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ABayesianFactorAnalysisModelWithGeneralizedPriorInformationDanielB.RoweDivisionofHumanitiesandSocialSciences,CaliforniaInstituteofTechnology,USAAbstractIntheBayesianapproachtofactoranalysis,availablepriorknowl-edgeregardingthemodelparametersisquanti edintheformofpriordistributionsandincorporatedintotheinferences.Theincor-porationofpriorknowledgehastheaddedconsequenceofeliminat-ingtheambiguityofrotationfoundinthetraditionalfactoranalysismodel.PreviousBayesianfactoranalysiswork(Press&Shigemasu1989,&Press1998,Rowe2000a,andRowe2000b),hasconsideredmainlynaturalconjugatepriordistributionsforthemodelparam-eters.AsismentionedinPress(1982),Rothenburg(1963)pointedoutthatthewithanaturalconjugatepriordistribution,theel-ementsinthecovariancematricesareconstrainedandthusmaynotberichenoughtopermitfreedomofassessment.Inthispa-per,generlizednaturalconjugatedistributionsareusedtoquantifyandincorporateavailablepriorinformationwhichpermitcompletefreedomofassessment.1IntroductionAfactoranalysisisperformedtoexplaintherelationshipamongasetofobservedvariablesintermsofasmallernumberofunobservedvariablesorlatentfactorswhichunderlietheobservations.Thissmallernumberofvariablescanbeusedto ndameaningfulstructureintheobservedvariables.Thisstructurewillaidintheinterpretationandexplanationoftheprocessthathasgeneratedtheobservations.IntheBayesianapproachtofactoranlysis(Press&Shigemasu1989,Rowe&Press1998,Rowe2000a,andRowe2000bhenceforthPS89,RP98,R00a,andR00b)theclassicalnormalsamplingmodelisassumed,butthedisturbancecovariancematrixisspeci edtobeafullpositivede nitematrix.Oneofthepriorassumptions(PS89)isthatthepriorexpectedvalueofthedisturbancecovariancematrixisdiagonalinordertorepresenttraditionalviewsofthefactormodelcontaining\commonand\speci cfactors.AsnotedinPress(1982),Rothenburg(1963)pointedoutthatthenaturalconjugatepriordistributionhascovariancematrixelementsthatareconstrainedandthusmaynotberichenoughtoassessthepriorparameters.Inthispaper,generalizednaturalconjugatepriordistributionsarespeci edfortheunknownmatriceswhichpermitcompletefreedomofassessment.Bayesianstatisticalmethodsnotonlyincorporateavailablepriorinfor-mationeitherfromsubstantiveexpertsorpreviousdata,butallowtheknowledgeregardingtheparametervaluestoaccumulateassubsequentdataisacquired.Inthenon-Bayesianfactoranalysismodel,thefactor2loadingmatrixisdeterminateuptoanorthogonalrotation.Typicallyaf-teranon-Bayesianfactoranalysis,anorthogonalrotationisperformedonthefactorloadingmatrixaccordingtooneofmanysubjectivecriteria.ThisisnotthecaseinBayesianfactoranalysis.Therotationisautomaticallyfound.ThemodelparametersareestimatedbybothGibbssampling(Geman&Geman,1984andGelfand&Smith,1990)anditeratedconditionalmodes(Lindley&Smith,1972andO’Hagen,1994)which ndposteriormarginalmeanandposteriorjointmodal(maximumaposteriori)estimatesrespec-tively.Theplanofthepaperistoreviewthemodelandtoadoptpriordistri-butionsinSection2.PresenttheconditionalposteriordistributionsalongwiththeGibbssamplingandICMalgorithmsinSection3.InSection4anexampleisdetailed,andestimatesfromboththeGibbssamplingandtheICMestimationmethodsarepresented.2Model2.1LikelihoodFunctionTheBayesianfactoranalysismodelis:(xjj ; ;fj)= + fj+ j;mp;(p 1)(p 1)(p m)(m 1)(p 1)(2.1)forj=1;:::;n,wherexjisthejthobservation, istheoverallpopulationmean, isamatrixofconstantscalledthefactorloadingmatrix;fjisthefactorscorevectorforsubjectj;andthe j’sareassumedtobemutually3uncorrelatedandnormallydistributedN(0; )variables.Inthetraditionalmodel, istakentobeadiagonalmatrixsothatcom-monandspeci cfactorscanreadilydistinguished.TheBayesianmodels,take tobeageneralsymmetric,positivede nitecovariancematrixwiththepropertyofbeingaprioridiagonalontheaverage,i.e.,E( )=adiagonalmatrix.Itisassumedthat , ,thefi’s,and areunobservableandthatthedistributionofeachxjcanbewrittenasp(xjj ; ;fj; )=(2 ) p2j j 12e 12(xj fj)0 1(xj fj):(2.2)Ifproportionalityisdenotedby\/andtheKronekerproductby then,thelikelihoodfor( ; ;F; )isp(Xj ; ;F; )/j j n2e 12tr 1(X en 0 F 0)0(X en 0 F 0)(2.3)wherethep-variateobservationvectorsonnsubjectsare,X0=(x1;:::;xn),thefactorscoresareF0=(f1;:::;fn),andtheerrorsofobservationareE0=( 1;:::; n).Thenotationp( )willgenericallydenoteadistributionwhichisdistinguishedbyitsargument.Theproportionalityconstantin(2.3)dependsonlyon(p;n)andnoton( ; ;F; ).2.2PriorsGeneralizednaturalconjugatepriordistributionsarespeci edfortheunknownparameterswhichpermitcompletefreedomofassessment.Thejointpriordistributionis:4p( ; ;F; )/p( )p( )p( )p(F);(2.4)wherep( )/j j 12e 12( 0)0 1( 0)(2.5)p( )/j j 12e 12( 0)0 1( 0);(2.6)p( )/j j 2e 12tr 1B; 2p;(2.7)p(F)/e 12trF0F(2.8)with ; ;B; 0andBadiagonalmatrix.Ageneralizednaturalconju-gatenormaldistributionisspeci edforthepopulationmeanwhere 0and aremeanandcovariancehyperparameterstobeassessed.Thevector =vec( 0)isspeci edtohavethegeneralizednaturalconjugatenormaldistributionwithmeanandcovariancehyperparameters 0=vec( 00)and .Thematrix followsanInvertedWishartdistribution,withhyper-parameters( ;B)whicharetobeassessed.It
本文标题:A Bayesian factor analysis model with generalized
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