecessary.buteven then,thebayesiananalysisgivesmoreinsightintotheuncertaintyassociated 11 =:072. withthemodels,itwasfoundforangerthat=:100andforhappinessthat consideredforthegivendataset.withrespecttotheuncertaintyassociated withthebestsolutionsandintowhichothervaluesforcanreasonablybe and:x14.similarly,forhappiness,thelogicalruleswithhighestposterior ruleswithposteriorprobabilityover.10,namely::x9,:x10,:x11,:x12, X11.Bywayofillustration,wediscusstheresultsofareanalysisofboth massonlyuseasubsetofthesixpredictors:x1,:x4,:x5,x9,x10,and Forthecriterionanger,onlyvepredictorsshowedupintheconjunctive tors.thisallowsustotheoreticallycomputetheposteriordistributionsand collectl=10000posteriordrawsforbothcriteria. inthepreviousanalysis,weusem=5sequencesand,afterconvergence,we criteriawithonlythesemanticprimitivesthatappearedrelevantaspredic- tocomparethistheoreticaldistributionwiththesimulateddistribution.like procedureworksne. happinessrespectively,boththesimulatedandthetheoreticalposteriordistribution.theresultsshowthatthesimulateddistributionisalwayscloseto thetheoreticaldistribution,fromwhichwemayconcludethattheestimation Table4displaysthemarginalposteriordistributionofforangerand subsection.weassumedthatwasconstantacrossobservations(see,eq(3)) ingoneassumptionthatimplicitlyunderliesthemodelappliedintheprevious Inthissection,theposterior-predictive-checkapproachisillustratedforcheck Modelchecking suchthatinthestudydiscussedabove,nodierencesamongthevesubjects logicalruleswithequalaccuracy. innumberofpredictionerrorsbetweenthevesubjects.therefore,atest involvedareallowed.or,otherwisestated,thesubjectsapplytherespective variablet(y)isdenedas: Individualdierencesinerrorratemaybequantiedbythevariance T(y;)=5Ph=1hDh(y;) 20 D(y;) i2 wheredhisthenumberofdiscrepanciesbetweenthe20-componentyand^y vectorofsubjecth.thelargerthevariationamongsubjects,thelargerthe ; lated.next,posteriorpredictivep-valueswerecomputedastheproportionof T(y;(l))andthereplicatedvalueT(y(l);(l))(l=1;:::;10000)werecalcu- y(10000)weresimulatedasdescribedinsection3.3andboththerealizedvalue valueoft. Forbothcriteriaangerandhappiness,10000replicateddatasetsy(1);:::;

12 12 Logicalrule Angry :X9^:X10^:X11^:X12^:X14 posteriorprobabilityposteriorprobability Exact Simulated :X9^:X10^:X12^:X14 :X10^:X11^:X12^:X14 :X9^:X10^:X14 :X9^:X10^:X11^:X other :X10^:X12^:X14 :X10^:X11^:X14 Happy <: <: :X1^:X5^X11 :X1^:X4^X9 :X4^X9 :X1^X9 X :X1^:X4^:X5^X11 :X5^X11 :X1^X11 :X5^X9 :X4^:X5^X :X1^:X4^:X5^X9 X11 :X4^:X5^X11 :X1^:X5^X9 :X1^:X4^X Table4:ExactandsimulatedposteriordistributionofforYangryandYhappy other :X4^X11 <: <: the10000simulationsforwhicht(y;(l))>t(y(l);(l)).foranger,theposteriorpredictivep-valueequals.566andforhappiness,itequals.624,which usingrelevantpredictorsonly isvisualizedinfigure1.figure1plotstheobservedversusthereplicated halfthenumberofpointsbelowtherstbisector.asaresult,itisconcludedthattheposteriorpredictivecheckprovidesnoevidenceforindividual valuesonthetestvariable:roughlyhalfthenumberofpointslieaboveand dierencesinaccuracy.

13 13 addingnormalrandomnumberstoeachpoint'scoordinates(withstandard theemotions\happy"and\angry."thexandycoordinatesarejitteredby deviation.001)inordertodisplaymultiplevalues. Figure1:PlotoftherealizedT(y;(l))versusthereplicatedT(y(l);(l))for Insomecases,onemayexpecttheprobabilityofafalsepositivetodier fromtheprobabilityofafalsenegativepredictionerror.forexample,ina 5medicalcontext,cautionmaycauseabiasinpredictingsuccessonadan- geroussurgerywhichmakesitunlikelythatfailureoccurswhensuccesswas AsdiscussedbyGelmanetal.(inpreparation),themodelcanbestraight- predictedtoby0and1,respectively. forwardlyexpandedbyallowingdierenterrorrates0and1forresponses Concludingremarks predicted,whereasthereversepredictionerroris(fortunately)morelikely. behelpfulindistinguishingbetweendisjunctiveandconjunctiveassociation than0.5.moreover,allowingtocoverthecompleterangefrom0to1may Booleanregressionmodelnoneedtorestrict(or0and1)tobesmaller junctive/conjunctivemodels(gelmanetal.,inpreparation),thereisforthe IncontrastwithmostotherBayesiangeneralizationsofdeterministicdis- insection2.1,itisclearthatifforeachxjboththeoriginalvariablexjand thecomplementedvariable:xjareincludedaspredictorsthenaconjunctive rules.fromourdiscussiononthedualityofconjunctiveanddisjunctiverules rulewitherrorrateisformallyequivalentwithadisjunctiverulewitherror 1.Theanalysesinsection4fortheillustrativeexampledidincludefor everypredictorboththeoriginalandthecomplementedversionandresulted intovaluesforthatare(considerably)smallerthan0.5.hence,forthis particularcaseaconjunctiveruleisfoundtobemoreappropriatethana disjunctiveone,whichisaresultthatcorrespondswithearliertheoriesand

14 thatwasestablishedonlyaposteriori. Asanalcomment,wenotethatboththedeterministicandBayesian 14 withtherespectivemodels)isexpectedtoincreasewiththenumberofobservations.andmoreinparticular,thedierencebetweenthebestandthe ind(y;)among,(i.e.,dierencesinnumberofdiscrepanciesassociated ofobservationsisverylarge.for,itistrueingeneralthatthevariance approachesforbooleanregressionseemstobelessusefulwhenthenumber secondbestmodelmostlikelyincreaseswiththenumberofobservations. millionobservationshas10%discrepanciesandhas10:01%discrepancies, FromEquations(6)and(7),whichmakeclearhowtheposteriordensitydependsonthenumberofdiscrepancies,itfollowsthatthelargerthenumber thenhasamuchhigherposteriordensitythan.howthisndingcan peaked.thisimpliesthatifamodelforsomedatasetwith,say,n=1 ofobservations,thesharperthe(marginal)posteriordistribution(for)is posteriordensity,isoneoftheobjectivesforfurtherresearch. bereconciledwiththeintuitionthatbothmodelsshouldhaveaboutequal

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16 Appendix:Derivingposteriordistributions 16 p(y)=x#2z1 Werstworkoutthepriorpredictivedistributionp(y): 0p(yj#;)p(#)p()d =X#2B(D#+1;n D#+1) 0D#(1 )n D#12k1d =X#212kD#!;(n D#)! 2k Z1 0B(D#+1;n D#+1)D#(1 )n D#d (n+1)! 1 Theintegralinthethirdstepbeingequalto1asitistheareaunderaBeta density. = 2k(n+1)X#21 1 nd# Fortheposteriordistributionof(;),westartfromEq.(4): p(;jy)=p(yj;)p(;) =D(1 )n D1 p(y) =(n+1)d(1 )n D 2k(n+1)P 1 P#21 D#) 2k Toderivethemarginalposteriordistributionof,isintegratedoutinthe jointposteriordistributionforandintheformulaabove. D#) p(jy)=z1 0p(;jy)d 0(n+1)D(1 )n D = #21 P(nD) 1 #21 P D#) d thelatterintegralbeing1asitisagaintheareaunderabetadensity. D#)Z1 0nD(n+1)D(1 )n Dd;