Bayesian probability - Wikipedia

Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is ... Bayesianprobability FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Forbroadercoverageofthistopic,seeBayesianstatistics. PartofaseriesonBayesianstatistics Theory Admissibledecisionrule Bayesianefficiency Bayesianepistemology Bayesianprobability Probabilityinterpretations Bayes'theorem Bayesfactor Bayesianinference Bayesiannetwork Prior Posterior Likelihood Conjugateprior Posteriorpredictive Hyperparameter Hyperprior Principleofindifference Principleofmaximumentropy EmpiricalBayesmethod Cromwell'srule Bernstein–vonMisestheorem Schwarzcriterion Credibleinterval Maximumaposterioriestimation Radicalprobabilism Techniques Bayesianlinearregression Bayesianestimator ApproximateBayesiancomputation MarkovchainMonteCarlo IntegratednestedLaplaceapproximations  MathematicsportalvteInterpretationofprobabilityBayesianprobabilityisaninterpretationoftheconceptofprobability,inwhich,insteadoffrequencyorpropensityofsomephenomenon,probabilityisinterpretedasreasonableexpectation[1]representingastateofknowledge[2]orasquantificationofapersonalbelief.[3] TheBayesianinterpretationofprobabilitycanbeseenasanextensionofpropositionallogicthatenablesreasoningwithhypotheses;[4][5]thatis,withpropositionswhosetruthorfalsityisunknown.IntheBayesianview,aprobabilityisassignedtoahypothesis,whereasunderfrequentistinference,ahypothesisistypicallytestedwithoutbeingassignedaprobability. Bayesianprobabilitybelongstothecategoryofevidentialprobabilities;toevaluatetheprobabilityofahypothesis,theBayesianprobabilistspecifiesapriorprobability.This,inturn,isthenupdatedtoaposteriorprobabilityinthelightofnew,relevantdata(evidence).[6]TheBayesianinterpretationprovidesastandardsetofproceduresandformulaetoperformthiscalculation. ThetermBayesianderivesfromthe18th-centurymathematicianandtheologianThomasBayes,whoprovidedthefirstmathematicaltreatmentofanon-trivialproblemofstatisticaldataanalysisusingwhatisnowknownasBayesianinference.[7]: 131 MathematicianPierre-SimonLaplacepioneeredandpopularizedwhatisnowcalledBayesianprobability.[7]: 97–98  Contents 1Bayesianmethodology 2ObjectiveandsubjectiveBayesianprobabilities 3History 4JustificationofBayesianprobabilities 4.1Axiomaticapproach 4.2Dutchbookapproach 4.3Decisiontheoryapproach 5Personalprobabilitiesandobjectivemethodsforconstructingpriors 6Seealso 7References 8Bibliography Bayesianmethodology[edit] Bayesianmethodsarecharacterizedbyconceptsandproceduresasfollows: Theuseofrandomvariables,ormoregenerallyunknownquantities,[8]tomodelallsourcesofuncertaintyinstatisticalmodelsincludinguncertaintyresultingfromlackofinformation(seealsoaleatoricandepistemicuncertainty). Theneedtodeterminethepriorprobabilitydistributiontakingintoaccounttheavailable(prior)information. ThesequentialuseofBayes'theorem:asmoredatabecomeavailable,calculatetheposteriordistributionusingBayes'theorem;subsequently,theposteriordistributionbecomesthenextprior. Whileforthefrequentist,ahypothesisisaproposition(whichmustbeeithertrueorfalse)sothatthefrequentistprobabilityofahypothesisiseither0or1,inBayesianstatistics,theprobabilitythatcanbeassignedtoahypothesiscanalsobeinarangefrom0to1ifthetruthvalueisuncertain. ObjectiveandsubjectiveBayesianprobabilities[edit] Broadlyspeaking,therearetwointerpretationsofBayesianprobability.Forobjectivists,whointerpretprobabilityasanextensionoflogic,probabilityquantifiesthereasonableexpectationthateveryone(evena"robot")whosharesthesameknowledgeshouldshareinaccordancewiththerulesofBayesianstatistics,whichcanbejustifiedbyCox'stheorem.[2][9]Forsubjectivists,probabilitycorrespondstoapersonalbelief.[3]Rationalityandcoherenceallowforsubstantialvariationwithintheconstraintstheypose;theconstraintsarejustifiedbytheDutchbookargumentorbydecisiontheoryanddeFinetti'stheorem.[3]TheobjectiveandsubjectivevariantsofBayesianprobabilitydiffermainlyintheirinterpretationandconstructionofthepriorprobability. History[edit] Mainarticle:Historyofstatistics§ Bayesianstatistics ThetermBayesianderivesfromThomasBayes(1702–1761),whoprovedaspecialcaseofwhatisnowcalledBayes'theoreminapapertitled"AnEssaytowardssolvingaProblemintheDoctrineofChances".[10]Inthatspecialcase,thepriorandposteriordistributionswerebetadistributionsandthedatacamefromBernoullitrials.ItwasPierre-SimonLaplace(1749–1827)whointroducedageneralversionofthetheoremandusedittoapproachproblemsincelestialmechanics,medicalstatistics,reliability,andjurisprudence.[11]EarlyBayesianinference,whichuseduniformpriorsfollowingLaplace'sprincipleofinsufficientreason,wascalled"inverseprobability"(becauseitinfersbackwardsfromobservationstoparameters,orfromeffectstocauses).[12]Afterthe1920s,"inverseprobability"waslargelysupplantedbyacollectionofmethodsthatcametobecalledfrequentiststatistics.[12] Inthe20thcentury,theideasofLaplacedevelopedintwodirections,givingrisetoobjectiveandsubjectivecurrentsinBayesianpractice. HaroldJeffreys'TheoryofProbability(firstpublishedin1939)playedanimportantroleintherevivaloftheBayesianviewofprobability,followedbyworksbyAbrahamWald(1950)andLeonardJ.Savage(1954).TheadjectiveBayesianitselfdatestothe1950s;thederivedBayesianism,neo-Bayesianismisof1960scoinage.[13][14][15]Intheobjectiviststream,thestatisticalanalysisdependsononlythemodelassumedandthedataanalysed.[16]Nosubjectivedecisionsneedtobeinvolved.Incontrast,"subjectivist"statisticiansdenythepossibilityoffullyobjectiveanalysisforthegeneralcase. Inthe1980s,therewasadramaticgrowthinresearchandapplicationsofBayesianmethods,mostlyattributedtothediscoveryofMarkovchainMonteCarlomethodsandtheconsequentremovalofmanyofthecomputationalproblems,andtoanincreasinginterestinnonstandard,complexapplications.[17]Whilefrequentiststatisticsremainsstrong(asdemonstratedbythefactthatmuchofundergraduateteachingisbasedonit[18]),Bayesianmethodsarewidelyacceptedandused,e.g.,inthefieldofmachinelearning.[19] JustificationofBayesianprobabilities[edit] TheuseofBayesianprobabilitiesasthebasisofBayesianinferencehasbeensupportedbyseveralarguments,suchasCoxaxioms,theDutchbookargument,argumentsbasedondecisiontheoryanddeFinetti'stheorem. Axiomaticapproach[edit] RichardT.CoxshowedthatBayesianupdatingfollowsfromseveralaxioms,includingtwofunctionalequationsandahypothesisofdifferentiability.[9]Theassumptionofdifferentiabilityorevencontinuityiscontroversial;HalpernfoundacounterexamplebasedonhisobservationthattheBooleanalgebraofstatementsmaybefinite.[20]Otheraxiomatizationshavebeensuggestedbyvariousauthorswiththepurposeofmakingthetheorymorerigorous.[8] Dutchbookapproach[edit] Mainarticle:Dutchbook BrunodeFinettiproposedtheDutchbookargumentbasedonbetting.AcleverbookmakermakesaDutchbookbysettingtheoddsandbetstoensurethatthebookmakerprofits—attheexpenseofthegamblers—regardlessoftheoutcomeoftheevent(ahorserace,forexample)onwhichthegamblersbet.Itisassociatedwithprobabilitiesimpliedbytheoddsnotbeingcoherent. However,IanHackingnotedthattraditionalDutchbookargumentsdidnotspecifyBayesianupdating:theyleftopenthepossibilitythatnon-BayesianupdatingrulescouldavoidDutchbooks.Forexample,Hackingwrites[21][22]"AndneithertheDutchbookargument,noranyotherinthepersonalistarsenalofproofsoftheprobabilityaxioms,entailsthedynamicassumption.NotoneentailsBayesianism.SothepersonalistrequiresthedynamicassumptiontobeBayesian.ItistruethatinconsistencyapersonalistcouldabandontheBayesianmodeloflearningfromexperience.Saltcouldloseitssavour." Infact,therearenon-BayesianupdatingrulesthatalsoavoidDutchbooks(asdiscussedintheliteratureon"probabilitykinematics"[23]followingthepublicationofRichardC.Jeffrey'srule,whichisitselfregardedasBayesian[24]).Theadditionalhypothesessufficientto(uniquely)specifyBayesianupdatingaresubstantial[25]andnotuniversallyseenassatisfactory.[26] Decisiontheoryapproach[edit] Adecision-theoreticjustificationoftheuseofBayesianinference(andhenceofBayesianprobabilities)wasgivenbyAbrahamWald,whoprovedthateveryadmissiblestatisticalprocedureiseitheraBayesianprocedureoralimitofBayesianprocedures.[27]Conversely,everyBayesianprocedureisadmissible.[28] Personalprobabilitiesandobjectivemethodsforconstructingpriors[edit] FollowingtheworkonexpectedutilitytheoryofRamseyandvonNeumann,decision-theoristshaveaccountedforrationalbehaviorusingaprobabilitydistributionfortheagent.JohannPfanzaglcompletedtheTheoryofGamesandEconomicBehaviorbyprovidinganaxiomatizationofsubjectiveprobabilityandutility,ataskleftuncompletedbyvonNeumannandOskarMorgenstern:theiroriginaltheorysupposedthatalltheagentshadthesameprobabilitydistribution,asaconvenience.[29]Pfanzagl'saxiomatizationwasendorsedbyOskarMorgenstern:"VonNeumannandIhaveanticipated...[thequestionwhetherprobabilities]might,perhapsmoretypically,besubjectiveandhavestatedspecificallythatinthelattercaseaxiomscouldbefoundfromwhichcouldderivethedesirednumericalutilitytogetherwithanumberfortheprobabilities(cf.p.19ofTheTheoryofGamesandEconomicBehavior).Wedidnotcarrythisout;itwasdemonstratedbyPfanzagl...withallthenecessaryrigor".[30] RamseyandSavagenotedthattheindividualagent'sprobabilitydistributioncouldbeobjectivelystudiedinexperiments.Proceduresfortestinghypothesesaboutprobabilities(usingfinitesamples)areduetoRamsey(1931)anddeFinetti(1931,1937,1964,1970).BothBrunodeFinetti[31][32]andFrankP.Ramsey[32][33]acknowledgetheirdebtstopragmaticphilosophy,particularly(forRamsey)toCharlesS.Peirce.[32][33] The"Ramseytest"forevaluatingprobabilitydistributionsisimplementableintheory,andhaskeptexperimentalpsychologistsoccupiedforahalfcentury.[34] ThisworkdemonstratesthatBayesian-probabilitypropositionscanbefalsified,andsomeetanempiricalcriterionofCharlesS.Peirce,whoseworkinspiredRamsey.(Thisfalsifiability-criterionwaspopularizedbyKarlPopper.[35][36]) Modernworkontheexperimentalevaluationofpersonalprobabilitiesusestherandomization,blinding,andBoolean-decisionproceduresofthePeirce-Jastrowexperiment.[37]Sinceindividualsactaccordingtodifferentprobabilityjudgments,theseagents'probabilitiesare"personal"(butamenabletoobjectivestudy). Personalprobabilitiesareproblematicforscienceandforsomeapplicationswheredecision-makerslacktheknowledgeortimetospecifyaninformedprobability-distribution(onwhichtheyarepreparedtoact).Tomeettheneedsofscienceandofhumanlimitations,Bayesianstatisticianshavedeveloped"objective"methodsforspecifyingpriorprobabilities. Indeed,someBayesianshavearguedthepriorstateofknowledgedefinesthe(unique)priorprobability-distributionfor"regular"statisticalproblems;cf.well-posedproblems.Findingtherightmethodforconstructingsuch"objective"priors(forappropriateclassesofregularproblems)hasbeenthequestofstatisticaltheoristsfromLaplacetoJohnMaynardKeynes,HaroldJeffreys,andEdwinThompsonJaynes.Thesetheoristsandtheirsuccessorshavesuggestedseveralmethodsforconstructing"objective"priors(Unfortunately,itisnotclearhowtoassesstherelative"objectivity"ofthepriorsproposedunderthesemethods): Maximumentropy Transformationgroupanalysis Referenceanalysis Eachofthesemethodscontributesusefulpriorsfor"regular"one-parameterproblems,andeachpriorcanhandlesomechallengingstatisticalmodels(with"irregularity"orseveralparameters).EachofthesemethodshasbeenusefulinBayesianpractice.Indeed,methodsforconstructing"objective"(alternatively,"default"or"ignorance")priorshavebeendevelopedbyavowedsubjective(or"personal")BayesianslikeJamesBerger(DukeUniversity)andJosé-MiguelBernardo(UniversitatdeValència),simplybecausesuchpriorsareneededforBayesianpractice,particularlyinscience.[38]Thequestfor"theuniversalmethodforconstructingpriors"continuestoattractstatisticaltheorists.[38] Thus,theBayesianstatisticianneedseithertouseinformedpriors(usingrelevantexpertiseorpreviousdata)ortochooseamongthecompetingmethodsforconstructing"objective"priors. Seealso[edit] Mathematicsportal Bertrandparadox—aparadoxinclassicalprobability DeFinetti'sgame—aprocedureforevaluatingsomeone'ssubjectiveprobability QBism—aninterpretationofquantummechanicsbasedonsubjectiveBayesianprobability Referenceclassproblem AnEssaytowardssolvingaProblemintheDoctrineofChances MontyHallproblem Bayesianepistemology References[edit] ^Cox,R.T.(1946)."Probability,Frequency,andReasonableExpectation".AmericanJournalofPhysics.14(1):1–10.Bibcode:1946AmJPh..14....1C.doi:10.1119/1.1990764. ^abJaynes,E.T.(1986)."BayesianMethods:GeneralBackground".InJustice,J.H.(ed.).Maximum-EntropyandBayesianMethodsinAppliedStatistics.Cambridge:CambridgeUniversityPress.CiteSeerX 10.1.1.41.1055. ^abcdeFinetti,Bruno(2017).TheoryofProbability:Acriticalintroductorytreatment.Chichester:JohnWiley&SonsLtd.ISBN 9781119286370. ^Hailperin,Theodore(1996).SententialProbabilityLogic:Origins,Development,CurrentStatus,andTechnicalApplications.London:AssociatedUniversityPresses.ISBN 0934223459. ^Howson,Colin(2001)."TheLogicofBayesianProbability".InCorfield,D.;Williamson,J.(eds.).FoundationsofBayesianism.Dordrecht:Kluwer.pp. 137–159.ISBN 1-4020-0223-8. ^Paulos,JohnAllen(5August2011)."TheMathematicsofChangingYourMind[bySharonBertschMcGrayne]".BookReview.NewYorkTimes.Archivedfromtheoriginalon2022-01-01.Retrieved2011-08-06. ^abStigler,StephenM.(March1990).Thehistoryofstatistics.HarvardUniversityPress.ISBN 9780674403413. ^abDupré,MauriceJ.;Tipler,FrankJ.(2009)."NewaxiomsforrigorousBayesianprobability".BayesianAnalysis.4(3):599–606.CiteSeerX 10.1.1.612.3036.doi:10.1214/09-BA422. ^abCox,RichardT.(1961).Thealgebraofprobableinference(Reprint ed.).Baltimore,MD;London,UK:JohnsHopkinsPress;OxfordUniversityPress[distributor].ISBN 9780801869822. ^McGrayne,SharonBertsch(2011).TheTheorythatWouldnotDie.[https://archive.org/details/theorythatwouldn0000mcgr/page/1010 ],p.10,atGoogleBooks. ^Stigler,StephenM.(1986)."Chapter 3".TheHistoryofStatistics.HarvardUniversityPress. ^abFienberg,Stephen.E.(2006)."WhendidBayesianInferencebecome"Bayesian"?"(PDF).BayesianAnalysis.1(1):5,1–40.doi:10.1214/06-BA101.Archivedfromtheoriginal(PDF)on10September2014. ^Harris,MarshallDees(1959)."Recentdevelopmentsoftheso-calledBayesianapproachtostatistics".AgriculturalLawCenter.Legal-EconomicResearch.UniversityofIowa:125(fn.#52),126.TheworksofWald,StatisticalDecisionFunctions(1950)andSavage,TheFoundationofStatistics(1954)arecommonlyregardedstartingpointsforcurrentBayesianapproaches ^AnnalsoftheComputationLaboratoryofHarvardUniversity.Vol. 31.1962.p. 180.Thisrevolution,whichmayormaynotsucceed,isneo-Bayesianism.Jeffreystriedtointroducethisapproach,butdidnotsucceedatthetimeingivingitgeneralappeal. ^Kempthorne,Oscar(1967).TheClassicalProblemofInference—GoodnessofFit.FifthBerkeleySymposiumonMathematicalStatisticsandProbability.p. 235.Itiscuriousthateveninitsactivitiesunrelatedtoethics,humanitysearchesforareligion.Atthepresenttime,thereligionbeing'pushed'thehardestisBayesianism. ^Bernardo,J.M.(2005)."Referenceanalysis".BayesianThinking-ModelingandComputation.HandbookofStatistics.Vol. 25.pp. 17–90.doi:10.1016/S0169-7161(05)25002-2.ISBN 9780444515391. ^Wolpert,R.L.(2004)."AconversationwithJamesO.Berger".StatisticalScience.9:205–218.doi:10.1214/088342304000000053. ^Bernardo,JoséM.(2006).ABayesianmathematicalstatisticsprimer(PDF).ICOTS-7.Bern. ^Bishop,C.M.(2007).PatternRecognitionandMachineLearning.Springer. ^Halpern,J.(1999)."AcounterexampletotheoremsofCoxandFine"(PDF).JournalofArtificialIntelligenceResearch.10:67–85.doi:10.1613/jair.536.S2CID 1538503. ^Hacking(1967),Section3,page316 ^Hacking(1988,page124) ^Skyrms,Brian(1January1987)."DynamicCoherenceandProbabilityKinematics".PhilosophyofScience.54(1):1–20.CiteSeerX 10.1.1.395.5723.doi:10.1086/289350.JSTOR 187470. ^Joyce,James(30September2003)."Bayes'Theorem".TheStanfordEncyclopediaofPhilosophy.stanford.edu. ^Fuchs,ChristopherA.;Schack,Rüdiger(1January2012).Ben-Menahem,Yemima;Hemmo,Meir(eds.).ProbabilityinPhysics.TheFrontiersCollection.SpringerBerlinHeidelberg.pp. 233–247.arXiv:1103.5950.doi:10.1007/978-3-642-21329-8_15.ISBN 9783642213281.S2CID 119215115. ^vanFrassen,Bas(1989).LawsandSymmetry.OxfordUniversityPress.ISBN 0-19-824860-1. ^Wald,Abraham(1950).StatisticalDecisionFunctions.Wiley. ^Bernardo,JoséM.;Smith,AdrianF.M.(1994).BayesianTheory.JohnWiley.ISBN 0-471-92416-4. ^Pfanzagl(1967,1968) ^Morgenstern(1976,page65) ^Galavotti,MariaCarla(1January1989)."Anti-RealisminthePhilosophyofProbability:BrunodeFinetti'sSubjectivism".Erkenntnis.31(2/3):239–261.doi:10.1007/bf01236565.JSTOR 20012239.S2CID 170802937. ^abcGalavotti,MariaCarla(1December1991)."ThenotionofsubjectiveprobabilityintheworkofRamseyanddeFinetti".Theoria.57(3):239–259.doi:10.1111/j.1755-2567.1991.tb00839.x.ISSN 1755-2567. ^abDokic,Jérôme;Engel,Pascal(2003).FrankRamsey:TruthandSuccess.Routledge.ISBN 9781134445936. ^Davidsonetal.(1957) ^Thornton,Stephen(7August2018)."KarlPopper".StanfordEncyclopediaofPhilosophy. ^Popper,Karl(2002)[1959].TheLogicofScientificDiscovery(2nd ed.).Routledge.p. 57.ISBN 0-415-27843-0–viaGoogleBooks.(translationof1935original,inGerman). ^Peirce&Jastrow(1885) ^abBernardo,J.M.(2005)."ReferenceAnalysis".InDey,D.K.;Rao,C.R.(eds.).HandbookofStatistics(PDF).Vol. 25.Amsterdam:Elsevier.pp. 17–90. 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Winkler,R.L.(2003).IntroductiontoBayesianInferenceandDecision(2nd ed.).Probabilistic.ISBN 978-0-9647938-4-2.Updatedclassictextbook.Bayesiantheoryclearlypresented Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Bayesian_probability&oldid=1095534332" Categories:BayesianstatisticsJustification(epistemology)ProbabilityinterpretationsPhilosophyofmathematicsPhilosophyofscienceHiddencategories:ArticleswithshortdescriptionShortdescriptionisdifferentfromWikidataCS1French-languagesources(fr) Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages العربيةCatalàЧӑвашлаDeutschEestiEspañolEuskaraفارسی한국어HrvatskiItalianoעבריתNederlands日本語NorsknynorskPolskiPortuguêsРусскийSimpleEnglishСрпски/srpskiSrpskohrvatski/српскохрватскиไทยTürkçeУкраїнська粵語中文 Editlinks