bayesian analysis utility prior and bayesian robustness

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1、Utility,Prior,andBayesianRobustnessWebeginthischapterwithadiscussionofrationahtyaxiomsforpreferenceandhowonemaydeducetheexistenceofautihtyandprior.LaterweexplorehowrobustorsensitiveisBayesiananalysistothechoiceofprior,utility,andmodel.Intheprocess,weintroducean

2、dexaminevariousquantitativeevaluationsofrobustness.3.1Utility,Prior,andRationalPreferenceWehaveintroducedinChapter2problemsofestimationandtestingasBayesiandecisionproblems.WerecallthecomponentsofageneralBayesiandecisionproblem.LetXbethesamplespace,0theparameter

3、space,f{x)thedensityofXand7r(^)priorprobabilitydensity.Moreover,thereisaspaceAofactions"a"andalossfunctionL{0,a).Thedecisionmaker(DM)chooses"a"tominimizetheposteriorrisk^{ax)=/L{e,a)iT{ex)dO,(3.1)whereT:{9X)istheposteriordensityof0givenx.Notethatgiventhelo

4、ssfunctionandtheprior,thereisanaturalpreferenceorderingai:

5、x).ThereisalongtraditionoffoundationalstudydatingbacktoRamsey(1926),inwhichonestartswithsuchapreferencerelationon.4x^satis•fyingcertainrationalaxioms(i.e

6、.,axiomsmodelingrationalbehavior)liketransitivity.Itcanthenbeshownthatsucharelationcanonlybeinducedasaboveviaalossfunctionandaprior,i.e.,3LandTTsuchthatai

7、on,onecanrecoverthesubjectivelossfunctionandprior.Ifthereisnosampledata.663Utility,Prior,andBayesianRobustnessthenTVwouldqualifyasasubjectivepriorfortheDM.Ifwehavedatax,alikelihoodfunctionf{x6)isgivenandweareexaminingapreferencerelationgivenx,thenalsoonecanded

8、ucetheexistenceofLandTTsuchthatai:

9、musthavea(subjective)priorifourpreferenceamongactionssatisfiescertainaxiomsaboutrationalbehavior.Together,theyjustify(3.2)andthrowsomelighton(3.3).Intheremainingsectionsweex