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1、9High-dimensionalProblemsRatherthanbeginbydefiningwhatismeantbyhigh-dimensional,webeginwithacoupleofexamples.Example9.1.(Stein'sexample)LetN{fipxi^^pxp=o-'^Ipxp)beap-variatenormalpopulationandXi=(Xn^...,X^p),i=1,...,nbeni.i.d.p-variatesamples.BecauseU=cr^/,wemayalter
2、nativelythinkofthedataaspin•dependentsamplesofsizenfrompunivariatenormalpopulationsN{/j.j,cr^),J=1,...,p.Theparametersofinterestarethe/ij's.Forconvenience,weini•tiallyassumecr^isknown.Usually,thenumberofparameters,p,islargeandthesamplesizenissmallcomparedwithp.Theseh
3、avebeencalledproblemswithlargep,smalln.NotethatninStein'sexampleisthesamplesize,ifwethinkofthedataasap-variatesampleofsizen.However,wecouldalsothinkofthedataasunivariatesamplesofsizenfromeachofpunivariatepopulations.Thenthetotalsamplesizewouldbenp.Thesecondinterpreta
4、•tionleadstoaclassofsimilarexamples.Notethattheobservationsarenotexchangeableexceptinsubgroups,inthissenseonemaycallthempartiallyexchangeable.Example9.2.Letf{xfij)^j=1,...,p,denotethedensitiesforppopulations,andXij^i=l,...,n,ji=l,...,pdenotepsamplesofsizenfromthesep
5、populations.AsinExample9.1,f{xfij)maycontainadditionalcommonparameters.Theobjectistomakeinferenceaboutthe/ij's.Inseveralpath-breakingpapersStein(1955),JamesandStein(1960),Stein(1981),Robbins(1955,1964),EfronandMorris(1971,1972,1973a,1975)haveshownclassicalobjectiveB
6、ayesorclassicalfrequentistmethods,e.g.,maximumlikelihoodestimates,willusuallybeinappropriatehere.SeealsoKieferandWolfowitz(1956)forapplicationstoexampleslikethoseofNeymanandScott(1948).TheseapproachesarediscussedinSections9.1through9.4,withstressontheparametricempiri
7、calBayes(PEB)approachofEfronandMorris,asextendedinMorris(1983).2569High-dimensionalProblemsItturnsoutthatexchangeabilityof/ii,...,/ipplaysafundamentalroleinalltheseapproaches.Underthisassumption,thereisasimpleandnaturalBayesiansolutionoftheproblembasedonahierarchical
8、priorandMCMC.MuchofthepopularityofBayesianmethodsisduetothefactthatmanynewexamplesofthiskindcouldbetreatedinaunifiedway.Becauseofth
