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What is Bayesian Analysis?

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Many people advocate the Bayesian approach because of its philosophical consistency. Various fundamental theorems show that if a person wants to make consistent ... Facebook Twitter WhatisBayesianAnalysis? by KateCowles,RobKass,andTonyO’Hagan WhatwenowknowasBayesianstatisticshasnothadaclearrunsince1763.AlthoughBayes’smethodwasenthusiasticallytakenupbyLaplaceandotherleadingprobabilistsoftheday,itfellintodisreputeinthe19thcenturybecausetheydidnotyetknowhowtohandlepriorprobabilitiesproperly.Thefirsthalfofthe20thcenturysawthedevelopmentofacompletelydifferenttheory,nowcalledfrequentiststatistics.ButtheflameofBayesianthinkingwaskeptalivebyafewthinkerssuchasBrunodeFinettiinItalyandHaroldJeffreysinEngland.ThemodernBayesianmovementbeganinthesecondhalfofthe20thcentury,spearheadedbyJimmySavageintheUSAandDennisLindleyinBritain,butBayesianinferenceremainedextremelydifficulttoimplementuntilthelate1980sandearly1990swhenpowerfulcomputersbecamewidelyaccessibleandnewcomputationalmethodsweredeveloped.ThesubsequentexplosionofinterestinBayesianstatisticshaslednotonlytoextensiveresearchinBayesianmethodologybutalsototheuseofBayesianmethodstoaddresspressingquestionsindiverseapplicationareassuchasastrophysics,weatherforecasting,healthcarepolicy,andcriminaljustice. Scientifichypothesestypicallyareexpressedthroughprobabilitydistributionsforobservablescientificdata.Theseprobabilitydistributionsdependonunknownquantitiescalledparameters.IntheBayesianparadigm,currentknowledgeaboutthemodelparametersisexpressedbyplacingaprobabilitydistributionontheparameters,calledthe“priordistribution”,oftenwrittenas    Whennewdata becomeavailable,theinformationtheycontainregardingthemodelparametersisexpressedinthe“likelihood,”whichisproportionaltothedistributionoftheobserveddatagiventhemodelparameters,writtenas    Thisinformationisthencombinedwiththepriortoproduceanupdatedprobabilitydistributioncalledthe“posteriordistribution,”onwhichallBayesianinferenceisbased.Bayes’Theorem,anelementaryidentityinprobabilitytheory,stateshowtheupdateisdonemathematically:theposteriorisproportionaltothepriortimesthelikelihood,ormoreprecisely,    Intheory,theposteriordistributionisalwaysavailable,butinrealisticallycomplexmodels,therequiredanalyticcomputationsoftenareintractable.Overseveralyears,inthelate1980sandearly1990s,itwasrealizedthatmethodsfordrawing samplesfromtheposteriordistributioncouldbeverywidelyapplicable. TherearemanyreasonsforadoptingBayesianmethods,andtheirapplicationsappearindiversefields.ManypeopleadvocatetheBayesianapproachbecauseofitsphilosophicalconsistency.Variousfundamentaltheoremsshowthatifapersonwantstomakeconsistentandsounddecisionsinthefaceofuncertainty,thentheonlywaytodosoistouseBayesianmethods.OtherspointtologicalproblemswithfrequentistmethodsthatdonotariseintheBayesianframework.Ontheotherhand,priorprobabilitiesareintrinsicallysubjective–yourpriorinformationisdifferentfrommine–andmanystatisticiansseethisasafundamentaldrawbacktoBayesianstatistics.AdvocatesoftheBayesianapproacharguethatthisisinescapable,andthatfrequentistmethodsalsoentailsubjectivechoices,butthishasbeenabasicsourceofcontentionbetweenthe`fundamentalist’supportersofthetwostatisticalparadigmsforatleastthelast50years.Incontrast,itismorethepragmaticadvantagesoftheBayesianapproachthathavefuelleditsstronggrowthoverthelast20years,andarethereasonforitsadoptioninarapidlygrowingvarietyoffields.PowerfulcomputationaltoolsallowBayesianmethodstotacklelargeandcomplexstatisticalproblemswithrelativeease,wherefrequentistmethodscanonlyapproximateorfailaltogether.Bayesianmodellingmethodsprovidenaturalwaysforpeopleinmanydisciplinestostructuretheirdataandknowledge,andtheyyielddirectandintuitiveanswerstothepractitioner’squestions. TherearemanyvarietiesofBayesiananalysis.ThefullestversionoftheBayesianparadigmcastsstatisticalproblemsintheframeworkofdecisionmaking.Itentailsformulatingsubjectivepriorprobabilitiestoexpresspre-existinginformation,carefulmodellingofthedatastructure,checkingandallowingforuncertaintyinmodelassumptions,formulatingasetofpossibledecisionsandautilityfunctiontoexpresshowthevalueofeachalternativedecisionisaffectedbytheunknownmodelparameters.Buteachofthesecomponentscanbeomitted.ManyusersofBayesianmethodsdonotemploygenuinepriorinformation,eitherbecauseitisinsubstantialorbecausetheyareuncomfortablewithsubjectivity.Thedecision-theoreticframeworkisalsowidelyomitted,withmanyfeelingthatstatisticalinferenceshouldnotreallybeformulatedasadecision.SotherearevarietiesofBayesiananalysisandvarietiesofBayesiananalysts.ButthecommonstrandthatunderliesthisvariationisthebasicprincipleofusingBayes’theoremandexpressinguncertaintyaboutunknownparametersprobabilistically.

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