Monte Carlo methods for posterior distributions associated with multivariate student's t data PDF
Preview Monte Carlo methods for posterior distributions associated with multivariate student's t data
MONTECARLOMETHODSFORPOSTERIORDISTRIBUTIONS ASSOCIATEDWITH MULTIVARIATESTUDENT’StDATA By DOBRINMARCHEV ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2004 Copyright2004 by DobrinMarchev Tothememoryofmygrandfather,Dimiter ACKNOWLEDGMENTS IwouldliketothankJimHobertforhissupportthroughoutthelast3years. Hisguidance,endlessquestionsandrigorouscriticismbroughtthenecessaryquality tomywork. Withouthisinsights,thisdissertationwouldhaveneverbeenthe same. IwouldalsoliketothankBrettPresnell,JimBooth,AlexTrindade,Murali RaoandMikeDanielsforservingonmycommitteeandspendingtheirvaluable timewithmydissertation. ToBrettPresnell,Iamparticularlythankfulfor stimulatingdiscussionsthroughoutmystudiesattheUniversityofFlorida. I learnedalotaboutprobability,mathematics,andLaTexfromhim. Aboveall,IthankPavlinaforherloveandforsharingwithmeallthe momentsofhappinessandgrief. Finally,Ithankmyparents,AngelandLiliana,whosupportedmeandpointed meintherightdirectionforthefirst25yearsofmylife. IV TABLEOFCONTENTS page ACKNOWLEDGMENTS iv LISTOFTABLES vii LISTOFFIGURES viii ABSTRACT ix CHAPTER 1 INTRODUCTION 1 1.1 PromtheNormaltotheStudent’stModel 1 1.2 Outline 4 1.3 HowtoSamplefromanIntractableDensity 4 2 MONTECARLOMETHODSFORTHESTUDENT’StMODEL.... 7 2.1 TheModel 7 2.1.1 Definition 7 2.1.2 ProprietyofthePosterior 8 2.2 MakingExactDrawsfromtheTargetPosterior 13 2.3 SimulationExamples 24 3 DATA-AUGMENTATIONALGORITHMS 26 3.1 Introduction 26 3.2 StandardDataAugmentation 27 3.3 MengandvanDyk’sExtensions 30 3.3.1 ConditionalAugmentation 31 3.3.2 MarginalAugmentation 32 3.3.3 MarginalAugmentationwithImproperPrior 34 3.4 ConnectionwithGroupTheory 39 3.5 UnderstandingtheArtworkofvanDykandMeng 47 3.6 OtherProblemswithvanDykandMeng’sLemma1 51 4 GEOMETRICERGODICITYOF FORTHEMULTIVARIATE STUDENT’StPROBLEM 55 4.1 Introduction 55 4.2 MarkovChainBackground 56 v 4.3 MultivariateStudent’stProblem 59 4.4 NumericalExample 66 4.5 ConcludingRemarks 70 REFERENCES 72 BIOGRAPHICALSKETCH 76 vi LISTOFTABLES Table page 2-1 Acceptanceratesoftherejectionsamplerwhend—2 25 2-2 Acceptanceratesoftherejectionsamplerwhend—3 25 vii LISTOFFIGURES Figure page 4-1 EmpiricalcomparisonofMarkovchainsandthetarget 69 viii AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulfillmentofthe RequirementsfortheDegreeofDoctorofPhilosophy MONTECARLOMETHODSFORPOSTERIORDISTRIBUTIONS ASSOCIATEDWITH MULTIVARIATESTUDENT’StDATA By DobrinMarchev August2004 Chair: JamesP.Hobert MajorDepartment: Statistics Let/denotetheposteriordensitythatresultswhenarandomsampleofsize nfromad-dimensionallocation-scaleStudent’stdistribution(withvdegrees offreedom)iscombinedwiththestandardnon-informativeprior. Weconsider severalMonteCarlomethodsforsamplingfromtheintractabledensity/,including rejectionsamplersandGibbssamplers. SpecialattentionispaidtotheMarkov chainMonteCarlo(MCMC)algorithmdevelopedbyvanDykandMengwho providedconsiderableempiricalevidencesuggestingthattheiralgorithmconverges tostationaritymuchfasterthanthestandarddata-augmentationGibbssampler. In additiontoitspracticalimportance,thisalgorithmisinterestingfromatheoretical standpointbecauseitisbasedonaMarkovchainthatisnotpositiverecurrent. WeformallyanalyzetherelevantmarginalchainunderlyingvanDykandMeng’s algorithm. Inparticular,weestablishdriftandminorizationconditionsshowing that,formany(d,u,n)triples,themarginalchainisgeometricallyergodic. Thisis thefirstgeneral,rigorousanalysisofanMCMCalgorithmbasedonanon-positive recurrentMarkovchain. Moreover,ourresultsareimportantfromapractical IX standpointsincegeometricergodicityguaranteestheexistenceofcentrallimit theoremsthatcanbeusedtocalculateMonteCarlostandarderrors;andbecause thedriftandminorizationconditionsthemselvesallowforthecalculationofexact upperboundsonthetotalvariationdistancetostationarity. Ourresultsare illustratedusingseveralnumericalexamples. x
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