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1、22BayesianStatisticsInsteadofasking‘whatdomydatashow?’,theBayesiananalystasks‘howdomydataalterourviewoftheworld?’.Itmaynotsoundlikemuch,butitisafundamentalchangeofoutlook.Theideaisthattheresultsofthenewstudyareassessedinthelightoftheexistingdata,toestablishanupdat
2、edassessmentofparametervaluesandtheiruncertainties.Therearenowtwomodelsratherthanone.Thereisamodelforwhatweknowalready:thisiscalledtheprior.Thenthereisamodelthatwe?ttoourdata:thisisthelikelihood.Thetwomodelscombinetogiveusanestimateoftheposterior.Weusetheposterior
3、distributiontomakestatisticalinferences.UnderitsBayesianinterpretation,probabilitymeasuresourcon?dencethatsomethingistrue.Whenthenewstudyissmall,andexistingknowledgeisextensive,thenweshouldnotexpectourworktomakemuchdifferencetooverallunderstanding.Forinstance,ifwe
4、studied150subjectsandfoundthatageatdeathwasnotaffectedbytheirsmokinghabits,wewouldbeilladvisedtoconcludethatthisresulthadgreatgenerality.Thisisbecauseweknowfromstudiesofmanythousandsofsubjectsovermanydecadesthatsmokingleadstoareductionofabout14yearsinmeanageatdeat
5、h.Smallstudiescanbehighlyinformative,ofcourse,especiallywhentheyareoriginalandaddressimportantquestions.Butthelargerandbetterdesignedthenewstudy,themorethatstudyshouldbecapableofalteringourpriorknowledge,andifthestudyisbigenoughandsuf?cientlywellconducted,thenitco
6、uldturnpriorexpectationonitshead.ABayesiananalysisworksinthreesteps:usingallthecurrentlyavailableinformationtocreateafullprobabilitymodel;formingaposteriorinferencebyconditioningthefullprobabilitymodelonthenewdata;criticizingthe?tofthemodeltothedataandevaluatin
7、gthepredictionsitmakes.YoushouldreadBayesianDataAnalysisbyGelmanetal.(2004)forbackground,examplesandcomputationalmethodsoneachofthesesteps.OneofthestrengthsoftheBayesianapproachisthatitdoesnotrelyontwoofthemostpeculiaraspectsofthefrequentistapproach.Beginnersoften
8、thinkthat:ifIrejectthenullhypothesisatthe5%level,thenthereisa5%chancethatthenullhypothesisistrue;ifIestablisha95%con?denceintervalforthevalueofaparame
