Bayesian optimization of time

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OpinionBayesianoptimizationoftimeperception123ZhuanghuaShi,RussellM.Church,andWarrenH.Meck1DepartmentofPsychology,Ludwig-Maximilians-Universita¨tMu¨nchen,Munich,Germany2DepartmentofCognitive,Linguistic,andPsychologicalSciences,BrownUniversity,Providence,RhodeIsland,USA3DepartmentofPsychologyandNeuroscience,DukeUniversity,Durham,NorthCarolina,USAPrecisetimingiscrucialtodecision-makingandbe-theBayesianapproachtooptimizationhasprovidedmanyhavioralcontrol,yetsubjectivetimecanbeeasilyimportantinsights,exactlyhowtheseprobabilisticdistri-distortedbyvarioustemporalcontexts.Applicationbutionsandinferencesmightbelinkedtotemporalproces-ofaBayesianframeworktovariousformsofcontex-singatamechanisticlevelremainsuncertain.tualcalibrationrevealsthat,contrarytopopularbelief,Inthisarticle,wereviewrecentprogressinunderstand-contextualbiasesintiminghelptooptimizeoverallingtheinfluenceofcontextualcalibrationonintervalperformanceundernoisyconditions.Here,wereviewtiming,withparticularfocuson‘central-tendency’andrecentprogressinunderstandingtheseformsoftem-‘modality’effects,aswellasthetime-ordererror(TOE).poralcalibration,andintegrateaBayesianframeworkWethencomparethedifferentexplanationsforcontextualwithinformation-processingmodelsoftiming.WecalibrationmadebyBayesianinferenceandtraditionalshowthattheessentialcomponentsofaBayesianinformation-processingmodelsofintervaltiming.Finally,frameworkarecloselyrelatedtotheclock,memory,weprovidearoadmapforintegratingaBayesianframe-anddecisionstagesusedbythesemodels,andthatworkwithexistingtimingmodelsandpointouttheimpli-suchanintegratedframeworkoffersanewperspectivecationsandpotentialchallengesassociatedwiththisondistortionsintimingandtimeperceptionthatareapproach.otherwisedifficulttoexplain.ContextualcalibrationoftimeperceptionIntroductionAsnotedabove,subjectivedurationscaneasilybedis-Humansareoftensurprisinglyaccurateattiminginter-tortedbyvariouscontextualfactors.Aclassicexampleofvalsinthesub-secondtominutesrangeduringdailycontextualcalibrationisVierordt’slaw[15],alsoknownroutines,aswellaspartofvocationalandrecreationalasthe‘central-tendency’effect.Whenparticipantsareactivities[1].Wemustjudgethecorrecttimetostrikethepresentedwitharangeofstimulusdurationsandarerightmusicalchord,adjustourrunningspeedtobeinthethenaskedtoreproducethosedurations,theytendtorightplaceattherighttimetocatchaflyball,andantici-overproduce‘short’durationsandunderproduce‘long’patewhentobeginpressingthebuttontoopenthedoordurations.This‘central-tendency’effecthasbeenwhenthesubwaytraincomestoastop.However,ourdemonstratedinnumerousstudies(forreviews,seesubjectiveexperienceoftimecanbehighlybiasedin[10,15]).Acommonexplanationofthiseffectisthatdifferentcontexts[2–8].Forexample,soundsareoftendurationjudgmentsarederivednotonlyfromcurrentjudgedlongerthanlights,evenwhentheyarebothofsensoryinputs,butarealsoinfluencedbytheacquiredthesamephysicaldurationandhavebeenmatchedforstatistics(e.g.,meanandstandarddeviation)oftheintensity[8].Traditionaltimingmodelssuggestthatthesedistributionofpreviouslyexperiencedstimulusdura-contextualeffectsareassociatedwiththedifferentialdecaytions[5,6,8–10,15].ofmodality-specificrepresentations,changesinthespeedInterestingly,the‘central-tendency’effecthasbeenob-ofthe‘internalclock’,and/or‘memorymixing’ofdifferentservedtovaryacrossdifferentgroupsofindividualswithtemporalrepresentations[5,8–10].Recently,researchersvariouslevelsofexperience,includingmusicalexpertise.haveusedBayesianinferencetoperformcomputational-Forexample,althoughindividualswithlittleornomusicallevelanalysisonvariousformsofcontextualcalibrationoftrainingexhibitthe‘central-tendency’effectforbothaudi-intervaltimingandrevealedthatsuchadaptationmaytoryandvisualstimuli,stringmusiciansshowverylowhelptoimproveoverallperformance[6,11–14].Althoughbiasesforauditorydurationreproduction.Moreover,ex-pertdrummersareabletoreproducebothauditoryandCorrespondingauthor:Meck,W.H.(meck@psych.duke.edu).Keywords:visualdurationsnearperfection[16].IncontrasttohealthyBayesianinference;Vierordt’slaw;contextualcalibration;modalitydifferences;memorymixing;scalartimingtheory.individuals,patientswithParkinson’sdisease(PD)are1364-6613/$–seefrontmattermorepronetocontextualmanipulationwhentestedoffß2013ElsevierLtd.Allrightsreserved.http://dx.doi.org/10.1016/j.tics.2013.09.009oftheirdopaminergicmedication,referredtoasatemporal‘migration’effect[17].WhenPDpatientsgivendopamine-replacementtherapy(e.g.,L-dopa+apomorphine)aretrainedtotime-specificstimulusdurations(e.g.,8sand556TrendsinCognitiveSciences,November2013,Vol.17,No.11 OpinionTrendsinCognitiveSciencesNovember2013,Vol.17,No.1121svisualstimuli)usingcorrectivefeedback,theyareable(A)0.2toKey:reproducethesestimulusdurationswiththesameaccuracyandprecisionasage-matchedcontrols.However,8sONwhen0.158sOFFthesesamePDpatientsaretestedoffoftheirmedi-cation,21sONastrong‘central-tendency’effectemerges,evenifcorrective21sOFFfeedbackisgiven.Undertheseconditions,the0.18sdurationisoverproducedandthe21sdurationisunderproducedwhenbothdurationsarepresentedwithinthesamesession,butnotwhentheyarepresentedin0.05separateMeanrelavefrequencysessions(Figure1A).Asecondexampleofcontextualdependenceisassoci-ated0withthepresentationorderoftwoormoredura-05101520253035tions,referredtoasthetime-ordererror(TOE)[4,18,19].Realme(s)Inthiscase,theorderofpresentationcanbiasthethresholdfordeterminingwhetherastimulusisshorter(B)25orKey:longerthantheprecedingduration[19]andchangethesensitivityofthedurationjudgment[18,20,21].TheONbiasesOFFinthethresholdcanbepositiveornegative20dependingonvariousexperimentalfactors,suchastherelativelengthsofthedurationsbeingcomparedand15theinter-stimulusintervals[19,22].Ameta-analysis[18]hasrevealedthatthesensitivityofdurationjudg-mentsisoftenworsewhenthecomparisondurationisSubjecveme(s)10presentedpriortothestandardduration,comparedtothereverseorder.TheusualexplanationfortheTOEissimilartoaccountsproposedforthe‘central-tendency’5effect,510152025namely,participantsadapttotheaveragestimu-lusdurationusedwithinanexperimentalsetting,there-Realme(s)bycausingtheperceiveddurationofthefirststimulusto‘gravitate’(C)towardthisremotestandard[23].Itisimpor-Key:tanttonote,however,that‘gravitationtothemean’doesPriornotrequirealargenumberoftrials,thatis,animmedi-Likelihood21sONatelyprecedingorsucceedingdistractorintervalissuffi-Posteriorcientforparticipantsto‘assimilate’comparedstimulusdurations[13,24].Anotherformofcontextualcalibrationistheclassicfindingthat‘soundsarejudgedlongerthanlights’[25].Whenauditoryandvisualstimuliareintermixedwithin21sOFFasession,participantsfrequentlyoverestimateauditorystimuliandunderestimatevisualstimuliofequivalentphysicaldurations[8,25–32].This‘modality’effectalsooccursforsimultaneouslypresentedmultimodaldura-51015202530tions[12,14,33–35].Forinstance,whendifferentaudito-ryRealme(s)andvisualstimuliarepresentedtogetherinanaudiovisualTRENDSinCognitiveSciencestemporalbisectiontask,theperceivedaudiovisualdurationappearstobeanintegratedesti-Figure1.Illustrationofthe‘migration’effectintemporalreproductionsofmateoftwodurationswithdominancebytheauditoryParkinson’sdisease(PD)patients.(A)Relativefrequencydistributionsforthestimulustwotargetdurations(8and21s)plottedasafunctionofthemedicationstates.The[14].peakfunctionsillustrateaccuracyandprecisionofdurationreproductioninPDpatientsONandOFFtheirdopaminergicmedication.IntheONmedicationstate,Traditionalapproachestocontextualcalibrationtheiraccuracyandprecisionarecomparabletoage-matchedcontrols.However,inAtheOFFmedicationstate,astrongcentral‘migration’effectemergesasindicatedwell-knownquantitativemodelthataddressescontex-byshiftsofthepeakfunctions.Adapted,withpermission,from[17].(B)Thetualinfluencesonperceptionistheadaptation-level(AL)subjectivepeaktimeplottedagainsttherealtimeasafunctionofthemedicationtheoryproposedbyHelson[36].Accordingtothistheory,states.IntheOFFmedicationstate,thepeaktimesarebiasedtowardstheasubjectivecenterofthedistributionofdurationsignals.Thedashedlineindicatesperceptofastimulusdependsonthebackgroundthecontext.estimateofthesubjectivecenterbasedonthesimplelinear-weightedaverageForexample,theperceivedluminanceofanmodelobject(Equation1).(C)SchematicillustrationofhowBayesianinferencemightbeissubjecttothesurroundingluminance.HelsonappliedfortheONandOFFstatesin21sdurationreproduction.ThewithdrawalofquantitativelythedopaminergicmedicationintheOFFstatemayincreaseuncertaintyinthedescribesthebackgroundcontextasthesensorypooledlikelihood.Thus,theunchangedpriorgainsmoreweightinthefinaleffectofallstimuli(i.e.,adaptationlevel).Aestimate,andtheposteriorshiftstowardstheprior.modifiedversionofALtheoryhasbeenusedtoexplainbiasesandsensitivitychangesinducedbythetemporalorderofpresentation[18,19].Inasimplifiedexample,557 OpinionTrendsinCognitiveSciencesNovember2013,Vol.17,No.11theperceivedsubjectiveintervaldˆisalinearweightedscalarproperty[38,39,41],allowingviolationsofthescalaraverageofthesensoryevidenceandcontext:propertytoserveasanindicatorofcontextualinfluencesontimingandtemporalmemory.Thisleadstothe‘memo-dˆ¼ð1wÞsþwdp;[1]ry-mixing’accountofmodalitydifferences[8–10],whichwheresuggeststhatcontextualcalibrationarisesfromthemixingsistherescaled(e.g.,logarithmic)valueoftheisofauditoryandvisualdurationswithinasharedmemorystimulusduration,dpthesubjectiveexpectancyofstim-ulusdistribution.Thisaccountassumesthatauditorystimulidurations,andwistheempiricallydeterminedweightingdrivetheclockstage(composedofapacemaker,switch,constantthatcanbeeitherpositiveornegative.isandaccumulator)atafasterratethanvisualstimuliand,Whentheweightwoftheinternalexpectancydpposi-tive,asaconsequence,theclockreadingstransferredintotheperceiveddurationdˆisattractedtowardthecentermemoryareproportionallylongerforauditorystimulithanofthedistribution.Thus,thesimple‘weightedaverage’forvisualstimuli.Consequently,whenthecurrentclockmodelalsopredictsVierordt’slawfordurationdiscrimination.readingiscomparedtoamemorysampleretrievedfromFigure1Bshowshowthislinear-weightedmodelthis‘mixed’distribution,clockreadingsforauditorysti-couldexplainthe‘migration’effectforthereproduc-tionmuliwillbejudged(onaverage)tobelongerandvisualofstimulusdurationsobservedinPDpatientswhentestedstimuli(onaverage)willbejudgedtobeshorterthantheoffoftheirdopaminergicmedication.Scalarmeanofthe‘mixed’distribution.Ifauditoryclockreadingstimingtheorywithitsclock,memory,anddeci-sionarecomparedonlywithauditorymemoriesandvisualstagesisthemostcommonheuristicusedtodescribetheclockreadingsarecomparedonlywithvisualmemories,cognitiveprocessesinvolvedinthecontextualcalibra-tionnomodalitydifferencesconsistentwithchangesinclockofdurationdiscrimination[5,8–10,37–40].Thehall-markspeedshouldbeobserved[8].Anexampleofthe‘modality’ofthismodelisthatthestandarddeviationoftemporaleffectintemporalbisectionisillustratedinFigure2A,estimatesincreaseslinearlywiththemeanofthewherethe‘short’(S)and‘long’(L)anchordurationsconsistdurationbeingestimated–whichisreferredtoasthescalarofbothauditoryandvisualstimuli,andintermediateproperty(Box1).ScalartimingtheorytypicallyassumescomparisondurationsofbothmodalitiesarerandomlythatamemorytranslationprocessinducestheBox1.Information-processing(IP)modelsofintervaltimingandthescalarpropertyOneofthebest-developedmodelsofintervaltimingisscalartimingtheory,whichbelongstoaclassofinformation-processingmodels(A)thatpositadedicated‘internalclock’[5,37–40,77].Accordingtoscalartimingtheory,thecognitiveprocessessupportingintervaltimingconsistofthreestages:clock,memory,anddecision(FigureI,Box2).Inordertorepresentatargetduration,apacemakeremits‘L’pulsesσlthatarepassedbyaswitchintoanaccumulator.Thevalueintheaccumulatorisassumedtobenormallydistributed[60],whichisthencomparedtotheexpectedtimesampledfromreference‘M’σmSubjecvemememory.Ifthesevaluesarecloseenoughatthedecisionstage,aresponse‘S’σsismade.Whentheentireresponsefunctionobtainedfrompeak-interval/temporalgeneralizationproceduresisplottedonarelativetimescaleformultipletargetdurations,thedifferentSMLresponsefunctionstypicallysuperimposewitheachother,whichRealmedemonstratesthatintervaltimingadheresstronglytoWeber’slaw[38,41,85–88].Thisiscalledthescalarpropertyofintervaltiming.Evidence(B)300forsuperimpositioninthepeak-intervalprocedureisillustrated1inFigureIforhumanparticipants.ScalartimingtheoryKey:assumesthescalarpropertyisintroducedbyamemorytranslation0.58sconstantk*[60,61,64–66],becausevariancesintheclockand12s2000decisionstagesareconsideredinsufficienttoaccountforthescalar00.511.5221sRelavemeproperty([38,39,41],butsee[54,77]).Thememorytranslationconstantk*isdrawnfromanormaldistributionN(mk*,sk*),suchthatthismultiplicationresultsinwidermemorydistributionsfor100longerdurationsthanforshortertargetdurations(FigureI).ResponsesperminuteSystematicviolationsofthescalarpropertycanoccurwhentimedintervalsareselectedfromparticularrangesortestedwithselected0clinicalpopulations(e.g.,PDpatients)[78,86,89,90].Theseviolations010203040ofthescalarpropertysuggestthatmultipletimingsystemsmayTime(s)subservedifferenttimescalesandthatcontextualinfluencesshouldTRENDSinCognitiveSciencesbeconsideredmorecarefullyintheoreticalmodelsoftimingandtimeperception.FigureI.Thescalarpropertyofintervaltimingandevidenceofsuperimpositionofpeak–intervalfunctions.(A)Scalartimingtheoryassumesthattheestimationerrorincreasesinproportiontothetargetinterval(grayarea),whichleadstothescalarpropertybeingexhibitedfor‘short’(S),‘medium’(M),and‘long’(L)targetdurations.(B)Evidenceofthescalarpropertyfromhumanparticipants.Meankeypressesperminuteareplottedasafunctionofsignalduration(s)forparticipantstrainedwith8,12,and21stargetdurations.Theinsetfigureshowsthattheresponsefunctionssuperimposeoneachotherwhentheyareplottedasafunctionofrelativetime.Adapted,withpermission,from[91].558 OpinionTrendsinCognitiveSciencesNovember2013,Vol.17,No.11(A)1ofdynamicupdatingofinternalreferents,In¼gIn1þð1gÞX1n;[2]0.75whereIisthedynamichistoryoftheinternalreferenceforthefirstintervalandX1nisthecurrentfirstinterval.WhenthesensoryestimateofthecurrentfirstintervalX1n0.5ispresented,themeanintervalreferenceInisthenp(‘long’)updated.Key:Intheseapproaches,theweightsofpriorhistoryandfeedbackareestimatedbyfittingagenerallinear0.25Auditorymodel,whichdoesnotindicatewhetherthesystemusesVisualanoptimalstrategyfordynamicupdating.Nevertheless,0whenEquation2combineswithBayesianinference,itis2345678referredtoastheapplicationofaKalmanfilter[43–45].Realme(s)TheginEquation2isreferredtoas‘Kalmangain’,whichisoptimallydeterminedandupdateddynamicallybythe(B)variancesoftheinternalreferenceandthesensoryesti-Key:mate.TheapplicationofaKalmanfilterhasbeensup-AuditoryportedbyanumberofempiricalstudiesthatexaminedtheVisual‘central-tendency’effectofdistancereproduction[43,46],sensorimotorcontrol[45],andmultimodalrecalibration[44,47].MaMavBayesianinferenceontemporalcontextualcalibrationMSubjecvemevThecombinationofalinear-weightedaveragemodelandscalartimingtheoryprovidesapowerfulapproachtoexplainvarious‘memory-mixing’phenomena[5,8–10,18],yettheseapproachesdonotinformusastowhatfactor(s)SPSEPSELquantitativelyavdeterminethelevelofcontextualcalibra-Realme(s)tion.RecentworkusingaBayesianapproachsolvesthisproblembyprovidingaquantitativepredictionoftheTRENDSinCognitiveSciencescontributionoftemporalcontextandthemechanismsthatFigure2.Illustrationofthe‘modality’effectintemporalbisection.(A)Groupinvolveprobabilitythesubjectiverepresentationofduration[6,16,48–ofa‘long’response,p(‘long’)functionsaveragedacrossparticipants51]for(Box2).ThebasiclogicoftheBayesianapproachisthatauditoryandvisualstimulusdurationspresentedina2svs8sbisectiontask.Adapted,withpermission,from[42].(B)SchematicillustrationforaBayesiansensorymeasurementsarenoisyanduncertain,andcom-inferenceaccountofthe‘modality’effectinwhichpulsesareintegratedatafasterbiningratethepriorknowledgeofthestatisticaldistributionofforauditorystimulithanforvisualstimuliduetodifferentialratesofopeningaandseriesofstimulusdurationscanbebeneficialforincreas-closingoftheswitchthatallowspulsestoflowfromthepacemakertotheaccumulator[8,9].Asaconsequence,theinternalreferenceofthemeandurationingtheprecisionofdurationestimates,althoughincorpo-betweenthe‘short’(S)and‘long’(L)anchordurationsislargerforauditorystimulirating(M)thepriormayleadtosystematicbiases.Inthisathanforvisualstimuli(Mv).Thesedifferentinternalrepresentationssense,correspondcontextualeffectsarestatisticallyoptimalandservetothesameexternaldurationindicatedbythemiddleverticaldashedline.Whentheauditoryandvisualdurationsarecombinedormixedtominimizeerror.Moreover,thetrade-offbetweenpreci-withinthesamememorydistribution,assumingthatMaandMvareindependentsionGaussians,andbiaswilldependonthemagnitudeofuncertaintytheinternalreferenceofthemixeddurations(Mav)isalinear-weightedandaveragetheselectedcostfunction[48].ofMaandMv.Basedonthismixedreference,theauditoryandvisualpointsofsubjectiveequality(PSE)areshiftedinoppositedirections–asindicatedAnimpressivedemonstrationofhowBayesianinferencebythefilledsquaresandcircles,respectively.canbeusedtopredictthe‘central-tendency’effectintemporalreproductionhasbeengivenbyJazayeriandShadlen[6].Intheirstudy,participantswereinstructedtoestimateasampledurationandreproduceitimmedi-intermixedatelyafterwards.Fordifferentblocksoftrials,however,withinasession[42].Asaresultofthisinter-mixingthesampledurationswereselectedfromthreedifferentofstimulusmodalities,thepointofsubjectiveequalityuniformdistributions(i.e.,‘short’,‘intermediate’,and‘long’(PSE)observedforauditorydurationsoccursatanranges)thatpartiallyoverlappedwitheachother.TheearliertimethanthePSEobservedforvisualdurations.Givenresultsrevealedastrong‘central-tendency’effectindura-thattheexternalworldisinconstantflux,themixingtionreproductionasillustratedinFigure3.JazayeriandoftemporalmemoriesislikelytoaccrueovertimeinShadlenthenusedaBayesianframeworktoshowhoworderfortheobservertoadapthis/hertimingbehavior.TaatgendifferenttypesofobservermodelswithvariationintheandVanRijn[5],forexample,foundthatthe‘central-tendency’applicationofthepriorandtheestimationerrormightbeeffectwasnotonlyinfluencedbythemixingusedindurationreproduction.Theyreportedthatthebest-ofmultipledurations,butalsofluctuatedasafunctionfittedmodelwastheoneusingaBayesleast-square(BLS)offeedbackandestimatesofthestimulusdura-tionsrule(i.e.,asquare-errorlossfunction;Box2).Thesuccessexperiencedonprevioustrials(recencyeffect).Simi-larly,oftheBLSrulesuggeststhatparticipantsusedthestatis-Dyjasetal.[18]suggestedthatTOEsassociatedwithsensitivityticalinformationprovidedbythedistributionofstimuluschangesindurationcomparisonsaretheresult559 OpinionTrendsinCognitiveSciencesNovember2013,Vol.17,No.11Box2.Bayesianinferenceoftimingandlinkagetoinformation-processingmodelsAnalogoustotheclassicinformation-processing(IP)modelsthatestimationerrorðDˆDÞ.Themostfrequentlyusedlossfunctionsinvolve2clock,memory,anddecisionstages,Bayesianinferencehasforthreemodelingbehaviorarethesquared-errorL¼DˆD[50,67],essentialcomponents:likelihood,priordistribution,andlossfunction02[45,48,92,93].AsillustratedinFigureI,twoframeworksare¼closelyorrelativesquared-errorL1D=Dˆfunctions[50,94].Thelinkedtoeachother.latterSupposeiscomparabletotheratioruleusedinclassicIPmodelstothatwehaveanexternaldurationD,withanassociatedachieveinternalthescalarproperty[38,95].clockreadingSthatrepresentsthenumberofpulsesintheWhenaccumulatorthelikelihoodandthepriorareindependentGaussians,thatattheendofD.ThelikelihoodfunctionP(SjD)istheis,probabilityP(S|D)N(ms,ss),P(D)N(mp,sp),theoptimalestimatebydistributionofobtainingaclock-readingSforagivenminimizingexternalthelossfunctionLis:durationD.Thespreadofthelikelihoodindicatestheuncertaintyofsensorymeasurement.Atthememorystage,sensorydˆ¼ð1wpÞmsþwpmp[II]estimatesofthetargetdurationupdatethepriordistributionP(D)in21=spreferencememory.Meanwhile,thepriorknowledgemayinfluencewherewp¼1=s22isproportionaltoitsinversedvariancespþ1=ss22thespssmemoryrepresentationofthecurrentclockreading.Accordingto(FigureI).Thevarianceofthisoptimalestimateiss22,whichisBayes’pþssrule,theprobabilityofhavinganexternaldurationDforathegivenminimumvarianceamongallpossiblelinearweightedcombina-clock-readingSisdeterminedbythesensorylikelihoodandthetionspriorbetweenthesensoryestimateandtheprior[92,93,96].WhenknowledgeoftargetdurationsP(D):therearetwoconditionalindependentlikelihoodfunctions,andthePðSjDÞPðDÞpriorisnotthefocusfactorandcanbeassumedtobeuniform,PðDjSÞ¼[I]PðSÞBayesianoptimizationisequivalenttomaximumlikelihoodestima-tion(MEL).TheoptimalestimateisalsoalinearweightedaverageofTheprobabilitydistributionP(DjS)isknownastheposteriorindividualprobability.sensoryestimates:Giventheposteriorprobability,theBayesianideal-dˆobserver¼wamþwbm[III]nexthastomakeanoptimaldecisionorchooseanactionabbasedwheremandmarethemeanestimatesoftwoindividualsignalonthelossfunction,afunctionthatspecifieshowtheabsystemdurations,andwandwaretheircorrespondingweightsthatareratestherelativesuccessorcostofaparticularresponseab[45,50,67],thatis,thecostsassociatedwithafunctionofproportionaltotheirinversedvariances.IPmodelBayesianinferenceμpClockstagePacemakerSwitchAccumulatorLikelihoodσsμpReferencememoryPriorσpdˆMemorystageWorkingmemoryPosteriorσdˆDecisionstageComparatorRDˆD<∈LossfunconL=(Dˆ–D)2D∼RTRENDSinCognitiveSciencesFigureI.Aninformation-processing(IP)modelofscalartimingtheoryandBayesianinferenceoftimeestimation.TheleftpanelshowsanIPmodeloftimeperceptioninvolvingclock,memory,anddecisionstages.TherightpanelillustratesthatthethreekeycomponentsofBayesianinferencearecloselymatchedtothethreestagesoftheIPmodel.Thesensorylikelihoodisderivedfromtheclockstage.Thepriorrepresentsthedurationsstoredinthereferencememory,whichisupdatedbycurrentestimates(dashedblackarrow).Theposteriorreflectstheprobabilitydistributionofthecurrentestimate,combiningtheclockreadingandtheinfluenceofthereferencememory(indicatedbythedashedredarrow).Inthedecisionstage,responsesaremadebasedonspecificcomparisonrules.ThegoalofBayesianinterferenceistominimizethelossfunction,whereasthecomparatoroftheIPmodelusesarelativediscriminationthreshold.durationspresentedwithinablockoftrialstominimizedecision-rulesusedintemporalbisectionarestillundertheiroveralltemporalreproductionerror[6].debate[52],oneproposalisthatacomparisonismadeThe‘modality’effect[8]inducedbythe‘memorymixing’betweenthecurrenttrial’sclockreadingandaninternalofauditoryandvisualsignalscanalsobequantitativelyreference(M)ofthemeanduration(e.g.,geometricmeanofmodeledusingBayesianinference.AlthoughtheexactSandL).Giventhatauditorystimulidrivetheclockfaster560 OpinionTrendsinCognitiveSciencesNovember2013,Vol.17,No.11(A)reliability(assumingMaandMvareindependentGaus-sians).WhentemporaljudgmentsaremadebasedonthismixedinternalreferenceMav,theauditoryPSEisshifted1,200earlier,whereasthevisualPSEisshiftedlaterintime.Asaresult,‘soundsarejudgedlongerthanlights’andthe1,023responsepatterncorrespondingtotheclassic‘modality’effectemergesasshowninFigure2A.OtherapplicationsofBayesianinferencetointerval847timinghaveshownhowtheassimilationofan‘entrain-ment’contextcanbeexplainedbyalinearweightedmodelKey:(EquationProduconme(ms)6711)inwhichtheweightsaredirectlyproportionalPriorcondiontothemeasuredWeberfractions[13].Thatis,theuncer-LongtaintiesIntermediateofthestimulusdurationandthebackground494contextShortdeterminetheircontributionsinthefinalestimate(Box2).SimilarBayesianapproacheshavebeen4946718471,0231,200appliedtomultimodal/sensorimotorintervalintegration[12,14,33,34].Forexample,Shiandcolleagues[12]testedSampleinterval(ms)whetherperceptualandmotortimingareintegratedinanauditoryreproductiontaskbycomparingpuremotorre-production,auditorydurationcomparison,andauditory(B)reproduction.Theyfoundthatparticipantsincorporatebothperceptualandmotortiminginanoptimalmannerintheauditoryreproductiontask.Itshouldbenoted,however,thatvariancesinsomeconditions,particularlyEsmatethoseforshort-intervalauditoryreproduction,arenotassmallasthemodel-predictedvariances,anddonotexhibittheexpectedimprovement.Interestingly,suchsuboptimalMeasurementoutcomeshavebeenconfirmedinseveralotherstudies[12,14,33–35],showingthattheobservedbehavioralvari-MeasurementPriorabilityisoftenlargerthanpredictedbythemodel[34,35].noisePosteriorLikelihoodApossiblereasonforthissuboptimalintegrationmaybethatsomeofthemodelingassumptions,suchasindepen-dentsensoryestimatesandGaussiannoise,arenotSampleintervalcompletelyfulfilled[53].Forexample,timeestimatesfromTRENDSinCognitiveSciencesdifferentsensorymodalitiesmaynotbestatisticallyinde-Figure3.Illustrationofthe‘central-tendency’effectintemporalreproduction.(A)pendent,assuggestedbyevidenceforacommontimingIndividualproductiontimesforindividualtrials(smalldots)andtheiraveragesformechanismeach[7,54,55].sampleinterval(largecirclesconnectedwiththicklines)areshownforthreepriorconditionsforatypicalparticipant.Averageproductiontimesdeviatedfromthelineofequality(diagonaldashedline)towardthemeanofthepriors(horizontalIntegratingBayesianinferencewithscalartimingtheorydashedlines).Prior-dependentbiaseswerestrongestforthe‘long’priorcondition.As(B)reviewedabove,scalartimingtheoryandBayesianSchematicrepresentationsofhowBayesianinferenceiscomputed.Upwardinferencearrowstacklecontextualcalibrationatdifferentlevels.inblackandgrayshowtwoexampleintervals.Verticaldashedlinesrepresentthenoise-perturbedmeasurementsassociatedwiththosesampleScalartimingtheoryfocusesoncognitivestructuresandintervals.Thelikelihoodfunction,theprior(distributionofexperimentalinformationintervals),flow,whereastheBayesianapproachempha-andtheposteriorareshownonthefarright.Thethickdotsonthesizesposteriorcomputationalprinciples.Oneofthecriticismsleviedandhorizontalarrowsshowtheoptimalestimatesbasedonaselectedcostfunction(here,square-errorfunction).Amongvariousobservermodels,aagainsttheBayesianapproachisthattheselectionoftheBayesianobserverassociatedwiththeBayesleast-square(BLS)wasthebestfit.likelihood,prior,andlossfunctionistooflexible,andAdapted,withpermission,from[6].sometimesadhoc.Manydatasetscanbefittedtosupporttheclaimofoptimalbehaviorbyselectingthe‘appropriate’thanvisualstimuli,theinternalreferenceforauditoryfunctions[48,56–58].Inordertoavoidthepitfallsofdurations(Ma)willbelargerthantheinternalreference‘BayesianFundamentalism’,JonesandLove[56]haveforvisualdurations(Mv).However,whenparticipantsusearguedthatBayesianmodelsmustbeintegratedwiththesamedecisioncriterionandbias,theinternaldiffer-moremechanisticapproachesiftheyaretoserveasgenu-encebetweenMaandMvcannotbeobservedifthetempo-inepsychologicaltheories.Inlightofthis,wecomparekeyralbisectiontaskisconductedseparatelywithinauditorycomponentsoftheBayesianapproachwithscalartimingorvisualmodalities.Thisisbecausebothinternalrefer-theory[59–63]inordertosuggesthowthesetwotheoreti-enceswillbecorrespondenttothesameexternal(auditorycalframeworksmightbeprofitablyintegrated.orvisual)duration(Figure2B).However,whenauditoryTherearethreeessentialelementsinaBayesianframe-andvisualdurationsarerandomlymixed,accordingtowork:thelikelihood,thepriorprobability,andthelossBayesianinference,theinternalreference(Mav)ofthefunctionforoptimization.Theseelementscanbestraight-mixeddurationswillbealinear-weightedaverageofMaforwardlymappedontotheprimaryinformation-proces-andMvwithweightsproportionaltotheircorrespondentsingstagesofscalartimingtheory:theclock,memory,and561 OpinionTrendsinCognitiveSciencesNovember2013,Vol.17,No.11decisionstages(Box2).Theclockstageisresponsibleforwhichprobabilisticrepresentationsareimplementedinthemeasurementofthedurationofanexternalevent,neuralcircuits[56,58,68].Forinstance,theobservationthatwhichissubjecttonoiseperturbation.TheBayesianlike-dopaminergicandcholinergicdrugshavedifferenteffectsonlihoodfunctionprovidesaprobabilitydescriptionofagiventheclockandmemorystagesofintervaltiming[61,69–71]measurement,conditionalonagivenphysicalduration.suggeststhatnovelpharmacologicaltechniquesmaypro-Scalartimingtheoryassumestwoseparatememoryrepre-videusefultoolsforstudyingtheneuralimplementationofsentations:aworkingmemorythatisabletotemporarilyprobabilisticrepresentations[72]–althoughthespecificstorethecurrentclockreadingandareferencememorydetailsremaintobetested.Interestingly,the‘migration’thatservesasalong-termstoreorreferenceofallrecordedeffectsobservedinPDpatientssuggestthatthelikelihoodclockreadingsrelevanttoaparticularcontext.FromthefunctionisflattenedbythewithdrawalofdopaminergicBayesianperspective,thepriorquantifiestheprobabilitymedication,whichisrequiredtomaintainmorenormaldistributionoftheinternalreferenceandtheposteriorfunctioningoftheclockstageinPDpatients.Thus,therepresentstheprobabilityofthememoryrepresentationunchangedpriorgainsmoreweightinthefinalestimateofthecurrentclockreading.Intheend,bothframeworks(Figure1C).Inthismanner,dopamine-deficientindividualsimplementsometypeofdecisionruleinordertogenerateaappeartobeabletobalanceperformancebyreducingtem-response.Inthisview,scalartimingtheoryprovidesaporaluncertaintyatthecostoftemporalaccuracy[11].rationalbasisforselectingtheappropriateBayesianfunc-IntegratingBayesianinferencewithscalartimingtheorytions,andtheBayesianframeworkprovidesprobabilisticcouldalsobebeneficialinexplainingotherformsofcontex-descriptionsoftemporalprocessing.tualcalibration,includingtheinfluencesofnon-temporalUnderlyingthelinkageofthetwoframeworks,however,factors(e.g.,backgroundintensity,speed/sequencestruc-areseveralkeydifferencesandanumberofimportantture)[4,73–75].constraints.Themaindifferencecomesfromtheimplemen-tationofthescalarproperty.ScalartimingtheoryassumesConcludingremarksthatthescalarpropertyisintroducedbythememorycon-Underordinarycircumstances,therepresentationsofstant,k*,appliedduringencoding[60,61,64–66].Bycon-eventdurationsare‘calibrated’byvariousformsoftempo-trast,thegeneralBayesianframeworkdoesnotprovideanyralcontext.Thesecontextualcalibrationsinclude‘migra-specificassumptionsconcerningthescalarproperty.How-tion’towardthecentraltendencyofadistributionever,inexperimentalapplicationsofBayesianinference(Vierordt’slaw),TOEs,andmodalitydifferences.Applica-[6,50],thescalarpropertyisoftenincorporatedintothetionofscalartimingtheorysuggeststhatcontextualcali-likelihooddistributioninordertoprovideconsistencywithbrationofeventdurationsoccursmainlyatthememorytheobtaineddata.Recentevidence[50]supportsthisas-stage[8–10].Ontheotherhand,recentBayesiansumption,suggestingthatthescalarpropertymayinfactapproachespointoutthatthelikelyreasonforsuchcali-originatefromtheclockstage.Thesecondmaindifferencebrationisanefforttoimprovetimedperformancebybetweenthetwoframeworksisinregardtomemoryupdat-reducingtheoverallerror[6,13,76].Inanefforttoresolveing.Inordertoaccountfortheeffectsofcontextualcalibra-theseapparentincompatibilities,wehaveshownthatthetion,thememorystageofscalartimingtheoryhasbeenthreeessentialcomponentsofaBayesianframework(i.e.,extendedfromtheoriginalworkingandreferencememorylikelihood,prior,andlossfunction)arecloselylinkedtothecomponents[38,39]toincludetheprocessesof‘memoryclock,memory,anddecisionstagesadvocatedbyscalarmixing’[8–10,40],anddynamicmemoryupdating[5,18].timingtheoryandincorporatedintoothertimingmodelsBycontrast,Bayesianinferenceoffersasimpleandconcise[40,69,77].ThematchedcounterpartsofaBayesianframe-approachforupdatingmemoryrepresentations,namelyworkcombinedwithscalartimingtheorynotonlyprovidesBayes’rule.However,selectionoftheappropriatelikeli-aforward-lookingperspectiveonintervaltiming,butalsohoodsandpriorsmustbedoneinconjunctionwithaspecificoffersquantitativepredictionsofdistortionsintemporalapplication.Thethirdandfinaldifferencethatwewillmemoryfornormalparticipants,aswellasforindividualsdiscussinvolvesthedecisionrulesappliedbythetwoframe-withneurologicalimpairments[78,79].Itisworthnoting,works.Scalartimingtheorystronglyfavorsaratio-rulehowever,thatmanyaspectsofintervaltimingremainapproach(i.e.,relativeerror–FigureI,Box2)[38,41,52]unsolved(Box3),evenforrelativelysimpletemporalfordurationcomparisons,largelybecauseitiscompatiblewiththescalarpropertyintroducedbythememorytransla-Box3.Questionsforfutureresearchtionconstant.IntheBayesianframework,thedecisionruleisHowcantheprobabilisticrepresentationsoftheBayesianlike-usedtominimizetheoverallerrorbasedonalossfunction,lihoodwhereasandpriorbeimplementedintemporalmemory?thelossfunctiondependsonaspecificapplication.TheBayesianinferencedoesnotexplicitlystatethesourceofthemostcommonlossfunctionusedbytheBayesianap-scalarproperty.Dothesensorylikelihood,prior,andlossfunctionproachisthesquare-errorfunction(Box2),whichhasbeenallexhibitthescalarproperty?shownCanBayesianinferencereadilyaccountfornon-temporalformsoftobeingoodagreementwithempiricalfindingscontextual[50,67].calibration(e.g.,stimulusintensityeffects)?Whatarethewaysinwhichgeneticand/orpharmacologicalObtainingabetterunderstandingoftheinteractionsprofilescanbeusedtounraveltheprobabilisticrepresentationsbetweentheBayesianframeworkandscalartimingtheoryandcomputationsusedintimeperception?shouldhelpustodevelopmorerobusttheoriesofintervalHowcanwefurtherintegratetheBayesianframeworkwithothertimingtheories,suchasthestriatalbeat–frequencymodelofintervalthatareabletohandlevarioustypesofcontextualtimingcalibration,[54,77]?whilealsosheddinglightonthemannerin562 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