bayesian analysis bayesian inference and decision theory

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1、BayesianInferenceandDecisionTheoryThischapterisanintroductiontobasicconceptsandimplementationofBayesiananalysis.Webeginwithsubjectiveprobabilityasdistinctfromclas•sicalorobjectiveprobabilityofanuncertaineventbasedonthelongrunrelativefrequencyofitsoccurrence.Subjectiveprobability,alongwithutil

2、ityorlossfunction,leadstoBayesianinferenceanddecisiontheory,e.g.,estima•tion,testing,prediction,etc.Elicitationofsubjectiveprobabilityisrelativelyeasywhentheobserva•tionsareexchangeable.Wediscussexchangeability,itsroleinBayesiananal•ysis,anditsimportanceforscienceasawhole.Inmostcasesinpractic

3、e,quantificationofsubjectivebelieforjudgmentisnoteasilyavailable.Itisthencommontochoosefromamongconventionalpriorsonthebasisofsomerelativelysimplesubjectivejudgmentsabouttheproblemandtheconventionalprobabilitymodelforthedata.Suchpriorsarecalledobjectiveornoninformative.Thesepriorshavebeencrit

4、icizedforvari•ousreasons.Forexample,theydependontheformofthelikelihoodfunctionandusuallyareimproper,i.e.,thetotalprobabilityoftheparameterspaceisinfinity.HereinChapter2,wediscusshowtheyareapplied;someanswerstothecriticismsaregiveninChapter5.InSection2.3ofthischapter,thereisabriefdiscussionoft

5、hemanyadvantagesofbeingaBayesian.2.1SubjectiveandFrequentistProbabilityProbabilityhasvariousconnotations.Historically,ithasbeenconnectedwithbothpersonalevaluationofuncertainty,asingamblingorotherdecisionmak•ingunderuncertainty,andpredictionsaboutproportionofoccurrenceofsomeuncertainevent.Thus

6、whenapersonsaystheprobabilityishalfthatthispar•ticularcoinwillturnupahead,thenitwillusuallymeanthatinmanytossesabouthalfthetimeitwillbeahead(aversionofthelawoflargenumbers).Butitcanalsomeanthatifsomeoneputsthisbetonhead-ifheadhewins302BayesianInferenceandDecisionTheoryadollar,ifnothelosesadol

7、lar-thegambleisfair.Thefirstinterpretationisfrequentist,thesecondsubjective.Similarlyonecanhavebothinterpretationsinmindwhenaweatherforecastsaysthereisaprobabilityof60%ofrain,butthesubjectiveinterpretationmattersmore.Ithelpsyoudecideifyouwill