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1、10SomeApplicationsThepopularityofBayesianmethodsinrecenttimesismainlyduetotheirsuc•cessfulapplicationstocomplexhigh-dimensionalreal-lifeproblemsindiverseareassuchasepidemiology,microarrays,patternrecognition,signalprocess•ing,andsurvivalanalysis.Thischapterpresentsafewsuchapplicationsto•gether
2、withtherequiredmethodology.Wedescribethemethodwithoutgoingintothedetailsofthecriticalissuesinvolved,forwhichreferencesaregiven.Thisisfollowedbyanapplicationinvolvingrealorsimulateddata.WebeginwithahierarchicalBayesianmodelingofspatialdatainSec•tion10.1.Thisisinthecontextofdiseasemapping,anarea
3、ofepidemio•logicalinterest.Thenexttwosections,10.2and10.3,presentnonparametricestimationofregressionfunctionusingwaveletsandDirichletmultinomialal•location.TheymayalsobetreatedasapplicationsinvolvingBayesiandatasmoothing.ForseveralrecentadvancesinBayesiannonparametrics,seeDeyetal.(1998)andGhos
4、handRamamoorthi(2003).10.1DiseaseMappingOurfirstapplicationisfromtheareaofepidemiologyandinvolveshierarchicalBayesianspatialmodeling.Diseasemappingprovidesageographicaldistribu•tionofadiseasedisplayingsomeindexsuchastherelativeriskofthediseaseineachsubregionoftheareatobemapped.Supposethatthear
5、eatobemappedisdividedintoNregions.LetOiandEiberespectivelytheobservedandexpectednumberofcasesofadiseaseintheithregion,i=1,2,...,A^.TheunknownparametersofinterestareOi,therelativeriskintheithre•gion,i=1,2,...,A^.HereEiisasimple-mindedexpectationassumingallregionshavethesamediseaserate(atleastaf
6、teradjustmentforage),videBanerjeeetal.(2004,p.158).TherelativeriskOiistheregionaleffectinamultiplicativemodelofexpectednumberofcases:E{Oi)=EiOi.liOi=1,wehaveE{Oi)=Ei.Theobjectiveistomakeinferenceabout^^'sacrossregions.Amongotherthings,thishelpsepidemiologistsandpublichealthprofessionals29010So
7、meApplicationstoidentifyregionsorclusterofregionshavinghighrelativerisksandhenceneedingattentionandalsotoidentifycovariatescausinghighrelativerisk.ThetraditionalmodelforOiisthePoissonmodel,whichstatesthatgiven(^1,...,^iv),Oi'sareindepen
