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TheAcquisitionandUseofCausalStructureKnowledge

BenjaminMargolinRottman

LearningResearchandDevelopmentCenterUniversityofPittsburgh3939O’HaraStreetPittsburghPA15260

InM.R.Waldmann(Ed.),OxfordHandbookofCausalReasoning.Oxford:OxfordU.P.

AuthorNote:ThisresearchwassupportedbyNSFBCS-1430439.IthankMichaelWaldmannandBobRehderforprovidinghelpfulcommentsandsuggestions.

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AbstractThischapterprovidesanintroductiontohowhumanslearnandreasonaboutmultiplecausalrelationsconnectedtogetherinacausalstructure.Thefirsthalfofthechapterfocusesonhowpeoplelearncausalstructures.Themaintopicsinvolvelearningfromobservationsvs.interventions,learntemporalvs.atemporalcausalstructures,andlearningtheparametersofacausalstructureincludingindividualcause-effectstrengthsandhowmultiplecausescombinetoproduceaneffect.Thesecondhalfofthechapterfocusesonhowindividualsreasonaboutthecausalstructure,suchasmakingpredictionsaboutonevariablegivenknowledgeaboutothervariables,oncethestructurehasbeenlearned.Someofthemostimportanttopicsinvolvereasoningaboutobservationsvs.interventions,howwellpeoplereasoncomparedtonormativemodels,andwhethercausalstructurebeliefsbiasreasoning.InbothsectionsIhighlightopenempiricalandtheoreticalquestions.Keywords:CausalStructureLearning,CausalReasoning,CausalBayesianNetworks1Introduction Inthepasttwodecades,psychologicalresearchoncasuallearninghasbeenstronglyinfluencedbyanormativeframeworkdevelopedbystatisticians,computerscientists,andphilosopherscalledCausalBayesianNetworks(CBN)orprobabilisticdirectedacyclicgraphicalmodels.ThepsychologicaladoptionofthiscomputationalapproachisoftencalledtheCBNframeworkorcausalmodels.TheCBNframeworkprovidesaprincipledwaytolearnandreasonaboutcomplexcausalrelationsamongmultiplevariables.

Forexample,Thrornley(2013)usedcausallearningalgorithmstoextractthecausalstructureinFigure1frommedicalrecords.Havingthecausalstructureisusefulforexpertssuchasepidemiologistsandbiologiststounderstandthediseaseandmakepredictionsforgroupsofpatients(e.g.,thelikelihoodofhavingcardiovasculardiseaseamong70yearoldsmokers).Itisalsousefulforscientistswhenplanningfutureresearch;whenresearchingcardiovasculardiseaseastheprimaryoutcomeitiscriticaltomeasureandaccountforsmokingstatusandage;itisnotimportanttomeasureorstatisticallycontrolforsystolicbloodpressure.Figure1:CausalStructureofCardiovascularDisease(adaptedfromThornley,2013)

Thoughcausalstructuresaresurelyusefulforscientists,thecausalmodelsapproachtocausalreasoninghypothesizesthatlaypeoplealsohaveanintuitiveunderstandingofcausalstructuresanduseaframeworksimilartoCBNstolearnandreasonaboutcausalrelations.Adoptinga“manasintuitivestatistician”or“intuitivescientist”approach(Peterson&Beach,1967),wecanalsocontemplatehowadoctormightdevelopasetofcausalbeliefsaboutcardiovasculardiseasesomewhatakintoFigure1.Ofcoursethedoctorlikelyhassomeknowledgeofspecificcausallinksfrommedicalschool

Sex

Cholesterol Ratio

Smoking

Cardiovascular Disease

Ethnicity

Diabetes

Age

Statin Use

Systolic Blood Pressure

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andresearcharticles.Buttheselinksmaybereinforcedorcontradictedbypersonalexperiencesuchasnoticingwhichpatientshavewhichsymptomsanddiseases,andtrackinghowpatients’symptomschangeafterstartingatreatment.DevelopingasetofcausalbeliefssuchasinFigure1wouldallowaphysiciantomakeprognosesandtreatmentplanstailoredtoindividualpatients. TheCBNframeworksupportsallofthesedifferentfunctions:learning,prediction,explanation,andintervention.TherestofthischapterwillexplainwhattheCBNframeworkentails,theevidencepertainingtohowpeoplelearnandreasonaboutcausalnetworks,andhowcloselyhumansappeartomimicthenormativeCBNframework. Theoutlineofthischapterisasfollows.IfirstexplainwhatCBNsare,bothnormativelyandasamodelofhumanlearningandreasoning.Thebulkofthefirsthalfofthischapterisdevotedtoevidenceabouthowpeoplelearnaboutcausalnetworksincludingthestructure,strength,andintegrationfunction.IthendiscussevidencesuggestingthatinsteadofusingthebasicCBNframework,peoplemaybeusingsomethingakintoageneralizedversionoftheCBNframeworkthatallowsforreasoningabouttime.Thesecondhalfofthechapterisdevotedtoevidenceonhowpeoplereasonabouttheircausalbeliefs.AttheendofthechapterIraisesomequestionsforfutureresearch.1.1WhatAreCausalBayesianNetworks? ACausalBayesNetworkisacompactvisualwaytorepresentthecausalrelationsbetweenvariables.Eachvariableisrepresentedasanode,andarrowsrepresentcausalrelationsfromcausestoeffects.Theabsenceofanarrowimpliestheabsenceofacausalrelation. ThoughCausalBayesianNetworkscapturethecausalrelationsamongvariables,theyalsosummarizethestatisticalrelationsbetweenthevariables.TheCBNframeworkexplainshowcausalrelationsshouldbelearnedfromstatisticalrelations.Forexample,givenadatasetwithanumberofvariables,theCBNframeworkhasrulesforfiguringoutthecausalstructure(s)thataremostlikelytohaveproducedthedata.Conversely,acausalstructurecanbereadsuchthatifthecausalstructureisbelievedtobetrue,itimpliescertainstatisticalrelationsbetweenthevariables;thatsomesetsofvariableswillbecorrelatedandotherswillnotbecorrelated.Inordertounderstandhowto“read”aCBN,itisimportanttounderstandtheserelationsbetweenthecausalarrowsandthestatisticalpropertiestheyimply.

First,itiscriticaltounderstandsomebasicstatisticalterminology.“Unconditionaldependence”iswhethertwovariablesarestatisticallyrelatedto(or“dependenton”)eachother(e.g.,correlated)withoutcontrollingforanyothervariables.Iftheyarecorrelated,theyaresaidtobedependent,andiftheyarenotcorrelated,theyaresaidtobeindependent.Conditionaldependenceiswhethertwovariablesarestatisticallyrelatedtoeachotheraftercontrollingforoneormorevariables(e.g.,whetherthereisasignificantrelationbetweentwovariablesaftercontrollingforathirdvariableinamultipleregression).ConditionalandunconditionaldependenceandindependencearecriticalforunderstandingaCBN,soitisimportanttobefluentwiththesetermsbeforemovingon.

Therearetwoproperties,theMarkovproperty,andthefaithfulnessassumption,thatexplaintherelationsbetweenthecausalarrowsinaCBNandthestatisticaldependenciesbetweenthevariablesinadatasetthatissummarizedbytheCBN(alsoseeRehder,thisvolumeaandb).TheMarkovpropertystatesthatonceallthedirectcausesof

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avariableXarecontrolledfororheldconstant,XisstatisticallyindependentofeveryvariableinthecausalnetworkthatisnotadirectorindirecteffectofX.

Forexample,considerFigure2b.TheMarkovassumptionstatesthatXwillbeindependentofallvariables(e.g.,Z)thatarenotdirectorindirecteffectsofX(XhasnoeffectsinFigure2b)oncecontrollingforalldirectcausesofX(YistheonlydirectcauseofX).SimilaranalysescanbeusedtoseethatXandZarealsoindependentconditionalonYinFigures2aand2c.

InregardstoFigure2d,theMarkovAssumptionimpliesthatXandZareunconditionallyindependent(notcorrelated).NeitherXnorZhaveanydirectcauses,soXwillbeindependentofallvariables(suchasZ),thatarenotadirectorindirecteffectofX(e.g.,Y).TheMarkovpropertyissymmetric;ifXisindependentofZ,ZisindependentofX-theyareuncorrelated. ThefaithfulnessAssumptionstatesthattheonlyindependenciesbetweenvariablesinacausalstructuremustbethoseimpliedbytheMarkovassumption(Glymour,2001;Spirtes,Glymour,&Scheines,1993).Statedanotherway,allvariablesinthestructurewillbedependent(correlated),exceptwhentheMarkovpropertystatesthattheywouldnotbe.ThismeansthatifwecollectaverylargeamountofdatafromthestructuresinFigures2a,2b,or2c,XandY,andYandZ,andXandZwouldallbeunconditionallydependent;theonlyindependenciesbetweenthevariablesariseduetotheMarkovassumption,thatXandZareconditionallyindependentgivenY.IfwecollectedalargeamountofdataandnoticedthatXandZwereunconditionallyindependent,thisindependencyinthedatawouldnotbe“faithful”toFigures2a,2b,or2c,implyingthatthedatadonotcomefromoneofthesestructures.ForFigure2d,theonlyindependencyimpliedbytheMarkovpropertyisthatXandZareunconditionallyindependent.Ifalargeamountofdatawerecollectedfromstructure2d,XandY,andZandYwouldbedependent(accordingtothefaithfulnessassumption).Figure2:FourCBNs

Insum,causalmodelsprovideaconcise,intuitive,visuallanguageforreasoningaboutcomplexwebsofcausalrelations.Thecausalnetworkdiagramintuitivelycaptureshowthevariablesarecausallyandstatisticallyrelatedtoeachother.Butcausalnetworkscandomuchmorethanjustdescribethequalitativecausalandstatisticalrelations;theycanpreciselycapturethequantitativerelationsbetweenthevariables.

Tocapturethequantitativerelationsamongvariables,causalnetworksneedtobespecifiedwithaconditionalprobabilitydistributionforeachvariableinthenetworkgivenitsdirectcauses.AconditionalprobabilitydistributionestablishesthelikelihoodthatavariablesuchasYwillhaveaparticularvaluegiventhatanothervariableX,Yscause,hasaparticularvalue.Additionally,exogenousvariables,variablesthathavenoknowncausesinthestructure,arespecifiedbyaprobabilitydistributionrepresentingthelikelihoodthattheexogenousvariableassumesaparticularstate.

Forexample,theCBNinFigure2awouldbespecifiedbyaprobabilitydistributionforX,aconditionaldistributionofYgivenX,andaconditionaldistributionofZgivenY.IfX,

X ZY

a

X ZY

b

X ZY

c

X ZY

d

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Y,andZarebinary(0or1)variables,thedistributionforXwouldsimplybetheprobabilitythatx=1,P(x=1).TheconditionalprobabilityofYgivenXwouldbetheprobabilitythaty=1giventhatx=1,P(y=1|x=1),andtheprobabilitythaty=1giventhatx=0,P(y=1|x=0),andlikewisefortheconditionalprobabilityofZgivenY.(Thereisalsoanotherwaytospecifytheseconditionaldistributionswith“causalstrength”parameters,whichwillbediscussedinlatersections,andsummarizedinSection3.6.SeealsoCheng&Lu,thisvolume;Griffiths,thisvolume;Rehder,thisvolume,aandb,formoredetailsaboutparameterizingastructurewithcausalstrengths.)

Ifthevariablesarenormally-distributedcontinuousvariables,thedistributionforXwouldbecapturedbythemeanandstandarddeviationofX.Then,theconditionaldistributionofYwouldbecapturedbyaregressioncoefficientofYgivenX(e.g.,theprobabilitythaty=2.3giventhatx=1.7),aswellasaparametertocapturetheamountoferrorvariance.

TheCBNinFigure2cwouldbespecifiedbyadistributionforY,andconditionalprobabilitydistributionsforXgivenYandZgivenY.TheCBNinFigure2dwouldbespecifiedbyprobabilitydistributionsforXandZ,andaconditionalprobabilitydistributionforZgivenbothXandY.Inthisway,alargecausalstructureisbrokendownintosmallunits.

Oncealltheindividualprobabilitydistributionsarespecified,Bayesianinferencecanbeusedtomakeinferencesaboutanyvariableinthenetworkgivenanysetofothervariables,forexample,theprobabilitythaty=3.5giventhatx=-0.7andz=1.1.CBNsalsosupportinferringwhatwouldhappenifonecouldinterveneandsetanodetoaparticularvalue.Beingabletopredicttheresultofaninterventionallowsanagenttochoosetheactionthatproducesthemostdesiredoutcome.

CBNshavebeentremendouslyinfluentialacrossawiderangeoffieldsincludingcomputerscience,statistics,engineering,epidemiology,managementsciences,andphilosophy(Pearl,2000;Spirtes,Glymour,&Scheines,2000).TheCBNframeworkisextremelyflexibleandsupportsmanydifferentsortsoftasks.Theycanbeusedtomakeprecisepredictions(includingconfidenceintervals),theycanincorporatebackgroundknowledgeoruncertainty(e.g.,uncertaintyinthestructure,oruncertaintyinthestrengthsofthecausalrelations)forsensitivityanalysis.Theycanbeextendedtohandleprocessesthatoccurovertime.AndsinceCBNsareanextensionofprobabilitytheory,theycanincorporateanyprobabilitydistribution(logistic,multinomial,Gaussian,exponential).Insum,theCBNframeworkisanextremelyflexiblewaytorepresentandreasonaboutprobabilisticcausalrelations.1.2WhatistheCausalBayesianNetworkTheoryofLearningandReasoning? Mostgenerally,thecausalmodeltheoryofhumanlearningandreasoningisthathumanslearnandreasonaboutcausalrelationsinwaysthataresimilartoformalCBNs.ThistheoryispartofabroadermovementinpsychologyofusingprobabilisticBayesianmodelsasmodelsofhigher-levelcognition.11Here“model”isservingtwopurposes.FirstprobabilisticBayesianmodelsareintendedtobeobjectivemodelsofhowtheworldworks(e.g.,Figure1isanobjectivemodelofcardiovasculardisease).Thesecondsenseofmodel,asusedbythepsychologist,isthatthesameprobabilisticmodelcouldalsoserveasamodelofhumanreasoning–treatingFigure1asarepresentationofhowadoctorthinksaboutcardiovasculardisease.

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Thebroadermovementofusingprobabilisticmodelsasmodelsofhigher-levelcognitionistypicallyviewedatMarr’scomputationallevelofanalysis–identifyingtheproblemtobesolved.Indeed,articlesappealingtocausalnetworkshavefulfilledthepromiseofacomputational-levelmodel–forexample,theyhavereframedtheproblemofhumancausalreasoningbyclarifyingthedistinctionbetweencausalstrengthvs.structure(Griffiths&Tenenbaum,2005),andbyidentifyingcausalstructurelearningasagoaluntoitself(AlisonGopniketal.,2004;Steyvers,Tenenbaum,Wagenmakers,&Blum,2003). ThoughtheflexibilityoftheCBNframeworkisobviouslyatremendousadvantageforitsutility,theflexibilitymakesitchallengingtospecifyaconstraineddescriptivetheoryofhumanlearningandreasoning.ThetheoreticalunderpinningoftheCBNframework(e.g.,learningalgorithms,inferencealgorithms)isanactiveareaofresearchratherthanastatictheory.Additionally,therearemanydifferentinstantiationsofhowtoapplytheframeworkinaspecificinstance(e.g.,alternativelearningalgorithms,alternativeparameterizationsofamodel). BecauseoftheflexibilityandmultifacetednatureoftheCBNframework,itisnotparticularlyusefultotalkabouttheCBNframeworkasawhole.Instead,inthecurrentchapterIfocusonthefitbetweenspecificaspectsoftheframeworkandhumanreasoning,withinaspecifictask.2Learning2.1LearningCausalStructure2.1.1LearningaCausalStructurefromObservations OneofthemostdramaticwaysthattheCBNframeworkhaschangedthefieldofhumancausalreasoningisbyidentifyingcausalstructurelearningasaprimarygoalforhumanreasoning(Steyversetal.,2003).Afundamentalprincipleoflearningcausalstructurefromobservationisthatitisoftennotpossibletoidentifytheexactcausalstructure;twoormorestructuresmayexplainthedataequallywell.Thisisessentiallyamoresophisticatedversionof“correlationdoesnotimplycausation.” Considerthe9observationsinTable1,whichsummarizesthecontingencybetweentwovariables,XandY.Thecorrelationbetweenthesetwovariablesis.79.JustknowingthatXandYarecorrelatedcannottelluswhetherXcausesY[X→Y]orwhetherYcausesX[X←Y].(TechnicallyitisalsopossiblethatathirdfactorcausesbothXandY,butIignorethisoptionforsimplicityofexplanation.)Onesimplewaytoseethisisthatunderbothofthesecausalstructures[X→Y]and[X←Y],wewouldexpectXandYtobecorrelated,sothefactthattheyarecorrelatedcannottellusanythingaboutthecausalstructure.AmoresophisticatedwaytounderstandhowitisimpossibletodeterminethetruecausalstructurejustfromobservingthedatainTable1istoseehowparameterscanbecreatedforbothcausalstructuresthatfitthedataequallywell.Thismeansthatthestructuresareequallylikelytohaveproducedthedata.

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TherightsideofTable1showsparametersfortherespectivecausalstructurethatfitthedataperfectly.Forexample,for[X→Y],weneedtofindthreeparameterstospecifythestructure.ThebaserateofX,P(x=1),canbeobtainedbycalculatingthepercentageoftimesthatX=1regardlessofY,((3+1)/9).P(y=1|x=1)issimplythepercentoftimesthatY=1giventhatX=1,andP(y=1|x=0)isthepercentoftimesthatY=1giventhatX=0.ParameterscanbededucedforX←Yinasimilarfashion.IfwesimulatedalargenumberofobservationsthatwewouldexpecttoseefromeachcausalstructurewiththeparametersspecifiedinTable1,wewouldfindthatbothstructureswouldproducedatawithproportionsthatlookssimilartothedatainTable1.Specifically,wewouldobservetrialsinwhichbothvariablesare1about3outof9times,trialsinwhichbothvariablesare0about5/9times,andtrialsinwhichX=1andY=0about1in9timesinthelongrun.Becausewewereabletofindparametersforthesestructuresthatproducedataverysimilartothedataweobserved,thesetwostructuresareequallylikelygiventheobserveddata.Table1:SampleDataforTwoVariablesX Y Numberof

ObservationsParametersthatfitthedataperfectly

foreachcausalstructureX→Y X←Y

1 1 3 P(x=1)=4/9P(y=1|x=1)=3/4P(y=1|x=0)=0

P(y=1)=3/9P(x=1|y=1)=1P(x=1|y=0)=1/6

1 0 10 1 00 0 5Thesamelogicalsoapplieswithmorevariables.ConsiderthedatainTable2withthreevariables.IfyouranacorrelationbetweeneachpairofvariablesyouwouldfindthatXandYarecorrelated(r=.25),YandZarecorrelated(r=.65),andXandZarecorrelated(r=.17)butareindependent(r=0)onceYiscontrolledfor.AccordingtotheMarkovandFaithfulnessassumptions,thispatternofdependenciesandconditionalindependenciesisconsistentwiththreeandonlythreecausalstructures;X→Y→Z,X←Y←Z,andX←Y→Z.

Table2showsparametersforeachofthesecausalstructuresthatfitthedataperfectly.IfwesampledalargeamountofdatafromanyofthethreestructuresinTable2withtheassociatedparameters,theproportionofthetypesof8observationsofX,Y,andZwouldbeverysimilartotheproportionsofthenumberofobservationsinTable2.ThesethreecausalstructuresaresaidtoformaMarkovclassbecausetheyareallequallyconsistentwith(orlikelytoproduce)thesetofconditionalandunconditionaldependenciesintheobserveddata.Thus,itisimpossibletoknowwhichofthesethreestructuresproducedthesetofdatainTable2.

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Table2:SampleDataforThreeVariablesX Y Z Numberof

ObservationsParametersthatfitthedataperfectly

foreachcausalstructureX→Y→Z X←Y←Z X←Y→Z

1 1 1 6 P(x=1)=1/2P(y=1|x=1)=3/4P(y=1|x=0)=1/2P(z=1|y=1)=2/3P(z=1|y=0)=0

P(z=1)=5/12P(y=1|z=1)=1P(y=1|z=0)=5/14P(x=1|y=1)=3/5P(x=1|y=0)=1/3

P(y=1)=5/8P(x=1|y=1)=3/5P(x=1|y=0)=1/3P(z=1|y=1)=2/3P(z=1|y=0)=0

1 1 0 31 0 1 01 0 0 30 1 1 40 1 0 20 0 1 00 0 0 6

Importantly,non-Markovequivalentcausalstructurescanbedistinguishedfromoneanotherwithobservations.Forexample,commoneffectstructuressuchasX→Y←ZareintheirownMarkovequivalenceclass,sotheycanbeuniquelyidentified.AccordingtotheMarkovassumptionfor[X→Y←Z],XandYaredependent,andZandYaredependent,butXandZareindependent.Therearenootherthree-variablestructureswiththisparticularsetofconditionalandunconditionaldependencies.ThismeansthateventhroughX→Y→Z,X←Y←Z,andX←Y→ZareallequallylikelytoproducethedatainTable2,X→Y←Zismuchlesslikely.SupposewetriedtofindparametersforX→Y←ZtofitthedatainTable2.ItwouldbepossibletochooseparameterssuchthatYandXarecorrelatedroughlyaroundr=.25,andthatYandZarecorrelatedroughlyaroundr=.65,matchingthedatainTable2fairlyclosely.Butcritically,wewouldfindthatnomatterwhatparameterswechose,XandZwouldalwaysbeuncorrelated,andthus,itwouldbeveryunlikelythatthedatafromTable2wouldcomefromX→Y←Z.

Insum,byexaminingthedependenciesbetweenvariablesitispossibletoidentifywhichtypesofstructuresaremoreorlesslikelytohaveproducedtheobserveddata.StructureswithinthesameMarkovequivalenceclassalwayshavetheexactsamelikelihoodofproducingaparticularsetofdata,whichmeansthattheycannotbedistinguished,butstructuresfromdifferentMarkovequivalenceclasseshavedifferentlikelihoodsofproducingaparticularsetofdata.

Steyversetal.,(2003)conductedasetofexperimentstotestwhetherpeopleunderstandMarkovequivalenceclassesandcouldlearnthestructurefrompurelyobservationaldata.First,theyfoundthatgivenaparticularsetofdata,participantswereabovechanceatdetectingthecorrectMarkovclass.Furthermore,peopleseemtobefairlygoodatunderstandingthatobservationscannotdistinguishX→YfromX←Y.AndpeoplealsoseemtounderstandtosomeextentthatcommoneffectstructuresX→Y←ZbelongtotheirownMarkovequivalenceclass. However,Steyversetal.’sparticipantswerenotgoodatdistinguishingchainandcommoncausestructuresevenwhentheywerefromdifferentequivalenceclasses(e.g.,X→Y→Zvs.X→Z→Y).Distinguishingthesestructureswasmademoredifficultintheexperimentbecausethemostcommontypeofobservationforallthesestructureswasforthethreevariablestohavethesamestate.Still,theparticipantsdidnotappeartousethe

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trialswhentwoofthevariablesshareastatedifferentfromathirdtodiscriminatecausalstructures(e.g.,theobservationX=Y≠ZismoreconsistentwithX→Y→ZthanX→Z→Y). GiventhatMarkovequivalenceclassissoimportantfortheoriesofcausalstructurelearningfromobservation,itissurprisingthatthereisnotmoreworkonhowwelllaypeopleunderstandMarkovequivalence.OneimportantfuturedirectionwouldbegiveparticipantsasetoflearningdatathatunambiguouslyidentifiesaparticularMarkov-equivalentclass,andtestthepercentofparticipantswho1)identifythecorrectclass,2)identifyallthestructuresintheMarkovclass,and3)includeincorrectstructuresoutsidetheclass.Suchanexperimentwouldhelpclarifyhowgoodorbadpeopleareatlearningcausalstructurefromobservations.Additionally,Steyversetal.usedcategoricalvariableswithalargenumberofcategoriesandnearlydeterministiccausalrelations,whichlikelyfacilitatedaccuratelearningbecauseitwasveryunlikelyfortwovariablestohavethesamevalueunlesstheyarecausallyrelated.ItwouldbeinformativetoexaminehowwellpeopleunderstandMarkovequivalenceclasseswithbinaryorGaussianvariables,whichwilllikelybeharder.Anotherquestionraisedbythisarticleistowhatextentheuristicstrategiesmaybeabletoexplainthepsychologicalprocessesinvolvedinthisinference.InthestudiesbySteyversetal.therearesomesimplerulesthatcandistinguishtheMarkovequivalenceclassesfairlysuccessfully.Forexample,uponobservingatrialinwhichX=Y=Z,X←Y→ZismuchmorelikelythanX→Y←Z,butuponobservingatrialinwhichX≠Y=Z,thelikelihoodsflip.ButinothertypesofparameterizationssuchasnoisybinarydataorGaussiandata,thisdiscriminationwouldnotbesoeasy.

EventhoughMarkovequivalenceisacorefeatureofcausalstructurelearningfromobservations,asfarasIknowthisstudybySteyversetal.istheonlystudytotesthowwellpeoplelearncausalstructurespurelyfromthecorrelationsbetweenthevariables.Thereareanumberofotherstudiesthathaveinvestigatedotherobservationalcuestocausality.Forexample,anumberofstudieshavefoundthatifXoccursfollowedbyY,peoplequicklyandrobustlyusethistemporalorderordelaycuetoinferthatXcausesY(Lagnado&Sloman,2006;Mccormack,Frosch,Patrick,&Lagnado,2015).Thisinferenceoccursdespitethatfactthatthetemporalordermaynotnecessarilyrepresenttheorderinwhichthesevariablesactuallyoccurred,butinsteaditmightreflecttheorderinwhichtheybecomeavailableforthesubjecttoobservethem.Anothercuethatpeopleusetoinfercausaldirectionarebeliefsaboutnecessityandsufficiency.Ifalearnerbelievesthatallcausesaresufficienttoproducetheireffects,thatwheneveracauseispresent,itseffectswillbepresent,thenobservingthatX=1andY=0impliesthatXisnotacauseofY,otherwiseYwouldbe1(Mayrhofer&Waldmann,2011).Insections2.1.4Iwilldiscussoneotherwaythatpeoplelearncausaldirectionfromobservation.Butthegeneralpointofallofthesestudiesisthatwhentheseothercuestocausalityarepittedagainstpurecorrelationsbetweenthevariables,peopletendtousetheseothercuestocausality(seeLagnado,Waldmann,Hagmayer,&Sloman,2007forasummary).

Insum,thoughthereissomeevidencethatpeopledounderstandMarkovequivalenceclasstosomeextent,thisunderstandingappearslimited.Furthermore,therearenotmanystudiesonhowwellpeoplelearncausalstructuresinabottom-upfashionpurelyfromcorrelationaldatawhencovariationistheonlycueavailable.Incontrast,whatisclearisthatwhenotherstrategiesareavailable,peopletendtousetheminsteadofinferringcausalstructurepurelyfromthedependenciesandconditionalindependencies.

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2.1.2LearningCausalStructurefromInterventions:ChoosingInterventionsAnotheracoreprincipleunderlyingcausalstructurelearningisthatinterventions

(manipulations)havethecapabilitytodiscriminatecausalstructuresthatwouldnotbedistinguishablefromobservation;thisisthesamereasonwhyexperimentsaremoreusefulthanobservationalstudies.GoingbacktotheexampleinFigure1,ifweweretryingtofigureoutthecausalrelationsbetweendiabetes(D),statinuse(S),andsystolicbloodpressure(BP),observationaldatawouldonlybeabletonarrowthepossibilitiesdowntothreestructures:D→S→BP,D←S←BP,andD←S→BP.However,ifwecoulddoarandomizedexperimentsuchthathalfthepatientstakeastatinandtheotherhalfdonot,wecouldinferthecausalstructure.IfD→S→BPisthetruecausalstructure,thenthepatientswhotakeastatinwouldhavelowerBPthanthosewhodonot,buttherewouldnotbeanydifferenceinDacrossthetwogroups.Incontrast,ifD←S←BPisthetruecausalstructure,therewouldbeadifferenceofDacrossthetwogroupsbuttherewouldnotbeadifferenceinBPacrossthetwogroups.Finally,ifD←S→BPisthetruecausalstructure,therewouldbeadifferenceinbothDandBPacrossthetwogroups.Therestofthissectionwillexplaininmoredetailhowinterventionscanbeusedtopreciselyidentifyacausalstructure,andhowhumansuseinterventions.

Thelanguageofcausalstructurediagramshasasimplenotationtorepresentinterventions.Whenaninterventionsetsthestateofavariable,allothervariablesthatwouldotherwisebecausesofthemanipulatedvariablearenolongercauses,sothoselinksgetremoved.Forexample,whenthepatientsinourexamplearerandomlyassignedtotakeastatin,eventhoughnormallyhavingdiabetesisacauseoftakingastatin,nowbecauseoftherandomassignmentdiabetesisnolongeracauseoftakingastatin.

Moregenerally,thereasonwhyinterventionscanmakecausalstructuresthatareinthesameMarkovequivalenceclassdistinguishableisthattheinterventionchangesthecausalstructure.Forthisreason,interventionsaresometimescalled“graphsurgery”(Pearl,2000).Figure3a-cshowsthreecausalstructuresthatarenotdifferentiablefromobservation.Figure3d-fandg-ishowthesamethreecausalstructuresundereitheraninterventiononYoraninterventiononX;theinodesrepresenttheintervention.(TheinterventiononYisanalogoustothepreviousexampleoftherandomizedexperimentabouttakingastatin;itcouldbeusefultocomparethesetwoexamplesforgenerality.)

UndertheinterventiononYallthreecausalstructuresnowhavedifferentdependencerelations.InGraphD,ZandYwouldstillbecorrelated,butneitherwouldbecorrelatedwithX.InGraphE,XandYwouldbecorrelated,butneitherwouldbecorrelatedwithZ.AndinGraphF,allthreevariableswouldbecorrelated,butXandZwouldbecomeuncorrelatedconditionalonY.Insum,interventionsonYchangethecausalstructuresuchthattheresultingstructuresnolongerfallwithinthesameMarkovequivalenceclass,sotheycanbediscriminated.Incontrast,aninterventiononXcandiscriminateGraphGfromH,butcannotdiscriminateGraphHfromI.ThismeansthataninterventiononXdoesnotprovideasmuchinformationfordiscriminatingthesethreecausalstructuresasdoesaninterventiononY.

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Figure3:ThreeCausalStructureswithDifferentTypesofInterventions.

Dopeoplechooseinterventionsthatmaximize“informationgain,”theabilitytodiscriminatebetweenmultiplepossiblestructures?Beforegettingtotheevidence,itisusefultoconsideranalternativestrategyforchoosinginterventionstolearnaboutcausalstructureasidefrommaximizinginformationgain;selectinginterventionsthathavethelargestinfluenceonothervariables.Forexample,consideragainthethreestructuresintheNoInterventionsrowinFigure3.InGraphA,Xinfluencestwovariablesdirectlyorindirectly,andinGraphsBandCXdoesnotinfluenceeitherothervariable,foratotal“centrality”ratingof2.Zhasthesamecentralityrating-2.Y,incontrast,influencesZinGraphA,XinGraphB,andXandYinGraphC,foratotalcentralityratingof4.Insum,lookingacrossallthreepossiblestructuresZismore“central”ormoreofa“rootcause.”IfalearnerchoosestointervenetomaximizetheamountofchangesinothervariablestheywilltendtointerveneonYinsteadofXorZ.

Sometimes,asintheexampleinFigure3theInformationGainstrategyandtheRootCausestrategyproducethesameinterventions;YisthemostcentralvariableandinterventionsonYhelpdiscriminatethethreestructuresthemost.However,sometimesthetwostrategiesleadtodifferentinterventions.Forexample,whentryingtofigureoutwhether[X→Z→Y]or[X→Y→Z]isthetruestructure,Xhasthehighestcentralityrating.However,interveningonXwouldnotdiscriminatethestructureswell(lowinformationgain)becauseforbothstructuresitwouldtendtoproducedatainwhichX=Y=Z.InterveningonYorZwouldmoreeffectivelydiscriminatethestructures.Forexample,interveningonYwouldtendtoproducedatainwhichX=Z≠Yfor[X→Z→Y]butwouldtendtoproducedatainwhichX≠Y=Zfor[X→Y→Z],effectivelydiscriminatingthetwostructures.TheRootCauseStrategy,interveningonX,canbeviewedasatypeofpositiveorconfirmatorytestingstrategyinthesensethatitconfirmsthehypothesisthatXhassomeinfluenceonYandZ,butdoesnotactuallyhelpdiscriminatebetweentheremaininghypotheses.

Coenenetal.(2015)testedwhetherpeopleusethesetwostrategiesandfoundthatmostpeopleuseamixtureofboth,thoughsomeappeartousemainlyoneortheother.InanotherexperimentCoenentestedwhetherpeoplecanshifttowardsprimarilyusingtheinformationgainstrategyiftheyarefirsttrainedonscenariosforwhichtherootcausepositivetestingstrategywasverypooratdiscriminatingthecausalstructures.Evenwithoutfeedback,overtimeparticipantsswitchedmoretowardsusinginformationgain.Theyalsotendedtousetherootcausestrategymorewhenansweringfaster.Insum,root

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causepositivetestingisaheuristicthatsometimescoincideswithinformationgain,anditappearsthatpeoplesometimescanovercometheheuristicwhenitisespeciallyunhelpful.

ThoughSteyversetal.(Steyversetal.,2003)didnotdescribeitinthesamewayasCoenenetal.(2015),theyactuallyhaveevidenceforafairlysimilarphenomenontothepositiveteststrategy.Theyfoundthatpeopletendedtointerveneonrootcausesmorethanwouldbeexpectedbypurelyusinginformationgain.Intheirstudy,participantsfirstsaw10observationallearningtrials,andthenchosethecausalstructurethattheythoughtwasmostplausible,forexample[X→Y→Z].Technicallysincethedatawasobservational,theycouldnotdistinguishmodelswithinthesameMarkovequivalentclassatthisstage.NexttheyselectedoneinterventiononeitherX,Y,orZandwouldget10moretrialsofthesameinterventionrepeatedly.Steyversetal.foundthattheirparticipantstendedtoselectrootcausevariablestointerveneupon.Iftheythoughtthatthechain[X→Y→Z]structurewasmostplausible,theymostfrequentlyintervenedonX,thenY,andthenZ.Thispatternfitsbetterwiththerootcauseheuristicthantheinformationgainstrategy,whichsuggestsinterveningonY.

Becausethisfindingcannotbeexplainedbyinformationgainalone,Steyversetal.createdtwoadditionalmodels;hereIonlydiscussRationalTestModel2.Thismodelmakestwoadditionalassumptions.First,itassumesthatparticipantshadadistortedsetofhypothesesaboutthepossiblecausalstructure.NormativelytheirhypothesisspaceforthepossiblecausalstructuresshouldhavebeentheMarkovequivalentclass;iftheyselected[X→Y→Z]asthemostlikelystructureafterthe10observationaltrials,theyshouldhavealsoviewed[X←Y←Z]and[X←Y→Z]asequallyplausible,andthegoalshouldhavebeentotrytodiscriminateamongthesethree.However,thismodelassumesthatinsteadoftryingtodiscriminatebetweenthethreeMarkovequivalentstructures,participantsweretryingtodiscriminatebetween[X→Y→Z],[X→Y;Z],[X;Y→Z],and[X;Y;Z];thelatterthreearethesubsetofthechainstructuresthatincludethesamecausallinksorfewerlinks.Icallthisassumptionthe“alternatehypothesisspaceassumption”inthatthesetofpossiblestructures(hypothesisspace)isbeingchanged.

Underthisnewhypothesisspace,whenXisintervenedupon,eachofthesefourstructureswouldproducedifferentpatternsofdata,makingitpossibletodeterminewhichofthesefourstructuresismostlikely;seeTable3.ThismeansthatXhashighinformationgain.Incontrast,whenYorZismanipulated,someofthestructuresproducethesamepatternsofdata,meaningthattheyprovidelessinformationgain.

Table3:MostCommonPatternofDataProducedbyAnInterventionOnAParticularNodeforaParticularStructure

AlternateHypothesisSpace

Assumption

AlternateHypothesisSpaceAssumption&OnlyAttendingtoVariableswithSameState

asManipulatedVariableStructureHypothesisSpace

InterventionOn: InterventionOn:X Y Z X Y Z

X→Y→Z X=Y=Z X≠Y=Z X=Y≠Z X=Y=Z Y=Z ZX→Y;Z X=Y≠Z X≠Y≠Z X=Y≠Z X=Y Y ZX;Y→Z X≠Y=Z X≠Y=Z X≠Y≠Z X Y=Z ZX;Y;Z X≠Y≠Z X≠Y≠Z X≠Y≠Z X Y ZHowInformativeThisInterventionIs High Medium Medium High Medium Low

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Steyversetal.introduceanotherassumptionaswell;theyassumethatpeopleonly

attendtovariablesthattakeonthesamestateastheintervened-uponvariableandignoreanyvariablesthathaveadifferentstatefromthemanipulatedvariable.ThecombinationofthetwoassumptionsisdetailedintherightcolumnofTable3.WhenXisinterveneduponitwouldproducethreedifferentpatternsofdataforthefourcausalstructures,whichmeansthatithasfairlyhighinformationgain;itcandistinguishallbutthebottomtwostructuresinTable3.ThereasonwhyaninterventiononXcannolongerdistinguishbetweenthebottomtwostructuresisbecauseoftheassumptionthattheothertwovariablesthatdonotequalX,YandZ,areignored.

WhenYisintervenedupon,itcannarrowdownthespaceof4structuresdownto2,amediumamountofinformationgain.WhenZisinterveneduponallthestructuresproducethesamepatternofdatasoaninterventiononZdoesnothelpatalltoidentifythetruestructure.Insum,thecombinationofthesetwohypothesesnowmakesitsuchthatinterveningonXismoreinformativethanY,whichismoreinformativethaninventingonZ.Thispatternmatchesthefrequencyofparticipants’interventions,whichweremostfrequentlyonX,thenonY,andlastlyonZ.

TherearetwokeypointsmadebythisanalysisofthesimilaritiesbetweenthefindingsofCoenenetal.andSteyversetal.First,eventhoughtheyapproachtheresultsfromdifferentperspectivesandtalkabouttheresultsindifferentways,theybothfoundthatpeopletendedtointerveneonrootcauses.Second,eventhoughSteyvers’modelhasrationalelementstoit,theresultingmodelisnotveryclosetotheidealmodel,forwhichYisthemostinformativeintervention.Finally,bycomparingdifferentmodelswithdifferentassumptionsitcanbeseenhowthetwoassumptionsmadebySteyversetal.effectivelyamounttothepositiveteststrategyputforthbyCoenenatal.Restated,thesamebehavioralpatternofinterveningprimarilyonrootcausescouldbeexplainedinmorethanoneway.

Bramleyetal.,(2015)alsostudiedanumberofimportantfactorsrelatedtolearningfrominterventions.Overall,theyfoundthathumanswerehighlyeffectivecausallearners,andabletoselectandmakeuseofinterventionsfornarrowingdownthenumberofpossiblestructures.Oneparticularfactorheintroducedwasthepossibilityofinterveningontwovariablessimultaneouslyratherthanjustone.Doubleinterventionsareparticularlyhelpfultodistinguishbetween[X→Y→ZandX→Z]vs.[X→Y→Z].Withasingleintervention,thesetwostructuresarelikelytoproduceverysimilaroutcomes.Forexample,aninterventiononXislikelytoproducedatainwhichX=Y=Z,aninterventiononYislikelytoproducedatainwhichX≠Y=Z,andaninterventiononZislikelytoproducedatainwhichX=Y≠Z.

However,consideradoubleinterventionsettingX=1andY=0.Underthesimplechain[X→Y→Z],Zismostlikelytobe0,butinthemorecomplexstructureinwhichZisinfluencedbybothXandY,Zhasahigherchanceofbeing1becauseX=1.Bramleyetal.foundthatdistinguishingbetweenthesetwotypesofstructureswasthehardestdiscriminationinthisstudyandproducedthemosterrors.61ofthesubjectsrarelyuseddoubleinterventions,whereas49weremorelikelytousethem,suggestingthatpeoplemaynotusedoubleinterventionsfrequentlyenoughforcausallearning.

Insum,therearemanyopenquestionsabouthowpeoplechooseinterventions.Thereissomeevidencethatpeopledouseinformationgainwhenselectinginterventions,

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buttherearealsoavarietyofheuristics(onlyusesingleinterventions,interveneonrootcauses)andorsimplifyingassumptions(focusonalimitedanddistortedhypothesisspace,onlyattendtovariableswiththesamevalueasthemanipulatedvariable).Clearlythereismoreworktobedonetohaveafullerandmorerobustunderstandingofhowhumanschooseinterventionstolearnaboutcausalstructures.2.1.3LearningCausalStructuresfromInterventions:InterpretingInterventionsandUpdatingBeliefsabouttheStructure Thepriorsectiondiscussedhowpeoplechooseinterventionstolearnacausalstructure.Thissectionexamineshowpeopleinterprettheoutcomeaftermakinganintervention.Fourpatternshavebeenproposedtoexplainhowpeopleinterprettheoutcomesofinterventions.ThefirstisthatifavariableXismanipulatedandanothervariableZassumesthesamestateofX,peopletendtoinferadirectlinkfromXtoZ.Thoughthisheuristicmakessensewhenlearningtherelationsbetweentwovariables,itcanleadtoincorrectinferencesincasesinvolvingthreeormorevariableslinkedtogetherinachainstructuresuchasX→Y→Zbecauseitcanleadpeopletoinferadditionallinksnotinthestructure.IfalearnerintervenesonXsuchthatitis1,andsubsequentlyYandZareboth1,thisheuristicimpliesthatX→YandX→Z.Indeed,peopleofteninferthatthereisanX→ZlinkaboveandbeyondX→Y→Z,evenincaseswhenthereisnotadirectlinkfromXtoZ(Bramleyetal.,2015;Fernbach&Sloman,2009;Lagnado&Sloman,2004;Rottman&Keil,2012).(Inreality,thecorrectwaytodeterminewhetherthereisaX→ZaboveandbeyondX→Y→ZistoseewhethertheprobabilityofZiscorrelatedwiththestateofXwithintrialsinwhichYis1orwithintrialsinwhichYis0,ortousedoubleinterventionsasexplainedpreviously.)

Thisheuristicisproblematicfortworeasons.Atatheoreticallevel,itsuggeststhatpeoplefailtopayattentiontothefactthatXandZareindependentconditionalonY.Asalreadydiscussed,attendingtostatisticalindependenciesiscriticalforunderstandingMarkovequivalenceclasses,andthisfindingsuggeststhatpeopledonotfullyunderstandtherelationsbetweenstatisticalindependenceandcausalMarkovequivalenceclass.Atamoreappliedlevel,addingthisadditionallinkX→Zcouldleadtoincorrectinferences(seesection3.2).Inparticular,wheninferringthelikelihoodthatZwillbepresentgiventhatYispresent,peopletendtothinkthatXhasaninfluenceonZaboveandbeyondY.InreferencetoasubsetofFigure1,eventhoughthetruecausalstructureisEthnicity→Smoking→CardiovascularDisease,thisheuristiccouldleaddoctorstoincorrectlypredictthatpeopleofcertainethnicitiesaremorelikelytodevelopcardiovasculardiseaseevenafterknowingtheirsmokingstatus,eventhoughethnicityhasnoinfluenceoncardiovasculardiseaseaboveandbeyondsmoking(accordingtoThornley,2013).Suchamisperceptioncouldleadpeopleofthoseethnicitiestofeelunnecessarilyworriedthattheirethnicitywillcausethemtohavecardiovasculardisease. Thesecondpatternofreasoningwasalreadydiscussedintheprevioussection.Steyversetal.,(2003)proposedthatwhenapersonintervenesonavariable,thattheyonlyattendtoothervariablesthatassumethesamestateasthemanipulatedvariable.Theprevioussectionexplainedhowthistendencywouldbiasreasonerstointerveneonrootcauses(Table3).Butthistendencywouldalsodecreasetheeffectivenessoflearningfrominterventions.IfoneintervenesonZandtheresultingobservationisX=Y≠Z,thefactthatXandYhavethesamestateshouldincreasethelikelihoodthatthereissomecausalrelationbetweenXandY;however,thisheuristicimpliesthatpeoplewouldnotlearnanything

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aboutXorYbecausepeopleonlyattendtovariableswiththesamestateastheintervened-uponvariable(Z).Insum,thissimplificationmeansthatpeopledonotextractasmuchinformationfrominterventionsastheycould. ThethirdandfourthhabitsofupdatingcausalbeliefsafterinterventionscomefromthestudybyBramleyetal’sstudy(2015).InthisstudyparticipantsmadeaseriesofinterventionsonX,YorZ,andaftereachinterventiontheydrewthecausalstructurethattheybelievedtobethemostplausiblestructuregiventheevidenceuptothatpoint.Theydiscoveredtwointerrelatedhabits.First,participantsupdatedtheirdrawingsofthecausalstructureslowly.Thiscanbeexplainedasaconservativetendency;peopleneedconsiderableevidencebeforeaddingordeletingacausalrelationtotheirsetofbeliefs.Thesecondpatternisthatwhendrawingthecausalstructuresparticipantswereinfluencedbythemostrecentinterventionandappearedtoforgetmanyoftheoutcomesofpriorinterventions.Thecombinationofthesetwohabits,conservatismandforgettingcanbeexplainedwithananalogytobalancingacheckbook.Aftereachtransactiononeupdatesthecurrentbalancebyaddingthemostrecenttransactiontothepriorbalance,butonedoesnotre-calculatethebalancefromallpastexperiencesaftereachtransaction.Keepingtherunningbalanceisawaytosimplifythecalculation.Likewise,storingarepresentationofthecausalstructureasasummaryofthepastexperienceallowsthelearnertogetbywithoutrememberingallthepastexperiences;thelearnerjusthastoupdatethepriorcausalstructurerepresentation.Inarelatedvein,FernbachandSloman(2009)foundthatpeoplehavearecencybias–theyaremostinfluencedbythemostrecentdata,whichissimilartoforgetfulness.Understandingtheinterplaybetweenallofthesehabitswillprovideinsightsintohowpeoplelearncausalstructuresfrominterventionsinwaysthatarecognitivelytractable.2.1.4LearningTemporalCausalStructures Sofarthischapterhasfocusedonhowpeoplelearnaboutatemporalcausalnetworksinwhicheachobservationisassumedtobetemporallyindependent.Intheexampleatthebeginningofthechapteraboutcardiovasculardisease,eachobservationcapturedtheage,sex,smokingstatus,diabetesstatus,andothervariablesofanindividualpatient.Thecausallinkbetweensmokingandcardiovasculardisease,forexample,impliesthatacrosspatients,thosewhosmokearemorelikelytohavecardiovasculardisease. However,oftenitisimportanttounderstandhowvariableschangeovertime.Forexample,aphysiciantreatingpatientswithcardiovasculardiseaseisprobablylessinterestedinpopulation-leveleffects,andinsteadismoreinterestedinunderstandinghowachangeinsmokingwouldinfluenceanindividualpatient’sriskofdevelopingcardiovasculardisease. TemporalversionsofCBNscanbeusedtorepresentlearningandreasoningaboutchangesovertime(Ghahramani,1998;Murphy,2002alsoseeRehder,thisvolumeb).TemporalCBNsareverysimilartostandardCBNs,excepteachvariableisrepresentedbyaseriesofnodesforeachtimepointt.Thecausalstructureisoftenassumedtobethesameacrosstime,inwhichcasethecausalstructureisrepeatedateachtimepoint.Additionally,oftenvariablesareassumedtobeinfluencedbytheirpaststate;positiveautocorrelationmeansthatifthevariablewashighattimet,itislikelytobehighatt+1.

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Figure4:ATemporalCBN

Forexample,Figure4showsacausalnetworkrepresentingtheinfluenceofusinganantihypertensiveonbloodpressure.1representsusinganantihypertensiveorhighbloodpressure,whereas0representsnotusinganantihypertensiveorhavingnormalbloodpressure.Insteadofjusthavingonenodethatrepresentsusinganantihypertensiveandanotherforbloodpressure,nowthestructureisrepeatedateachtimepoint.Additionally,theautocorrelationcanbeseenwiththehorizontalarrows.Allthingsbeingequal,ifapatient’sbloodpressureishigh,itwilltendtostayhighforperiodsoftime.Likewise,ifapatientstartsusinganantihypertensive,theymightcontinuetouseitforawhile.

LikeallCBNs,temporalCBNsfollowthesamerulesandconventions.Hereinsteadofusinginodestorepresentinterventions,Iusedtexttoexplaintheintervention(e.g.,aphysicianprescribedanantihypertensive).TheinterventionsarethereasonthatsomeoftheverticalandhorizontalarrowsareremovedinFigure4becauseaninterventionmodifiesthecausalstructure.TheMarkovconditionstillholdsinexactlythesamewayasintemporalCBNs.Forexample,apatient’sbloodpressure(BP)atage73isinfluencedbyhisBPat72,buthisBPat71doesnothaveaninfluenceonhisBPat73aboveandbeyondhisBPatage72. Causallearningfrominterventionsworksinessentiallythesamewayintemporalandatemporalcausalsystems.InFigure4itiseasytolearnthatusinganantihypertensiveinfluencesbloodpressure,notthereverse.Whenthedrugisstarted,thepatient’sBPdecreases,andwhenthedrugisstopped,thepatient’sBPincreases.Butwhenanotherintervention(e.g.,exercising)changesthepatient’sbloodpressure,itdoesnothaveaneffectonwhetherthepatientusesastatin. Oneinterestingaspectabouttemporalcausalsystems,isthatispossibletoinferthedirectionofacausalrelationshipfromobservations,whichisnotpossiblewithatemporalsystems.ConsiderthedatainFigure5;thedirectionofthecausalrelationisnotshowninthefigure.Thereisanasymmetryinthedata;sometimesXandYchangetogether,andsometimesYchangeswithoutXchanging,butXneverchangeswithoutYchanging.ColleaguesandIhavefoundthatbothadultsandchildrennoticethisasymmetryanduseittoinferthatXcausesY(Rottman&Keil,2012;Rottman,Kominsky,&Keil,2014;Soo&Rottman,2014).ThelogicisthatYsometimeschangesonitsown,implyingthatwhatevercausedYtochangedidnotcarryovertoX;YdoesnotinfluenceX.Furthermore,sometimesXandYchangetogether.SincewealreadybelievethatYdoesnotinfluenceX,onewayto

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Age7170 72 73 74 75 76

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77Stopped taking

medicine because of fatigue

started exercising

stopped exercising

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explainthesimultaneouschangeinXandYisthatachangeinXcausedthechangeinY.2ThisisonewayinwhichhumancausallearningseemsmoreakintolearningtemporalCBNratherthananatemporalCBN;thetemporalaspectofthisdataiscriticalforinferringthecausaldirection.Figure5:ExampleofLearningCausalDirectionfromTemporalData

AnumberofotherphenomenafitwellintothetemporalCBNframework.Consider

thedatainFigure6ObservedData.Inthissituation,subjectsknowthatCisapotentialcauseofE,notthereverseandthegoalistojudgetheextentthatCinfluencesE.OverallthereisactuallyzerocorrelationbetweenCandE.ThefaithfulnessassumptionstatesthattheonlyindependenciesinthedataarisethroughtheMarkovassumption.IfCandEareunconditionallyindependent,itmeansthatCcannotbeadirectcauseofE.Instead,anotherpossibility(PossibleStructure1inFigure6)isthatsomeunobservedthirdvariableUisentirelyresponsibleforE.

Figure6:LearningaboutanInteractionwithanUnobservedFactor

However,whenfacedwithdatalikeinFigure6,peopledonotconcludethatCis

unrelatedtoE;insteadtheynoticethatthereareperiodsoftimeinwhichChasapositiveinfluenceonE(Times0-3),andotherperiodsoftimeinwhichChasanegativeinfluenceonE(Times4-7).TheysubsequentlytendtoinferthatCdoesactuallyhaveastronginfluenceonE,butthatthereissomeunobservedfactorthatisfairlystableovertime,andCandtheunobservedfactor(U)interacttoproduceE(Rottman&Ahn,2011).Thisexplanationis2Technically,thereasonwhyitispossibletolearnthedirectionofthecausalrelationistheautocorrelation,thebeliefthatYt→Yt+1andthatXt→Xt+1.Thus,thelearnerisreallydiscriminatingbetween[Yt→Yt+1←Xt+1←Xt]and[Yt→Yt+1→Xt+1←Xt],whichareindifferentMarkovequivalenceclasses.IthankDavidDanksforpointingthisout.

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representedinFigure6PossibleStructure2.Inthisstructure,bothCandUinfluenceE,andthereisanarkbetweenthetwolinks,whichrepresentsaninteraction;inthiscasetheinteractionisaperfectcross-oversuchthatEis1ifbothCandUare1orbothare0.ThereasonpeopleappeartomakethisinferenceaboutthecrossoverinteractionwithanunobservedcauseratherthaninferringthatCisunrelatedtoEisbecausethedataaregroupedintodistinctperiodssuchthatthereareperiodsduringwhichthereissometimesapositiverelationandothertimesanegativerelationbetweenCandE.ThisallowsthereasonertoinferthatsomeunobservedfactorUmustaccountfortheswitch.Ifthesame8trialswererandomizedthenpeopletendtoinferthatonlyUisacauseofE,notC.Thisinferenceagainsuggeststhatpeopletendtorepresentcausalsystemsastemporallyextended(thatvariablessuchasUtendtobeautocorrelated)ratherthanatemporal(seeRottman&Ahn,2009,foranotherexample).

Elsewhere,colleaguesandIhaveargued(Rottmanetal.,2014)thatmanyofthecausallearningphenomenathathavebeenusedasevidencethatpeoplelearnaboutcausalrelationsinwaysakintoCBNsareevenbetterexplainedbytemporalCBNs.Forexample,onestudyfoundthatchildrencanlearnaboutbidirectionalcausalrelationsinwhichtwovariablesbothcauseeachother(Schulz,Gopnik,&Glymour,2007).Bidirectionalcausalstructurescanonlyberepresentedthroughtemporal,notatemporalcausalnetworks(Griffiths&Tenenbaum,2009;Rottmanetal.,2014). Inconclusion,thereisgrowingevidencethat,atleastincertainsituations,peopleappeartobelearningsomethingsimilartoatemporalcausalnetwork,andthetemporalaspectofreasoningallowsthemtoinferquitesophisticatedcausalrelationsthatwouldotherwisebeimpossibletolearn.2.2LearningabouttheIntegrationFunction AnotheraspectofaCBNthatmustbelearnedinadditiontothestructureistheintegrationfunction;thewaythatmultiplecausescombinetoinfluenceaneffect(alsoseeGriffiths,thisvolume,Rehder,thisvolumeaandb).Forexample,inregression,thepredictorsaretypicallyassumedtocombinelinearly.TheCBNframeworkallowsforthepossibilitythatcausescanpotentiallycombineinanyconceivableway,andhumansareextremelyflexibleaswell.Forexample,Waldmann(2007)demonstratedthatpeoplenaturallyreasonaboutcausesthatareadditive(e.g.,theeffectoftakingtwomedicinesisthesumofthetwoindividualeffects)andaverages(e.g.,thetasteoftwochemicalsmixedtogetheristheaverageofthetwo).Furthermore,peopleusebackgroundknowledge(e.g.,aboutmedicinesandtaste)todecidewhichtypeofintegrationfunctionismoreplausibleinagivensituation. Mostresearchoncausallearninghasfocusedonbinaryvariables.Themostprominentintegrationfunctionforbinaryvariables,calledNoisy-OR,describessituationsinwhichtherearemultiplegenerativecauses(Cheng,1997;Pearl,1988).Itstipulatesthattheprobabilityoftheeffectbeingabsentisequaltotheprobabilitythatallthecauseshappentosimultaneouslyfailtoproducetheeffect.Iftherearetwocauses,eachofwhichproducetheeffect50%ofthetimeontheirown(acausalstrengthof.5),thenbothwouldfailsimultaneously25%ofthetime;theeffectshouldoccur75%ofthetime.Iftherearethreecauses,eachofwhichproducestheeffect50%ofthetime,thenallthreewouldsimultaneouslyfail53=12.5%ofthetime;theeffectwouldbepresent87.5%ofthetime.AnanalogousintegrationfunctioncalledNoisy-And-Notcanbeusedtodescribeinhibitorycausesthatcombineinasimilarfashion.Itisnotdifficulttoimagineothersortsof

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integrationfunctions,andthefollowingstudieshaveexaminedhowpeoplelearnabouttheintegrationfunctionfromdata. Beckersetal.(2005;seealsothechapterinthisvolumebyBoddez,DeHouwer,&Beckers)studiedhowbeliefsabouttheintegrationfunctioninfluencelearning.Inonestudyparticipantsfirstlearnedabouttwocauses,GandH,bothofwhichproduceanoutcomeof1onetheirown.Inthe“additive”conditiontheysawthatGandHtogetherproduceanoutcomeof2,whichisconsistentwithanintegrationfunctioninwhichtwocausesaddtogether.InanotherconditiontheysawthatGandHtogetherproduceanoutcomeof1.Thisisinconsistentwiththenotionthatthetwocausesaddtogether;insteaditsuggestssomesortof“sub-additive”integrationfunctioninwhichtheeffectcanneverbehigherthan1.SubsequentlyparticipantsinbothconditionsexperiencedablockingparadigminwhichtheylearnthatAbyitselfproducesanoutcomeof1,andAplusXproducesanoutcomeof1.InthesubadditiveconditionparticipantsstillthoughtthatXmightbeacausebecausetheybelievedthattheeffectcouldnevergohigherthan1.Incontrast,intheadditiveconditiontheyconcludedthatXwasnotacause;ifitwas,thenpresumablytheeffectwouldhavebeen2.

LucasandGriffith(Lucas&Griffiths,2010)investigatedasimilarphenomenon,thatinitialtrainingabouthowcausescombineinfluenceswhethersubjectsinterpretthatavariableisacauseornot.Theyfirstpresentedpeoplewithdatathatsuggestedthatthecausesworkedconjunctively(multiplecauseswereneededtobepresentfortheeffecttooccur),orthroughthenoisy-ORfunction(asinglecausewassometimessufficienttoproducetheeffect).Afterwards,participantssawacauseDneverproducetheeffect,andsawthattwocausesincombination,DandF,producedtheeffect.ParticipantsintheconjunctiveconditiontendedtoconcludethatbothDandFwerecauses,whereasparticipantsinthenoisy-ORconditiontendedtoinferthatonlyFwasacause.

Insum,theseresultsshowthatpeoplequicklyandflexiblylearnabouthowcausescombinetoproduceaneffectandtheintegrationrulethattheylearndramaticallyinfluencessubsequentreasoningaboutthecausalsystem.2.3LearningCausalStrength Sofarthischapterhasfocusedonhowpeoplelearncausalstructure,andtoalesserextentintegrationfunctions.Oneotherimportantcomponentofcausalrelationsiscausalstrength,ourinternalmeasurementofhowimportantacauseis.Forexample,ifamedicineworksverywelltoreduceasymptom,ithashighcausalstrength,butifitdoesnotreducethesymptomatallithaszerocausalstrength. PriortotheCBNframework,theoriesofcausalstrengthlearningwerebasedonsimplemeasuresofthecontingencybetweenthecauseandeffect.Forexample,theΔPmodelcomputesthestrengthoftheinfluenceofacause(C)onaneffect(E)bytheextentofthedifferenceintheprobabilityoftheeffectwhenthecauseispresentvs.absent;P(e=1|c=1)-P(e=1|c=0)(Cheng&Novick,1992;Jenkins&Ward,1965).Thissamecontrastiscalculatedatasymptotebyoneofthemostinfluentialmodelsconditioningasawaytocapturehowstronglyacueandoutcomebecomeassociatedbyananimal(Danks,2003;Rescorla&Wagner,1972).Thissamemodelhasalsobeenproposedasamodelofcausallearning,theideabeingthatthestrongerthatacueisassociatedwithanoutcome,thestrongerthathumanswouldinferthatthecuecausestheoutcome(DavidRShanks&Dickinson,1987).

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WiththeintroductionoftheCBNframeworkanumberoftheoriesofcausallearningwereproposedthatincorporatedifferentsortsoftop-downcausalbeliefsintothelearningprocess.AnumberofotherchaptersdiscusscausalstrengthlearningincludingthosebyGriffiths,ChengandLu,andPerales,Catena,MaldonadoandCándido.Thus,IbrieflydiscusstheconnectionsbetweentheCBNframeworkandtheoriesofcausalstrengthlearning,whileleavingthedetailstothoseotherchapters.2.3.1ElementalCausalInduction:LearningCausalStrengthBetweenTwoVariables OneofthemostimportantdevelopmentsofmodelsofcausalstrengthlearningisthePower-PCmodel(Cheng,1997).ThismodelbuildsofftheΔPmodelbyincorporatingcausalbeliefsandassumptions.ThismodelassumesthatonegenerativecausecombinesthroughtheNoisy-ORintegrationfunctionwithanotherunobservedcause.Forexample,imaginethattheeffectEoccurs25%ofthetimewithouttheobservedcauseC;P(e=1|c=0)=.25.Wecanattributethis25%tosomebackgroundcausethathasastrengthof.25.Further,imaginethattheobservedcausehasastrengthof2/3.Whentheobservedcauseispresent,theeffectshouldoccur75%ofthetimeifCandthebackgroundcausecombinethroughanoisy-ORfunction;P(e=1|c=1)=.75.(Theeffectwouldfailwithaprobabilityof1/3×3/4=¼).

Chengusedthissortoflogic,inreverse,todeducethatifanobservedcausecombineswithabackgroundcausethroughanoisy-ORintegrationfunction,thecorrectwaytocalculatecausalstrengthinvolvesdividingΔPbyP(e=0|c=0).Considernowtheprobabilitiesjustpresented,withoutknowingthecausalstrength:P(e=1|c=1)=.75andP(e=1|c=0)=.25.AccordingtoΔP,thecausalstrengthis.5;thecausesraisestheprobabilityoftheeffectby.5.AccordingtoPower-PC,thecausalstrengthofCis(.75-.25)/(1-.25)=.67;thecauseincreasestheeffectby2/3rds(from.25to.75).Insum,byspecifyingasetofpriorbeliefsaboutthecausalrelation,Chengspecifiedhowcausalstrengthshouldbeinducedgiventhosebeliefs. AnotherinfluentialdevelopmenttocausalstrengthlearningistheCausalSupportmodel.GriffithsandTenenbaum(2005)proposedthatwhenpeopleestimatecausalstrength,whattheyareactuallydoingisnotjudgingthemagnitudeoftheinfluenceofthecauseontheeffect,similartoeffectsizesininferentialstatistics,butratherjudgingtheextenttowhichthereisevidencethatthereisanycausalrelationornot,similartothefunctionofap-valueinhypothesistesting.Atatheoreticallevel,thismodeliscalculatedbydeterminingtherelativelikelihoodthatthetruecausalstructureis[C→E←U],thatbothCandanunobservedfactorUinfluenceEvs.thatthetruecausalstructureis[C;E←U],thatCdoesnotinfluenceEandEisdeterminedbyanunobservedfactorU.Thus,causalsupporttreatscausalstrengthlearningasdiscriminatingbetweentwopossiblecausalstructures,oneinwhichCactuallyisacauseofE,andoneinwhichCisnotacauseofE.

CausalSupporthasanumberofbehavioralimplications,butthemostobviousoneandeasiesttothinkaboutissamplesize.WhereasΔPandPower-PCareunaffectedbysamplesize,CausalSupportisinfluencedbysamplesize.GoingbacktotheanalogyofCausalSupportasap-valuewhereasΔPandPower-PCareeffectsizemeasures,ifthereisalargeenoughsamplesizeitispossibletohaveaverylowp-value(confidentthatthereisacausalrelation)eveniftheeffectsizeissmall. Insum,PowerPCandCausalSupportwerebothmotivatedbyunderstandingcausalitythroughaCBNperspective,involvingtop-downbeliefsabouthowanobservedcausecombineswithotherunobservedfactors.

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2.3.2InferringCausalStrength:ControllingforOtherCauses Theprevioussectionfocusedonhowpeopleinfercausalstrengthgivenobservationsofjustasinglecauseandeffect,elementalcausalinduction.However,oftentherearemorethantwovariables.Wheninferringthestrengthofonecauseonaneffectitisimportanttocontrolforcertaintypesofthirdvariables(andnotothers),dependingonthecausalstructure.ConsiderFigure1.Whenstudyingthestrengthoftheeffectofanewdrugoncardiovasculardisease,itisimportanttocontrolforageandsmokinghabits,eitherstatisticallyorthroughthedesignofthestudy.Oneshouldnotcontrolforstatinusebecauseitisnotadirectcauseofcardiovasculardisease.Figure7:PossibleThirdVariableswhenLearningtheCausalRelationfromCtoE

Moregenerally,considertryingtolearnifthereisacausallinkfromapotentialcauseCtoapotentialeffectE,andifso,howstrongtherelationis.Figure7presents8differentthirdvariables(S-Z);thequestioniswhichofthesevariablesshouldbecontrolledfor.Forreadersfamiliarwithmultipleregression,youcanthinkofCasonepredictorintheregressionthatyouareprimarilyinterestedin,andEistheoutcomevariable.Thequestionaboutcontrollingforalternativevariablesiswhichofthesevariablesshouldbeincludedaspredictorsorcovariatesintheanalysis?ThefollowingbulletssystematicallyexplaineachofthethirdfactorsandwhetheritshouldbecontrolledforwheninferringthestrengthofConE:

• VandXareconfoundsandmustbecontrolledforwheninferringtherelationofConE.IftheyarenotcontrolledfortherewouldbeaspuriouscorrelationbetweenCandEevenifthereisnocausalrelationbetweenCandE.(XrepresentsthecasewhensomeunobservedfactorcausesbothCandX.)

• WrepresentsanalternativemechanismfromCtoE.InordertotestwhetherthereisadirectinfluenceofConEaboveandbeyondWitmustbecontrolledfor.

• Yisanoisevariable.AccountingforitincreasesourpowertodetectarelationbetweenCandE.

• UandZshouldnotbecontrolledfor.Thelogicisabitopaque(Eells,1991,p.203),butconsiderthesimplecasethatEdeterministicallycausesZsuchthattheyareperfectlycorrelated.ControllingforZexplainsallthevarianceinE,andtherewillbenoleftovervarianceforCtoexplain.ControllingforZandUcandistorttheapparentrelationbetweenCandE.

• SandTneverneedtobecontrolledfor.WithlargesamplesizesitdoesnotmatterifSandTarecontrolledforornotwheninferringtheinfluenceofConE.ThereasonisthateventhoughSandTarecorrelatedwithC,sinceSandTarescreenedofffromE

C

W

E

S

Z

T VU X Y

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(SandTareindependentofEaftercontrollingforC),theywillnothaveanypredictivepowerinaregressionaboveandbeyondC.However,withsmallsamplesizes,mostlikelySandTwillnotbeperfectlyuncorrelatedwithEcontrollingforC,inwhichcasetheycanchangetheestimatedinfluenceofConE.Thus,theyshouldnotbecontrolledfor.

Insum,theoverallruleisthatwheninferringthestrengthofarelationofConE,thirdvariablesthatarebelievedtobepotentialdirectcausesofEshouldbecontrolledfor;othervariablesshouldnotbecontrolled(Cartwright,1989;Eells,1991;Pearl,1996).Thisrulenicelydovetailswithhowcausalstructuresaredefined;eachvariableismodeledusingaconditionalprobabilitydistributionincorporatingallofitsdirectcauses. Remarkably,avarietyofresearchsuggeststhatpeoplehavetheabilitytoappropriatelycontrolforthirdvariableswheninferringcausalstrength.Infact,researchonthistopicwasthefirstresearchonwhetherpeopleintuitivelyusebeliefsaboutcausalstructurewhenreasoningaboutcausality(Waldmann&Holyoak,1992;Waldmann,1996,2000).MichaelWaldmannandcolleaguescalledthistheorytheCausalModeltheory;theideawasthatwheninferringcausalstrength,peopleusebackgroundknowledgeaboutthecausalstructure(“model”)todeterminewhichvariablestocontrolfor.Inthefirststudyonthistopic,ascenariowiththreevariablesX,Y,andZwassetup.Basedonthecoverstorythethreevariableswereeithercausallyrelatedinacommoneffectstructure[X→Y←Z]orinacommoncausestructure[X←Y→Z].Inthecommoneffectcondition[X→Y←Z],thegoalforparticipantswastodecidetheextenttowhichXandZwerecausesofY;normativelypeopleshouldcontrolforalternativecauses(e.g.,controlforXwhendeterminingwhetherZisacauseofY).Inthecommoncausecondition[X←Y→Z],thegoalforparticipantswastodecidetheextenttowhichXandZareeffectsofY;normativelythesetwodecisionsshouldbemadeseparately(e.g.,oneshouldignoreXwhendeterminingtheinfluenceofYonZ). Afterthecoverstorymanipulatingthebelievedcausalstructure,participantsfirstexperiencedasetofdatainwhichXandYwereperfectlycorrelated;Zwasnotdisplayed.ThistrainingmadeitseemthatthereisastrongcausalrelationbetweenXandY.ThentheyexperiencedasetofdatainwhichX,Y,andZwereallperfectlycorrelated;nowZisaredundantpredictorofYbecauseXisentirelysufficienttopredictY.Insum,participantsexperiencedtheexactsamedata,andtheonlydifferencebetweenthetwoconditionswastheirbeliefaboutthecausalstructure. Inthecommoneffectcondition[X→Y←Z],participantscontrolledforXwheninterpretingwhetherZwasacauseofY,andconsequentlyconcludedthatZisnotacauseofYbecauseXisentirelysufficienttopredictwhetherYwaspresentorabsent.Incontrast,inthecommoncausecondition[X←Y→Z],participantsdidnotcontrolforX,andconcludedthatYwasacauseofbothXandZ.

Subsequently,anumberofotherstudieshavealsoshownthatpeoplecontrolforalternativecauses(V-YinFigure7)ofthemaineffectandnotalternativeeffectsofthemaincause(TinFigure7)(Goodie,Williams,&Crooks,2003;Spellman,Price,&Logan,2001;Waldmann,2000).ThereisevenworksuggestingthatpeopledonotcontrolforvariableslikeSandZ(Waldmann&Hagmayer,2001);however,therehasnotbeenresearchonwhetherpeoplecontrolforvariableslikeU.

Insum,whenlearningaboutacausalrelationbetweenCandE,peoplehavesomecoreintuitionstocontrolforvariablesthattheybelievetobealternativecausesofE,and

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nototherroles,whichiscriticalforcorrectcausallearning(Glymour,2001).Thisresearchissomeofthemostdramaticshowinghowtop-downbeliefsaboutcausalstructureinfluencelearning,andconsequentlyissomeofthestrongestevidencethathumancausalreasoninginvolvesstructureddirectionalrepresentationsbeyondjustassociationsbetweenvariables(Waldmann,1996).3ReasoningwiththeCausalStructure Sofarthischapterhasfocusedonhowpeoplelearnaboutacausalnetwork;thestructureofthenetwork,theparametersorcausalstrengths,andthefunctionalform.Theremainderofthechapterishowpeopleusethisknowledge(seealsoOaksfordandChater,thisvolume).GoingbacktoFigure1,onemightdesiretoexplainwhetheraperson’scardiovasculardiseasewascausedbyhisage,orhissmoking.Onemightdesiretopredictwhetherhiscardiovasculardiseasewillgetworseasheages.Andonemightdesiretoknowwhichintervention,stoppingsmokingorstartingtotakeastatinwouldhavethelargestinfluenceonhiscardiovasculardiseaseinordertochoosetheactionwiththegreatestrewards. Thoughthissecondhalfofthechapterfocusesonreasoningaboutthecausalnetworkratherthanlearning,itisimpossibletocompletelydivorcelearningandreasoning.Intherealworldwelearnaboutcausalrelationsbothfromfirst-handexperiencewithdata(e.g.,didstartingthestatinlowermybloodpressure)andalsofromcommunicatedknowledge(e.g.,fromfamilymembers,teachers,doctors,newspaperarticles).Researchinpsychologyhasusedbothpersonalexperienceandcommunicatedknowledge,oftenincombination,toteachsubjectsaboutthecausalstructurebeforetheyreasonaboutthestructure.Typicallywordsandpicturesareusedtoconveythecausalstructuretoparticipants,althoughthestructuralinformationissometimesconveyedthroughorsupplementedwithexperienceddata.Iftheparticipantslearnanythingabouttheparameters(causalstrengths)ofthecausalstructure,itisusuallyconveyedthroughdata-drivenexperience,thoughsometimestheparametersareconveyedtextually.Theintegrationfunctionisoftennotmentionedatall,thoughsometimesitismentioned.

Oneofthechallengeswithstudyinghowwellpeoplereasonaboutcausalstructuresisthatapparentflawsinreasoningcaneitherbeexplainedasreasoningbiases,oraspoor,biased,orinsufficientlearningaboutthecausalstructure.Itisnotclearhowtocleanlydifferentiatethetwobecausecheckingthatthecausalstructureislearnedappropriatelyinvolvesquestionsthataretypicallyviewedasreasoningaboutthecausalstructure.Thissetsupadifficultsituationbecauseanyobservedreasoningbiascanpotentiallybeexplainedawaybyclaimingthattheresearcherfailedtosufficientlyconveythecausalstructuretotheparticipants.HereIdonottrytosolvethisproblem,butinsteadjustpresenttheempiricalfindingsofhowcloselyreasoningappearstofitwiththecausalstructurespresentedtosubjects.Theseconclusionsarebasedonamuchmorethoroughanalysisoftheliteraturethancanbepresentedhere(Rottman&Hastie,2014),thoughthischapterincludessomenewlypublishedevidence.3.1ReasoningbasedonObservationsvs.Interventions InSection2,IexplainedhowtheCBNframeworktreatsobservationsandinterventionsverydifferentlyforlearningacausalstructure.Interventionschangethecausalstructurebyremovinglinksfromvariablesthatwerepreviouslycausesofthemanipulatedvariable.Forexample,giventhestructureX→Y→Z,ifYisintervenedupon,YgetsseveredfromXresultingin[X;Y→Z].UnderaninterventiononY,Xwouldbe

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statisticallyindependentoruncorrelatedfromY,eventhoughZwouldstillbedependentuponY. Practically,giventhestructureX→Y→Z,ifareasonercanobservethestateofY,theycanmakeapredictionaboutbothXandZ.Inthetypesofsituationstypicallystudiedinthelabwithbinaryvariablesandpositivecausalrelations,ifYisobservedas1,thenXandZarebothlikelytobe1aswell.However,ifareasonerintervenesonYandsetsitsvalueto1,thenZislikelytobe1,butthisinterventionwouldhavenoinfluenceonX,sothebestestimateofXissimplyitsbaserate.Insum,interventionsonlyinfluencevariablesdown-streamfromthemanipulatedvariable,notup-stream(butseeHiddleston,2005foranalternativeapproach,andalsoseethechapterbyOveronwhether“if...then”conditionalsareinterpretedasinterventions). Anumberofresearchershavefoundthatpeoplediscriminatebetweenobservationsandinterventionswhenmakinginferencesbasedonacausalstructure.SlomanandLagnado(2005)setupsimpleverbaldescriptionsinwhichoneevent(X)causestheother(Y),andfoundthatwhenYwasobservedtohaveaparticularvalue,Xwouldbeinferredtohavethesamevalue,butwhenYwasintervenedupontohaveaparticularvalue,Xwasinferredtohaveitsnormaldefaultvalue.Insum,whenitwasmadeveryclearwhethertherewasanobservationvs.anintervention,subjects’judgmentslargelyfollowedtheprescriptionsoftheCBNframework.Incontrast,whenmoreambiguouslanguageisusedsuchthatthevalueofavariablecouldbeknowneitherthroughanobservationoranintervention,thentheresponseslookedmoremuddy(seealsoRips,2010). Anothersetofstudiestookthisbasicfindingastepfurtherbydemonstratingthatthisdifferencebetweeninterventionsvs.observationsalsoholdsincontextsinwhichparticipantsaretoldthecausalstructureandthenlearntheparameters(e.g.,thebaseratesandthecausalstrengths)fromexperience.Considerasetofstudiesthatinvestigatedreasoningonadiamondstructure[X←W→YandX→Z←Y](Meder,Hagmayer,&Waldmann,2008,2009;Waldmann&Hagmayer,2005).Thesestudiesareuniqueforinvolvingmorethanthreevariables,andalsoforhavingtwocausalroutesW→X→ZandW→Y→Z.Despitethecomplexitiesinvolvedinthesestudies,theparticipantsshowedremarkablesubtletyinreasoningaboutthecausalstructures,anddistinguishingbetweeninterventionsandobservationsdifferently.

ConsiderobservingalowvalueofX,andtryingtoinferthevalueofZ.InthediamondstructuretherearetworoutesfromXtoZ:X←W→Y→ZandX→Z.DuetothesetworoutesXandZshouldbestronglycorrelated,andthusZshouldbequitelowwhenXisobservedtobelow.Incontrast,ifXisinterveneduponandsettoalowvalue,therouteX←W→Y→Zisdestroyed–thelinkfromWtoXiscut.TheX→Zrouteisstillopen,sothepredictedvalueofZisstilllow,butitshouldnotbeaslowaswhenXisobserved.Infact,thisistheexactpatternofreasoningthatwasobserved;theinferenceofZafteranobservationofXwaslowerthanafteraninterventiononX.Thisfindingfurthersuggeststhatpeoplereasonaboutobservationsbothdown-streamandup-stream,buttheyreasonaboutinterventionsonlydown-stream.Thisresearchalsoshowshowpeoplecanreasonaboutobservationsandinterventionsonmorecomplexstructures. Sofarthissectionhasfocusedon“perfect”interventionsinwhichtheinterventioncompletelydeterminesthestateofthemanipulatedvariables,andcompletelyseversallotherinfluences.However,ofteninterventionsarenotperfect.Forexample,afterprescribingapatientanantihypertensivetotreathighbloodpressure,thepatientmaynot

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actuallytakeit,ormaynottakeitexactlyasprescribed(e.g.,asfrequentlyastheyshould,attherightdose).Furthermore,evenifthepatientdoestakethemedicineasprescribed,themedicinedoesnotguaranteethatallpatientswillhavea120/80bloodpressure.Patientswhoinitiallyhadveryhighbloodpressureswillprobablystilltendtohavehigherbloodpressuresthanthosewhoinitiallyhadmoderatelyhighbloodpressures.Or,perhapsthemedicineonlysucceedstobringthebloodpressureintoanormalrangeforacertainpercentageofpatients,butnotforothers.Intheseways,takinganantihypertensiveisan“imperfect”interventiononbloodpressure;apatient’sbloodpressureisnotcompletelydeterminedbytheintervention.Insuchcasesofimperfectinterventions,reasoningup-streamiswarrantedtosomeextent,similartoobservations.Unfortunately,therehasbeenfairlylittleworkexamininghowpeoplereasonaboutimperfectinterventions(Meder,Gerstenberg,Hagmayer,&Waldmann,2010;Meder&Hagmayer,2009).

Insum,theexistingresearchhasfoundthatpeopledodistinguishbetweeninterventionsandobservationswhenreasoningaboutcausalsystems,inparticularthatinterventionsonlyinfluencevariablesdown-streamfromtheintervened-uponvariable.Animportantdirectionforfutureresearchistoexaminehowpeoplereasonaboutimperfectinterventions.Thisseemsespeciallyimportantgiventhatmanyoftheactionsor“interventions”humansperformarenotperfectinterventions.3.2DoPeopleAdheretotheMarkovConditionwhenReasoningaboutCausalStructures? RecallthattheMarkovconditionstatesthatonceallthedirectcausesofavariableZarecontrolledfororheldconstant,ZisstatisticallyindependentofeveryvariableinthecausalnetworkthatisnotadirectorindirecteffectofZ.Forexample,inthestructureX→Y→Z,ZisconditionallyindependentofXonceY(theonlydirectcauseofZ)isheldconstant.PeoplehaveoftenbeenfoundtoviolatetheMarkovassumption;theirinferencesaboutthestateofZareinfluencedbythestateofXevenwhentheyalreadyknowthestateofY(Mayrhofer&Waldmann,2015;Park&Sloman,2013;Rehder&Burnett,2005;Rehder,2014;Rehder,thisvolumeb;Walsh&Sloman,2008).Specifically,peopletendtoinferthatP(z=1|y=1,x=1)>P(z=1|y=1,x=0)eventhoughtheyshouldbeequivalent.Likewise,theyuseZwheninferringXevenafterknowingthestateofY.Goingbacktosection2.1.3,suchamistakecouldleadadoctortoincorrectlybelievethatethnicityhasaninfluenceoncardiovasculardiseaseaboveandbeyondsmokingevenwhenthetruecausalstructureisEthnicity→Smoking→CardiovascularDisease. ThereareavarietyofpossibleexplanationsforwhyinferencesviolatetheMarkovcondition,andmostoftheexplanationshaveattemptedtofindrationalizationsfortheviolations,reasonsthatsuchjudgmentswouldmakesenseaccordingtotheCBNframeworkassumingsomemodificationtothestructureduetopriorknowledge.Forexample,ifsubjectsbelievethatthereissomeothercausallinkbetweenXandZ(e.g.,X→Z,X←Z,orX←W→Z)inadditiontothecausalstructuretoldtothembytheexperimenter(X←Y→Z),suchadditionalinformationcouldjustifytheirinferences.ThreespecificproposalsarethatpeopleinferanunobservedfactorthatinhibitsbothXandZ,anunobservedfactorthatinfluencesX,Y,andZ,oranintermediarymechanismMsuchthatZcausesA,whichinturncausesXandY.Differentarticlesinthelistabovehavesupporteddifferentaccounts.Forexample,BurnettandRehder(2005)arguedfortheaccountinwhichanunobservedfactorinfluencesX,Y,andZ.ParkandSloman(2013)foundthatpeopleonlymaketheMarkovviolationwhenthemiddlevariableispresent,notabsent;

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P(X=1|Y=1,Z=1)>P(X=1|Y=1,Z=0)butthatP(X=1|Y=0,Z=1)=P(X=1|Y=0,Z=0).ThisfindingismostconsistentwiththeaccountthatpeopleinferanunobservedfactorthatinhibitsXandZ.TheyalsofoundthatthesizeoftheMarkovviolationwaslargerwhenparticipantsbelievedthatthetwoeffects(XandZ)arebothcausedthroughthesamemechanism(e.g.,YcausesmechanismA,whichinturncausesXandZ),thanthroughseparatemechanisms(e.g.,X←A←Y→B→Z,whereAandBarethetwomechanismsthatexplainhowXandZareeachcausedbyY).MayrhoferandWaldmann(2015)havealsofoundevidencethatpeopleinferanunobservedinhibitoryfactorthatinfluencesmultipleeffectsofthesamecause.AndtheyfurtherfoundthatthesizeoftheMarkovviolationwasinfluencedbywhetherthecausesandeffectsweredescribedasagentsvs.patients(e.g.,cause“sending”informationtoeffectvs.effect“reading”informationfromcause).

Rehder(2014)foundsomesupportforboththeunobservedinhibitorandtheonevs.twomechanismaccounts,thoughmoregenerallyhefoundthatnoneoftheserationalizationsprovideaparsimoniousandcomprehensiveexplanationforallthereasoningerrors.Hearguedthatitisindeedhighlylikelythatpeopleembellishcausalstructuresgiveninexperimentswithadditionalnodesandlinksbasedontheirownpriorknowledge.However,Rehderproposedthatinadditiontoanyembellishmentsduetobackgroundknowledge,somejudgmentsfollowedanassociative-styleofreasoningthatdoesnotobeytheMarkovassumption.Heproposedtakinganindividual-differencesapproachtounderstandingwhycertainpeoplearemorelikelytouseanassociativestyleofreasoning. OnesurprisingaspectabouttheworkonwhetherpeopleupholdtheMarkovconditionisthattherehavebeenveryfewstudiesinwhichpeoplelearntheparametersofthecausalstructurethroughtrial-by-trialexperience,andthenmakejudgments.3GivingparticipantsstatisticalexperiencewiththecorrelationsbetweenthevariablesprovidesthemwithdirectevidencethatXandZarestatisticallyindependentgivenY.ParkandSloman(2013)conductedoneexperimentofthissort.TheirparticipantsinferredthatP(z=1|y=1,x1)>P(z=1|y=1,x=0),thoughP(z=1|y=0,x=1)=P(z=1|y=0,x=0);aviolationoftheMarkovconditiononlywheny=1.Asdiscussedabove,thispatternactuallyfitstheproposalthatpeopleinferanunobservedinhibitorycauseofbothXandY.However,themodifiedstructurewiththeunobservedinhibitorycauseisstillunfaithfultothedatathattheyobserved;inthelearningdataXandZwereindependentwheny=1.Thisraisesaquestionforfutureresearch:ifbeingtoldthestructureandexperiencingdatafaithfultothestructureisnotsufficienttostampoutviolationsoftheMarkovassumption,whatis?3.3QualitativeandQuantitativeInferenceswhenReasoningaboutCausalStructures RottmanandHastie(2014)reviewedinferencesonmanydifferenttypesofcausalstructuresincludingonelink[X→Y],chains[X→Y→Z],commoncause[X←Y→Z]commoneffect[X→Y←Z],anddiamond[X←W→YandX→Z←Y]structures.Foreachofthesestructureswereviewedevidenceabouthowwellpeoplemakeinferencesononevariablegivendifferentobservedcombinationsoftheothers(e.g.,XgivenknowledgeaboutY,orYgivenknowledgeofXandZ,etc.).3Inthesectionsaboveonlearningcausalstructures,whenthetruestructureisX→Y→Z,peopletendtoalsoinferthelinkX→Z,suggestingthattheyarenotfullyawareoftheconditionalindependence.Thissectionfocusesonreasoningaboutthecausalstructureratherthanlearning,thoughofcoursetheyarerelated.

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Weconcludedthatforalmostallthecausalstructures(seethesectionbelowonexplainingawayforanexception)theinferencestendtogointherightdirection.Forexample,forthechain[X→Y→Z],ifbothcausalrelationsbetweenX→YandY→Zwerepositiveorbothwerenegative,peopletendedtoinferapositiverelationbetweenXandZ.Butifoneofthelinkswaspositiveandtheothernegativepeopleinferanegativecausalrelation(Baetu&Baker,2009).

Thepreviouslymentionedstudiesinvolvinginterventionsandobservationsonadiamondstructure[X←W→YandX→Z←Y]alsorevealhowsensitivepeoplearetotheparametersofthestructure(Mederetal.,2008,2009).Thesestudiessystematicallymanipulatedthebaseratesofsomeofthevariables,andalsothestrengthsofsomeofthecausallinks.Eventhoughthecausalstructuresinvolved4variables,andtheinferencerequiredreasoningwithtworoutesfromXtoZ,allofthesemanipulationshadinfluencesonsubjects’inferencesinthepredicteddirections.Insum,reasoninghabitsoftencorrespondtothequalitativepredictionsoftheCBNframework.

Yet,despitethequalitativecorrespondencebetweenhumaninferencesandthenormativejudgmentsbasedontheCBNframework,thequantitativecorrespondenceisnotsotight.Forexample,inoneconditionwheninferringtheprobabilityofZgivenXforthestudyabove,thenormativeanswerwas12.5%,yetsubjectsansweredonaverage37%.Giventhat50isthemiddleofthescale,37%isactuallyconsiderablyclosertoadefaultof50%thanthenormativeanswer.Thispatternofconservativeresults,judgmentstooclosetothecenterofthescalewasverycommonacrossmanystudiesreviewedinRottmanandHastie(2014).Forexample,forbothchain[X→Y→Z]andcommoncause[X←Y→Z]structurespeopledotypicallyinferacorrelationbetweenAandC,however,oftenthecorrelationisconsiderablyweakerthanthecorrelationinthedatathatthesubjectsobserved(Baetu&Baker,2009;Bes,Sloman,Lucas,&Raufaste,2012;Hagmayer&Waldmann,2000;Park&Sloman,2013).Therearemultiplepossibleinterpretationsofsucheffectssuchasresponsebiasesormemoryerrors(Costello&Watts,2014;Hilbert,2012)orpotentiallypriorsontheparameters(Lu,Yuille,Liljeholm,Cheng,&Holyoak,2008;Yeung&Griffiths,2011).Moreevidenceisneededtounderstandwhytheseeffectsoccur,andalsotounderstandtheaccuracywhenreasoningwithmorethan3or4variables.3.4ReasoningaboutExplainingAwaySituations

Theprevioussectionalreadyaddressedquantitativeinferencesoncausalnetworks,andtheconclusionisthatforthemostpartpeoplearefairlygoodatmakinginferences,thoughthereisaconservativebias.However,thereisonetypeofinferencecalledexplainingawaythatstandsoutasparticularlydifficult.ExplainingawayinferencesinvolvejudgmentsofP(x=1|y=1,z=1)andP(x=1|y=1,z=0)onacommoneffectstructure[X→Y←Z].ThereasonthatexplainingawayinferencesaresochallengingisthatoncethestateofYisknown,XandZactuallybecomenegativelydependent,sothenormativepatternofinferenceisP(x=1|y=1,z=1)<P(x=1|y=1,z=0).Thisisunlikeanyothertypeofinference.Forexample,onachainstructure[X→Y→Z],positiverelationsbetweenXandYandYandZmeanthatthereisapositiverelationbetweenXandZ;P(x=1|z=1)>P(x=1|z=0),andbecauseoftheMarkovassumptionP(x=1|y=1,z=1)=P(x=1|y=1,z=0).

IntermsofFigure1[smoke→cardiovasculardisease←age],explainingawaycouldinvolveinferringtheprobabilitythatsomeonesmokesgiventheirageandknowingthattheyhavecardiovasculardisease.Outofpatientswhohavecardiovasculardisease,

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knowingthatagivenpatientisoldmeansthatitislessnecessarytoinferthattheysmokeinordertoexplainthecardiovasculardisease;oldage“explainsaway”thecardiovasculardisease.Ifthepatientisyoungitbecomesmorenecessarytoinferthattheysmoke-otherwisewhatexplainsthecardiovasculardisease?Insum,whenthetwocauseshaveapositiveinfluenceontheeffect,thecausesbecomenegativelyrelatedcontrollingfortheeffect.

Priorevidencedidnotdecisivelyidentifyhowwellpeopleexplainaway(Morris&Larrick,1995;Sussman&Oppenheimer,2011).Thenewestandclearestevidencesuggeststhatpeoplehaveconsiderabledifficultieswhenmakingexplaining-awayjudgments(Rehder,2014).Thoughsometimespeoplegetthedirectionoftheinferencecorrect,P(x=1|y=1,z=1)<P(x=1|y=1,z=0),theyoftenareambivalentaboutthedirectionoftheinference,andsometimesthinkthatZwouldhaveapositiveeffectonX,P(x=1|y=1,z=1)>P(x=1|y=1,z=0).Rehderproposedthatthistypeofreasoningismoreakintoanassociativespreading-activationnetworkthancausalreasoning.ReidHastieandIhavealsorecentlycollecteddataonexplainingaway;unlikethepreviousresearchwegaveparticipantslearningdatasothattheycouldreasonfromexperienceratherthanjustfromthecausalstructure,andsothattheyalsohavedirectevidencethatP(x=1|y=1,z=1)<P(x=1|y=1,z=0).WesometimesfoundexplainingawaythatwasmuchweakerthannormativelypredictedbytheCBNframework,andothertimesinferencepatternsintheoppositedirectionfromexplainingaway.

Thechallengepeoplehavewithexplainingawayissomewhatmysterious.Therearenoothertypesofcausalinferencethatgivereasonerssomuchtrouble,yetatthesametimeexplainingawayhasalsobeentoutedasafundamentalstrengthofhumanreasoning(Jones,1979;Kelley,1972;Pearl,1988,p.49).Therearealsootherresultsinwhichexplainingawaydoesoccur.Oppenheimeretal.(2013)createdstoriestoelicitexplainingaway.Forexample,participantsweretoldaboutananimalwiththreefeatures–feathers,layseggs,andcannotfly–andaskedtoratehowlikelythisanimalistobeanostrich.Beinganostrichisaplausibleexplanationforwhythisbirdcannotfly.Otherparticipantsweregiventhesamethreefeatureswithoneadditionalfeature,thatithasabrokenwing,whichisanalternativecausefornotbeingabletofly.Theseparticipantsjudgedthelikelihoodofbeinganostrichaslowerthantheparticipantswhowerenotgiventhisfeature,suggestingexplainingaway(seealsoOppenheimer&Monin,2009).Sosometimespeopledogetthedirectionoftheinferencecorrect.4(Thisstudydidnothavenormatively-correctquantitativeanswerstocomparehumaninferencesagainst,anditalsotestsacomparisonof).

Anadditionalcomplexityisthatexplainingawayisrelatedtoanotherphenomenon.ExplainingawayinvolvesinferringtheprobabilityofXgivenknowledgeofYandZonthestructure[X→Y←Z].AnothermuchstudiedtopicisinferringthecausalstrengthofXonY.Asalreadydiscussed,peopleknowthattheymustcontrolforZwheninferringthecausal4Thisstudyisdifferentfromtheonesaboveintwoways.First,thisstudydidnothaveanormativelycorrectquantitativeanswertocomparehumaninferencesagainst.Second,thisstudyteststhecomparisonP(ostrich|feathers,layseggs,cannotfly,brokenwing)vs.P(ostrich|feathers,layseggs,cannotfly),notP(ostrich|feathers,layseggs,cannotfly,nobrokenwing).ThisisanalogoustoP(x=1|y=1,z=1)vs.P(x=1|y=1)insteadofP(x=1|y=1,z=0),soitisaslightlydifferentcomparison.

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strengthofXonY.However,whenZisaverystrongcauseofY,itisnotuncommonforpeopletoinferthatthestrengthofXisveryweak,weakerthanitactuallyis;sometimesthisiscalled“discounting”(Goedert&Spellman,2005).Thisdiscountingeffectisrelatedtoexplainingawayinthatbothphenomenarequireunderstandingthattwocausesarecompetingtoexplainaneffect.

Insum,thereisconflictingevidenceastowhen,whether,andhowmuchpeopleexplainaway.Despitethefactthatexplainingawayhasbeenstudiedfor40years,thereisstillimportantworktobedonetoreconcilethesefindings.3.5DoCausalRelationsBiasReasoning?

Itisafairlycommonviewinpsychologythatitisitiseasierforpeopletoreasonfromcausestoeffectsthanfromeffectstocauses(Pennington&Hastie,1993;White,2006),andthishypothesisissupportedbyevidencethatcausetoeffectjudgmentsaremadefasterthaneffecttocausejudgments(Fernbach&Darlow,2010).Thequestioninthissectioniswhethercognitiveeasehasaninfluenceontheinferencesthemselves.

TverskyandKahneman(1980)foundthatcausalinferencesarehigherwhenreasoningfromcausestoeffects.Similarly,Besetal.(2012)foundthatwhenmakinginferencesonthechain[X→Y→Z],inferencesofP(z=1|x=1)werehigherthanP(x=1|z=1).Additionally,bothoftheseinferenceswerehigherthaninferencesP(z=1|x=1)orP(x=1|z=1)onacommoncause[X←Y→Z]structure.Thesedifferencesareespeciallyinstructivebecausetheirparticipantsreceivedtrial-by-trialtraining,accordingtowhichalltheinferencesmentionedaboveshouldhavebeenequivalent.TheyspeculatethatmakinginferencesbetweenXandZonthecommoncauseisharderbecauseonemustreasonaboutcausalrelationsgoingintwodifferentdirections,andthisincreaseddifficultycouldlowerthefinaljudgment.

Thisstudyreachesaverydifferentconclusionthanmostoftherestofthearticlespresentedinthischapter.Theconclusionisthatstrengthoftheinferencesisdeterminedbytheeaseofexplaininghowthetwovariablesareconnected,andthatthiscognitiveeaseoverwhelmstheprobabilitiesparticipantsexperience.Eventhoughtheexplanationsforthesefindingsappealtocausalstructureandcausaldirection,theyareinconsistentwiththeCBNframework;theCBNframeworkpredictsthatalltheinferencesmentionedabovewouldbeequalgiventheparametersusedinthestudy.

Thoughtheeffectsofcausaldirectionwerefoundconsistentlyacrossthreeexperiments,thereareotherresultsthatdonotentirelyfitwiththestorythatcause-to-effectjudgmentsarehigherthaneffect-to-causejudgments.First,Fernbachetal.(2011,p.13)failedtoreplicatethestudybyTverskyandKahneman(1980).Morebroadly,Fernbachetal.havefoundthatinferencesfromcausestoeffectstendtobelowerthanthenormativestandard,butinferencesfromeffectstocausestendtoberoughlynormative(Fernbach,Darlow,&Sloman,2010;Fernbachetal.,2011;Fernbach&Rehder,2013;seealsoRehder,thisvolumeb).Theexplanationisthatwhenreasoningfromcausestoeffects,peoplesometimesforgetthatalternativecausescouldproducethetargeteffectasidefromthemaincause,thoughtheydonotforgetaboutalternativecauseswhenreasoningfromtheeffecttoatargetcause.

Thereissometensionbetweenthesetwosetsoffindings;Besetal.foundthateffect-to-causejudgmentsaretoolow(lowerthancause-to-effectjudgments),whereasFernbachetal.foundthatcause-to-effectjudgmentsaretoolow.However,theseresults

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cannotbedirectlycomparedbecausetheydifferonavarietyofdimensions.5Fernbachetal.usedrealworldcoverstories,askedparticipantstheirbeliefsabouttheparametersofthecausalstructure,andthenusedthoseparameterstocalculatethenormativeanswers.Becauseofthisapproach,Fernbachetal.couldnotdirectlycomparethecause-to-effectandeffect-to-causeinferencesandinsteadcomparedeachinferencetothenormativestandardforthatinference.6Incontrast,Besetal.(Experiment3)gaveparticipantstrial-by-triallearningdata;becausethelearningdataweresymmetricthecause-to-effectandeffect-to-causeinferencescouldbedirectlycompared(althoughthecoverstorylabelsforthevariableswerenotcounterbalanced).

Insum,thoughitisintuitivethatitiseasiertoreasonfromcausestoeffectsratherthanviceversa,itisstillunclearweatherorhowcognitivefluencyandneglectofalternativecausesmanifestinjudgments;itisnotclearexactlywhetherorwhencause-to-effectjudgmentsarehigherthaneffect-to-causejudgments.Itisespeciallyimportanttocometoconsensusontheseresults,orexplainwhydifferentpatternsofreasoningarefoundindifferentsituations,becausebothofthepatternsoffindingsimplydeviationsfromtheCBNframework.3.6AlternativeRepresentationsforCausalReasoning SofarthischapterhaspresentedtheCBNframeworkasasinglemethodoflearningcausalstructuresandmakinginferences.However,likemostsophisticatedmodelingtools,thereareactuallymanychoicesthatthemodelercanmake.AssumingthathumancognitiverepresentationsofcausalityaresomehowsimilartotherepresentationofaCausalBayesiannetwork(directedrepresentationsofcausality,parameterstocapturethestrengthofcausalrelationsandbaserates),thesechoicescorrespondtodifferentcognitiverepresentationsofthetaskandbackgroundknowledge.Anaccuratedescriptionofcausalreasoningrequiresclarifyingtherepresentationsbeingused.InthenexttwosectionsIdiscusssomerepresentationaloptions,andwhethertheycanbeempiricallydistinguished.

ConsiderthecasethatyouaretoldthatXandZbothcauseY[X→Y←Z],youexperienceasetoflearningtrialsthatinstantiatethestatisticalrelationsbetweenthesevariables,andaresubsequentlyaskedtoinferP(x=1|y=1,z=1).Figure8detailsfourpossibleprocessesformakingthejudgment.

Thefirstroute,thedashedline,involvesmakingtheinferencedirectlyfromtheexperienceddata.Wheneveralearnerexperiencesdatathatinstantiatesthecausalstructureitispossibletocometothecorrectinferencebyfocusingontheexperienceddata5IthankMichaelWaldmannforhighlightingthesedifferences.6Assumingaworldinwhichcausesandeffectshavethesamebaserates,onaverage,Fernbachetal.’sfindingsimplythatcause-to-effectjudgmentswouldbelowerthaneffect-to-causejudgments.However,Fernbachetal.actuallyassumeaworldinwhicheffectshavehigherbaseratesthancausesonaverage.Fernbachetal.(2011,p.13)claimthatanormativeCBNanalysisshowsthatinferencesofP(effect=1|cause=1)shouldbehigherthanP(cause=1|effect=1)65%ofthetimewhenintegratingacrosstheentireparameterspacewithuniformpriors.Thereasonforthisfindingisduetothefactthattheyassumedthattherearealternativefactorsthatcangenerateeffectsbutnotinhibiteffects.Thissameanalysisshowsthateventhoughcauseshaveabaserate.5onaverage,effectshaveabaserateof.625.Sotheiranalysisisonlyappropriateinworldsinwhichtherearenoinhibitoryfactors.

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andignoringthecausalstructure.Forexample,inordertocalculateP(x=1|y=1,z=1),areasonerjustneedstorememberthetotalnumberofobservationsinwhichallthreevariableswere1,P(x=1,y=1,z=1),anddividethisbythetotalnumberofobservationsinwhichy=1andz=1ignoringX,N(y=1,z=1);seeFigure8.Thisreasoningprocesscanbethoughtofassimilartoexemplarmodelsofcategorization;inferenceisperformedbyrecallingspecificexemplars. Theremainingthreeoptionsallinvolveelaboratingthecausalstructurewithdifferentkindsofparameters,andinferenceisperformedthroughacomputationontheparameters.Thoughinsomewaystheinferenceitselfseemsmorecomplicated,thecognitivebenefitisthatthelearneronlyneedstostorethestructureandtheparameters,notalltheindividualinstances.ThedifferencebetweenthesethreeoptionsishowtheyrepresenttheconditionalprobabilitydistributionofY,theprobabilityofYgiventhecausesXandZ.ThisconditionalprobabilitydistributionisdenotedasP(Y=y|X=x,Z=z),whichmeanstheprobabilitythatYisinaparticularstate(y=0or1),giventhatXandZareeachinparticularstates,xandz.

Representation1,involvescalculatingtheconditionalprobabilitydistributionP(Y=y|X=x,Z=z)directlyfromtheexperienceddata.Forexample,theprobabilitythaty=1giventhatx=1andz=1,iscalculateddirectlyfromrows1and3fromtheexperiencetable.Inferencecanthenproceedthroughsimpleprobabilitytheory(Figure8).Heckerman(1998)providesatutorialonthisapproach,andprovidescitationstootherexactandapproximateinferencealgorithms. Representation2doesnotdirectlyrepresenttheconditionalprobabilitydistributionP(Y=y|X=x,Z=z),butinsteadassumesthatpeoplespontaneouslyinfercausalstrengthsfromthelearningdata.SX→YandSZ→YrefertothestrengthofXonYandZonY,respectively.Themostpopularwaytorepresentcausalstrengthsinthenormativepsychologicalliteratureisusingcausalpowertheory,whichassumesthatcausescombinethroughaNoisy-ORfunction(Cheng,1997;Novick&Cheng,2004alsoseeSections2.2and2.3).Thisapproachalsorequiresthelearnertoestimatetheprobabilitythattheeffectispresentwithoutanyofitscauses,P(Y=1|x=0,z=0).Thecausalstrengthsandthefunctionalform(Noisy-OR)subsequentlyallowareasonertodeducetheconditionaldistributionP(Y=y|X=x,Z=z),whichwouldbeusedformakingtheinferenceP(x=1|y=1,z=1).ThecriticaldifferencebetweenRepresentation1vs.2isthatRepresentation2embodiestheassumptionthatXandZcombinethroughaNoisy-ORfunctionanddonotinteract(Novick&Cheng,2004);theNoisy-ORassumptionisthereasonwhyRepresentation2hasonly5parametersinsteadofthe6parametersinRepresentation1. Representation3isverysimilartoRepresentation2;however,insteadofrepresentingtheparameterP(Y=1|x=0,z=0),anadditionalbackgroundcauseBisaddedthatexplainsthecaseswhenY=1butXandZare0.InFigure8,Bisassumedtoalwaysbepresent,andtohaveastrengthof1/3.

Thequestionraisedbythesefouroptionsiswhethersomesortofrepresentationofcausalstructureandstrengthmediatestheprocessofmakinganinferencebasedonexperienceddata,orwhethertheinferenceismadedirectlyfromtheexperienceddata(dashedline).Ifindeedsomesortofcausalstructurerepresentationmediatestheinference,whichformofrepresentationgetsused?Allfourapproachesmaketheexactsamepredictions,sotheyaredifficulttodistinguishempirically.

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Idonotknowofanystudiesthataddressthefirstquestion,whetheracausalstructurerepresentationmediatestheprocessofmakinganinferencebasedonexperiencedata.However,therearesomestudiesthathaveattemptedtodistinguishthenatureoftheCBNrepresentation,specificallythedifferencebetweenRepresentations2vs.3.Figure8:FourPossibleProcessesforMakinganInference

Note:Nreferstothenumberoftrialsofobservationsofaparticulartype.

KrynskiandTenenbaum(2007)studiedhowwellpeoplemakeinferencesonthefamousmammogramproblem.Inthisproblem,participantsaretoldthatbreastcancer

0 0 0 80 1 60

400 16110401 0

1 30 18011

1 911#ZYX

X

Y

Z

P(y=1|x=1,z=1)=3/4P(y=1|x=1,z=0)=2/3P(y=1|x=0,z=1)=1/2P(y=1|x=0,z=0)=1/3

P(x=1)=.5 P(z=1)=.5

Functional Form = Noisy OR, no interactionP(y=1|X=x,Z=z)= 1-(1-SX→Y)x(1-SZ→Y)z(1-SB→Y)

X

Y

ZP(x=1)=.5 P(z=1)=.5

B

SB→Y=1/3

SZ→Y=1/4SX→Y

=1/2

P(b=1)=1

P(y=1|x=0,z=0)=1/3,Functional Form = Noisy OR, no interactionP(y=1|X=x,Z=z)= 1-(1-SX→Y)x(1-SZ→Y)z(1-P(y=1|x=0,z=0))

X

Y

ZP(x=1)=.5 P(z=1)=.5

SZ→Y=1/4

SX→Y=1/2

CBN Representation 1:Conditional distribution is represented as a table of all combinations of the causes.

CBN Representation 2: Conditional distribution is represented as the Noisy-OR functional form combining causal strengths for known causes, and a parameter for Y when all known causes are absent.

CBN Representation 3: Conditional distribution is represented as the Noisy-OR functional form combining causal strengths. Reasoner infers a background cause B if Y is present without X or Z.

Joint Probability Distribution:The memory for the number of observations of each type of event. When there are three binary variables, there are 8 possible event types.

P(x=1|y=1,z=1) = P(x=1,y=1,z=1) / P(y=1,z=1) = P(x=1,y=1,z=1) / (P(x=1,y=1,z=1) + P(x=0,y=1,z=1)) = P(y=1|x=1,z=1)P(x=1)P(z=1) / (P(y=1|x=1,z=1)P(x=1)P(z=1) + P(y=1|x=0,z=1)P(x=0)P(z=1)) = 3/4*1/2*1/2 / (3/4*1/2*1/2 + 1/2*1/2*1/2) = 3/5

P(x=1|y=1,z=1) = N(x=1,y=1,z=1) / N(y=1,z=1) = N(x=1,y=1,z=1) / (N(x=1,y=1,z=1) + N(x=0,y=1,z=1)) = 9/(9+6) = 3/5

X

Y

Z

Causal Structure:

Given information: Reasoner is told the causal structure and receives trial-by-trial experience of the multivariate distribution, which can be summarized as a joint probability table. Reasoner is asked to infer P(x=1|y=1,z=1).

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(cause)almostalwaysresultsinapositivemammogramtest(effect),andtheyaretoldthebaserateofbreastcancer.Theyarealsotoldthatmammogramshavefalsepositives6%ofthetime.Critically,thisfalsepositiverateisframedeitherasinherentrandomness(Representation2,whichhasaparametertorepresenttheprobabilityoftheeffectwhentheknowncauseisabsent),orduetoabenigncyst(anexplicitbackgroundcauselikeinRepresentation3).KrynskiandTenenbaumfoundthatparticipants’judgmentsabouttheprobabilityofbreastcancergivenapositivemammogramwereconsiderablymoreaccuratewhenthefalsepositiveratewasframedasbeingcausedbyabenigncyst,suggestingthatRepresentation3maybethemostintuitive.

AnumberofrecentstudieshelptoclarifythisfindingbyKrynskiandTenenbaum.First,thoughthisfacilitationofBayesianrespondingbyacausalframinghassometimesbeenfound,theeffecthasnotalwaysbeenconsistent(Hayesetal.,2015;Hayes,Newell,&Hawkins,2013;McNair&Feeney,2014,2015).Thereappeartobetwomainreasonsfortheinconsistency.First,thecausalframinghasabiggerinfluenceforparticipantswhohavehighermathematicalabilities(McNair&Feeney,2015).Second,thefacilitationeffectisoftenseeninareductioninextremeoverestimations(calledbaserate‘neglect’);however,thefinaljudgmentsareoftenlower,closertothenormativeresponse,butstillnotquite‘normative’(McNair&Feeney,2014).AplausibleexplanationforthiseffectwasputforthbyHayes,Hawkins,andNewell(2015;2014),whofoundthatthecausalframingincreasestheperceivedrelevanceofthefalsepositiveinformation.Theyconcludedthatthecausalframingmainlyhasaninfluenceontheattentionpaidtothefalsepositiverateandpossibletheconstructionofarepresentationoftheproblem,butdoesnotnecessarilyhelpparticipantstoactuallyusethefalsepositiverateinanormativewaywhencalculatingtheposteriorinference.

Insum,itseemslikehavingexplicitalternativecauses(Representation3)mayfacilitateaccuratecausalinference.Thatsaid,thisfindingraisesaworryingprospectthatcausalreasoningisapparentlyfragileenoughthatitcanbeharmedbyasmallframing.Ifcausalreasoningisrobustwhycan’tpeopletranslatebetweentheserepresentationsbymentallygeneratinganalternativecausetorepresentthefalsepositiverate?

Morebroadly,thepurposeofthisanalysisinFigure8wastoshowthattheCBNframeworkcanbeinstantiatedinmultiplepossibleways.Differentarticlespresentdifferentversions.Eventhoughtheyallmakesimilarifnotidenticalpredictions,thesealternativeversionspresentdifferentcognitiveprocessesinvolvedinmakingtheinference.Inordertomovefromacomputational-leveltheorytoanalgorithmic-leveltheoryitwillbenecessarytofurtherclarifytherepresentationsandinferenceprocess.Itisespeciallycriticaltoclarifywhetheracausalstructurerepresentationmediatescausalinferencewhenareasonerhasexperiencedlearningdatabecauseinsuchinstancesitispossibletomakeinferencesdirectlyfromtherememberedexperienceswithoutthinkingaboutthecausalstructureatall.3.7EvenMoreComplicatedAlternativeModelsforCausalReasoning TheprevioussectiondiscussedfourpossibleimplementationsoftheCBNframework.However,inrealitytherearemanymorepossibilities.AfullyBayesiantreatmentoflearningandinferenceallowsforawayforpriorknowledgetoinfluencethelearningandinferenceprocesses.Inregardstoacausalstructure,therearethreepossiblerolesofpriorinformation;priorbeliefsaboutthenetwork,abouttheintegrationfunction,andaboutthestrengthsorparameters.

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First,whereasRepresentations2and3inFigure8bothassumeoneparticularfunctionalform,theNoisy-OR,inrealitylearningisnotthissimple.Section2.2onfunctionalformsalreadycoveredexperimentsonhowpeoplelearnthespecificwayinwhichmultiplecausescombinetoproduceaneffect,andhowthisbeliefshapesfurtherlearningandreasoningaboutthecausalsystem.(Beckers&Miller,2005;Lucas&Griffiths,2010;Waldmann,2007).Thus,afullyBayesianversionofFigure8wouldallowformultiplepossibleintegrationfunctionsandpriorsonthosefunctions. Second,theparametersinFigure8werecalculatedbyusingpointestimates.Forexample,theparameterP(y=1|x=0,z=0)forRepresentations1and2,andtheSB→YparameterinRepresentation3,areallgivenasexactly1/3inFigure8,whichwascalculatedbycomparingrows6and8inthedatatable.Ifapointestimateoftheparametersisused,thenallfourapproachesproduceexactlythesameinferences.Alternatively,anotheroptionisthatpeoplerepresentuncertaintyaboutalloftheparametersbasedontheamountofdataexperienced.Ifthissecondapproachisused,thenRepresentation1willmakesomewhatweakerinferencesthanRepresentations2and3,becauseRepresentation1requiresinferringanadditionalparameter.Additionally,peoplemayhavepriorbeliefsaboutcausalstrengthsthatmaybiasthelearningandinferenceprocess.Forexample,Luetal.arguedthatpeoplebelievecausestobesparseandstrong(Luetal.,2008).GiventhedatainFigure8,thesparseandstrongpriorspullthestrengthsdownward;insteadofastrengthof.50,thesparseandstrongpriorswouldproduceastrengthestimateof.43andwithmoredatatheestimategetscloserto.50.Incontrast,YeungandGriffiths(Yeung&Griffiths,2015)foundthatpeoplehavepriorssuchthattheybelievethatmostcandidatecausesareverystrong.Ifpeoplehadsuchpriorsitwouldresultincausalstrengthestimatesabove.50.Priorsonstrengthwouldhaveadown-streaminfluenceoninference;thestrongerthecausalstrengthbeliefs,thestrongertheinferencesshouldbe. Third,peopleoftenhavepriorbeliefsaboutthecausalnetwork.Luetal.’ssparseandstrongpriorsuggeststhatpeoplebelievethatfewercausesaremorelikelythanmanycauses(Luetal.,2008).Inarelatedvein,Mederetal.(2014)proposedthatwhenperforminganinference,eveniftoldacausalstructure,peoplemayentertainthepossibilitythatanothercausalstructurecouldactuallybethetruestructure,whichcaninfluencethejudgment.Inparticular,Mederetal.toldparticipantsthestructure[X→Y],hadthemobservecontingencydatasothattheycouldlearnthestatisticalrelationbetweenXandY,andthenhadthemmakeaninferenceofP(x=1|y=1).TheyfoundevidencethatwhenthecausalstrengthofXonYisfairlyweak,peoplemaynotbelievethestructure[X→Y]andinsteadentertainthepossibilitythatXandYmaybeunrelated.Thisgeneralapproach,thatpeoplemayentertainthepossibilitythatthecausalstructurepresentedbytheexperimentermaynotactuallybethetruecausalstructurehasalsobeenusedtoexplainviolationsoftheMarkovassumption(seeSection3.2).Oneproblemwiththisaccount,however,isthatwhentherearemorethantwovariablesitisunclearwhatsetofalternativestructuresisentertained,andconsideringmultiplepossibilitieswouldquicklybecomecognitivelyunwieldy. Insum,allowingforthepossibilitythatpeoplethinkaboutmultiplepossiblestrengths,functionalforms,andcausalstructuresmakestheCBNframeworkveryflexible,andonacase-by-caselevelitseemsplausiblethatpeoplemayactuallyhavepriorsforanyoftheseaspectsofthenetwork.However,incorporatingallofthesepriorsmakesthe

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reasoningtaskmuchharderthananyoftheoptionsinFigure8,anditseemsunlikelythatpeoplearealwaysengagedinreasoningwithallthesepriorssimultaneously.Thus,itwillbeimportanttounderstandwhenpeoplemakeuseofthepriorsandhowwelltheyincorporatepriorswithobserveddataformakinginferences.4FinalQuestions,FutureDirections,andConclusions ThroughoutthechapterIhavehighlightedquestionsandfuturedirections.InthissectionIrepeatsomeofthosequestionsandaddsomenewones.Ibelievethatthesequestionsarecriticalforhavingathoroughandaccurateunderstandingofhumancausallearningandreasoning.

1) Thoughrecentlytherehavebeenmoreattemptstoexploreotherfunctionalforms,thevastmajorityofresearchontheCBNframeworkhasinvestigatedbinaryvariablesthatcombinethroughaNoisy-ORfunction.Therehasbeenverylittletheorizingaboutwhatcausalstrengthmeans,forexample,whencausesandoreffectsaremultilevel(Pacer&Griffiths,2011;Rottman,2016;White,2001).Forexample,isthehumaninterpretationofcausalstrengthformultilevel(e.g.,Gaussian)variablesanalogoustoeffectsizemeasuresforlinearregression?Whatistherelationbetweenfunctionlearningandcausalstrengthlearning?Dopeoplefaceanychallengesorusedifferentheuristicswhenlearningcausalstructuresfrommultilevelratherthanbinaryvariables?Insum,causalreasoningisextremelydiverse,anditwillbecriticaltobroadenourexperimentalparadigmstocapturethisdiversity.

2) Oneofthegoalsofcognitivepsychologyistounderstandtherepresentationsthatpeopleuseforthought.AsFigure8demonstrates,therearemultiplepossiblerepresentationsforhowpeoplereasonaboutcausalstructures,andmanyoftheserepresentationsmakeexactlythesame(orverysimilar)predictions.Clarifyingwhichsortsofrepresentationsareusedwillhelpdevelopamoreprecisedescriptiveaccountofcausalreasoning.

3) SofartheCBNframeworkhasbeenframedasacomputational-leveltheoryofhumancausalreasoning.However,thecomputationsinvolvedininferringacausalstructurefromdata,ormakinginferencesonanetwork(e.g.,Figure8)areverycomplex.Thus,animportantgoalistodevelopaprocess-levelaccountofhowpeopleactuallyperformtheseinferences.Anumberoftheoristshaveproposedvariousheuristicsforcausallearning,whichoftencomeclosetotheoptimalsolution,andoftenhaveequalorbetterfittoparticipants’inferences(Bramleyetal.,2015;Coenenetal.,2015;Lagnado&Sloman,2004;Rottman&Keil,2012;Rottmanetal.,2014;Steyversetal.,2003).YetsofarthisheuristicsapproachhasbeendisconnectedandhasoftentakenthebackseattoproofofconceptdemonstrationsthattheCBNframeworkcanmodelhumanlearning.Moreattentiontohowtheseinferencesareactuallymadethroughaprocess-levelaccountwillhelpprovidepsychologicalinsightintothisfascinatingandcomplexreasoningprocess.

4) Lastly,allofthestudiesonhumancausalreasoninggiveparticipantstoyexamplesandsampledatainshortperiodsoftime.Itisunclearhowwellthisresearchstrategycapturesactualcausalreasoningintherealworld,whichinvolveslong-termaccumulationofdataandmanymorevariables.Anidealapproachwouldbetofindareal-worlddomaininvolvingcausesandeffectsthatincludesrecordsofexperiences.Forexample,ahighlyaccurateelectronicmedicalrecordssystemmight

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inthefuturepermitustotrackadoctor’sexperienceswithallthevariablesinFigure1toseeifthedoctor’sjudgmentsfitcloselywithhisorherpersonalexperiences.

ThecausalBayesiannetworkframeworkhasentirelyreshapedthelandscapeofresearchoncausalitytothepointthatitisnowrareseearticlesthatinvestigatecausallearningwithoutmentioningtheCBNframework.Whereasresearchoncausalreasoningusedtobeprimarilyaboutinferencesbetweenasinglecauseandeffect,nowthecentralquestionsareaboutlargercausalstructures.Thus,thenewfocusisonhowpeoplelearnthestructureanddeterminecausaldirectionality,howpeoplesimplifycomplexstructuresintosmallerunitsusingtheMarkovassumption,andhowvariousbeliefscapturedinthenetworksuchastheintegrationfunctioninfluencelearningandreasoning.EvenolderquestionssuchaselementalcausallearninghavebenefittedtremendouslyfromtheCBNframeworkbyreinterpretingstrengthasaparameterinthecausalnetwork.

Onthedescriptiveside,themostimportantfactabouthumancausalreasoningisthathumansareremarkablygoodcausalreasoners;weadeptlyincorporatemanydifferentbeliefswhenlearningandreasoning(e.g.,integrationfunctions,autocorrelation,causaldirectionality),wecanlearnaboutquitecomplicatedcausalrelations(e.g.,unobservedcausesthatinteractwithobservedcauses),andweoftendosowithremarkablylittledata.TheintroductionoftheCBNframeworkhasrevealedmanyofthesecapacitiesthatwerepreviouslyunknownandhasalsoraisedimportantquestionssuchashowsuchashowtodevelopaprocess-levelaccountofthesesophisticatedinferences,howcloselydotherepresentationsoftheCBNframeworkmapontotheactualrepresentationsthatweuseforcausalreasoning,howcausalreasoningoccurswithmorediversesortsofstimuliandinmorenaturalisticenvironments.Answeringthesequestionswillnotonlyhelpusdevelopamoreaccurateandcompletepictureofhumancausalreasoningbutmayalsoidentifywaystohelppeoplebecomeevenbettercausalreasoners.

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References:Baetu,I.,&Baker,aG.(2009).Humanjudgmentsofpositiveandnegativecausalchains.

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