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# Convergence Rates and Regeneration of the Block Gibbs Sampler for Bayesian Random Effects Models

## Material Information

Title: Convergence Rates and Regeneration of the Block Gibbs Sampler for Bayesian Random Effects Models
Physical Description: 1 online resource (77 p.)
Language: english
Creator: Tan, Aixin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

## Subjects

Subjects / Keywords: asymptotic, convergence, drift, ergodicity, geometric, minorization, variance
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: Markov chain Monte Carlo (MCMC) methods have received considerable attention as powerful computing tools in Bayesian statistical analysis. The idea is to produce Markov chain samples to estimate characteristics of complicated Bayesian posterior distributions. We consider the widely applicable Bayesian one-way random effects model. If the standard diffuse prior is used, there is a simple block Gibbs sampler that can be employed to explore the intractable posterior distribution, $\pi$. Indeed, the sampler produces a Markov chain output $\Phi$ that has invariant distribution $\pi$. Then the sample averages of $\Phi$ are used to estimate expectations with respect to $\pi$. Consider specifying a different prior, such as the reference prior, for the model. This results in a more complex posterior distribution, $\pi^*$. Constructing an MCMC algorithm for $\pi^*$ is much harder than that for $\pi$. So we resort to the importance sampling technique, which uses weighted averages of $\Phi$ to estimate expectations with respect to $\pi^*$. Basic Markov chain theory implies that both types of MCMC estimators mentioned above are valid in the following sense. They converge almost surely to their respective estimands as the Markov chain sample size grows to infinity. Nevertheless, it is always important to ask what is an appropriate sample size when running an MCMC algorithm. To answer this question for the above block Gibbs sampler, we develop a regenerative simulation method that yields simple, asymptotically valid Monte Carlo standard errors. These standard errors can be used to construct confidence intervals for the quantities of interest. Then one method to determine when to stop the Markov chain is to run the simulation until the width of the intervals are below user-specified values. The regenerative method rests on the assumption that the underlying Markov chain converges to its stationary distribution at a geometric rate. We use the drift method to show that, unless the data set is extremely small and unbalanced, the block Gibbs Markov chain is geometrically ergodic. We illustrate the use of the regenerative method with data from a styrene exposure study. In the final chapter of this dissertation, we discuss a slightly different topic. We study the convergence rate of Markov chains underlying a simple class of two-variable Gibbs samplers. These Markov chains live in a common state space that constitutes points on the diagonal and the subdiagonal of $\mathbb{N}\times\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers. It is shown that a geometric tail decay of the target distribution is almost necessary and sufficient for the corresponding chain to be geometrically ergodic. The argument involves indirect evaluation of the norm of Markov chain operators.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Aixin Tan.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024910:00001

## Material Information

Title: Convergence Rates and Regeneration of the Block Gibbs Sampler for Bayesian Random Effects Models
Physical Description: 1 online resource (77 p.)
Language: english
Creator: Tan, Aixin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

## Subjects

Subjects / Keywords: asymptotic, convergence, drift, ergodicity, geometric, minorization, variance
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: Markov chain Monte Carlo (MCMC) methods have received considerable attention as powerful computing tools in Bayesian statistical analysis. The idea is to produce Markov chain samples to estimate characteristics of complicated Bayesian posterior distributions. We consider the widely applicable Bayesian one-way random effects model. If the standard diffuse prior is used, there is a simple block Gibbs sampler that can be employed to explore the intractable posterior distribution, $\pi$. Indeed, the sampler produces a Markov chain output $\Phi$ that has invariant distribution $\pi$. Then the sample averages of $\Phi$ are used to estimate expectations with respect to $\pi$. Consider specifying a different prior, such as the reference prior, for the model. This results in a more complex posterior distribution, $\pi^*$. Constructing an MCMC algorithm for $\pi^*$ is much harder than that for $\pi$. So we resort to the importance sampling technique, which uses weighted averages of $\Phi$ to estimate expectations with respect to $\pi^*$. Basic Markov chain theory implies that both types of MCMC estimators mentioned above are valid in the following sense. They converge almost surely to their respective estimands as the Markov chain sample size grows to infinity. Nevertheless, it is always important to ask what is an appropriate sample size when running an MCMC algorithm. To answer this question for the above block Gibbs sampler, we develop a regenerative simulation method that yields simple, asymptotically valid Monte Carlo standard errors. These standard errors can be used to construct confidence intervals for the quantities of interest. Then one method to determine when to stop the Markov chain is to run the simulation until the width of the intervals are below user-specified values. The regenerative method rests on the assumption that the underlying Markov chain converges to its stationary distribution at a geometric rate. We use the drift method to show that, unless the data set is extremely small and unbalanced, the block Gibbs Markov chain is geometrically ergodic. We illustrate the use of the regenerative method with data from a styrene exposure study. In the final chapter of this dissertation, we discuss a slightly different topic. We study the convergence rate of Markov chains underlying a simple class of two-variable Gibbs samplers. These Markov chains live in a common state space that constitutes points on the diagonal and the subdiagonal of $\mathbb{N}\times\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers. It is shown that a geometric tail decay of the target distribution is almost necessary and sufficient for the corresponding chain to be geometrically ergodic. The argument involves indirect evaluation of the norm of Markov chain operators.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Aixin Tan.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024910:00001

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1BayesianInferenceandIntractableIntegrals ................. 12 1.2ClassicalMonteCarloMethods ........................ 16 1.3MCMCMethods ................................ 19 2ONMARKOVCHAINLIMITTHEORY ...................... 25 2.1BackgroundKnowledgeonGeneralStateSpaceMarkovChains ...... 25 2.2MCMCEstimatorsandTheirCentralLimitTheorems ........... 27 3BLOCKGIBBSSAMPLINGFORBAYESIANRANDOMEFFECTSMODELS 35 3.1TransitionFunctionoftheBlockGibbsSampler ............... 35 3.2GeometricErgodicityoftheBlockGibbsSampler .............. 38 3.3MinorizationfortheBlockGibbsSampler .................. 46 3.4AnExample:StyreneExposureData ..................... 49 4ONTHEGEOMETRICERGODICITYOFTWO-VARIABLEGIBSSAMPLERS 54 4.1Introduction ................................... 54 4.2ASpecialKindofDiscreteTwo-variableGibbssampler ........... 56 4.3ASucientConditionforXtobeGeometric ................ 57 4.4ASucientConditionforXtobeSubgeometric .............. 58 4.5AnAttempttoCharacterizeGeometricErgodicityofX 62 APPENDIX AProprietyofthePosteriorsandFiniteMomentConditions ............ 66 B-irreducibility .................................... 68 CBounding1and3below1 ............................. 69 C.1Bounding1(s) ................................. 69 C.2Bounding3(s) ................................. 71 DValidChoicesoftheDistinguishedPoint ...................... 72 6

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....................................... 74 BIOGRAPHICALSKETCH ................................ 77 7

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Table page 1-1UnbalanceddatacongurationsthatimplygeometricallyergodicblockGibbschains ......................................... 23 3-1Averagestyreneexposurelevelofworkers ..................... 49 3-2Summarystatisticsforthestyreneexposuredata ................. 49 3-3ResultsbasedonR=5;000regenerations ..................... 52 3-4ResultsbasedonR=40;000regenerations ..................... 52 8

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Figure page 3-1SRQplots ....................................... 52 3-2Traceplots ...................................... 53 9

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Inthenalchapterofthisdissertation,wediscussaslightlydierenttopic.WestudytheconvergencerateofMarkovchainsunderlyingasimpleclassoftwo-variableGibbssamplers.TheseMarkovchainsliveinacommonstatespacethatconstitutespointsonthediagonalandthesubdiagonalofNN,whereNisthesetofnaturalnumbers.Itisshownthatageometrictaildecayofthetargetdistributionisalmostnecessaryandsucientforthecorrespondingchaintobegeometricallyergodic.TheargumentinvolvesindirectevaluationofthenormofMarkovchainoperators. 11

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Liu 2001 ; RobertandCasella 2004 ).ErgodicaveragesoftheMarkovchaincanthenbeusedtoestimatetheposteriormeans.ThislatterapproachiscalledtheMarkovchainMonteCarlo(MCMC)method. BasicMarkovchaintheoryimpliesthat,undersimpleregularityconditions,MCMCestimatorsconvergealmostsurelytothequantitiesofinterestastheMarkovchainsamplesizegrowstoinnity.Nevertheless,anyestimatorthatweuseinpracticeisbasedonanitenumberofsamples.Hence,thereisalwaysaMonteCarloerrorassociatedwithit.Weneverknowtheexactvalueoftheerror,butwecanstudyitssamplingdistribution.Thisknowledgeprovidesusatooltomeasurethelevelofcondencewecanhaveintheestimator.WecanalsousethisinformationtodecideanappropriateMarkovchainsamplesizedependingonhowcondentweneedtobeofthisestimator.ThemaingoalofthisdissertationistoexplaintheaboveidearigorouslyandcarryitoutforthefollowingBayesianmodel. 12

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Hill ( 1965 )and TiaoandTan ( 1965 ).ABayesianversionofthemodelrequiresapriordistributionfor(;2;2e),callitp;2;2e.Actually,sincethevectorofrandomeects,=(1;:::;q),isunobserved,ittooisviewedasaparameterwitha\built-in"priordensitygivenby(j;2;2e)=qYi=1221 2expn1 22(i)2o: whereL(;;2;2e;y)=qYi=1mYj=122e1 2expn1 22e(yiji)2o 1{1 ). Weconsidertwotypesof(improper)priorsfor(;2;2e).Thersttypehasdensity whereaandbareprespeciedhyper-parameters.Weusea;btodenotetheposteriordensityunderpa;b.Themembersofthisareconditionallyconjugatepriors;thatis,foreachparameter,thepriorandthefullconditionaldensityhavethesameform.Forexample,theprioron2hasaninverse-gammaform,andthecorrespondingfullconditionaldensity,(2j2e;;),isaninverse-gammadensity.Aswewillseelater,conditionallyconjugate 13

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2;0isapopularchoiceforreasonswenowdescribe. Accordingto Gelman ( 2006 ),thechoiceofpriorfor(;2e)isnotcrucialsincethedataoftencontainagooddealofinformationabouttheseparameters.Ontheotherhand,thereistypicallyrelativelylittleinformationinthedataconcerning2,sothechoiceofpriorforthisparameterismoreimportantandsubtle.Acommonlyusedpriorfor2isaproperinversegammaprior,whichisaconditionallyconjugateprior.Whenlittleornopriorinformationconcerning2isavailable,the(shapeandscale)hyper-parametersofthispriorareoftensettoverysmallvaluesinanattempttobe\non-informative".However,inthelimit,asthescale-parameterapproaches0withtheshapeparametereitherapproaching0orxed,notonlydoesthepriorbecomeimproper,butthecorrespondingposterioralsobecomesimproper.Consequently,theposteriorisnotrobusttosmallchangesinthese(somewhatarbitrarilychosen)hyper-parameters.Thisproblemhasledseveralauthors,including Daniels ( 1999 )and Gelman ( 2006 ),torecommendthattheproperinversegammapriornotbeused.Incontrast,Gelman(2006)illustratesthattheimproperprior21 2workswellunlessqisverysmall(say,below5).Combiningthispriorwithauniformprioron;log(2e)leadstop1 2;0.vanDykandMeng(2001)callp1 2;0thestandarddiusepriorandwewillrefertoitassuchthroughoutthisdissertation. However,seriousBayesiansdisagreeaboutthesuitabilityofthestandarddiuseprior.Indeed, Bernardo ( 1996 )statesthat\...theuseof\standard"improperpowerpriorsonthevariancesisawelldocumentedcaseofcarelesspriorspecication..." Bernardo goesontorecommendtheso-calledreferencepriorof BergerandBernardo ( 1992 ),whichisthesecondpriorthatweconsider.Whenthedatasetisbalanced,i.e.,whenmi=mfori=1;:::;q,thereferencepriortakestheform 2;(1{3) 14

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Beforeweproceedwithanyoftheabovepriors,acriticalstepistocheckthattheircorrespondingposteriorsaregenuineprobabilitydistributions.Resultsin HobertandCasella ( 1996 )showthattheposteriora;bisproperifandonlyifallthreeofthefollowingconditionshold:a<0;a+q 2;anda+b>1M A thatq3isalsoanecessaryandsucientconditionfortheproprietyofr. Makinginferencethroughtheposteriordistributionoftenreducestocomputingexpectationswithrespecttotheposteriordensity.Unfortunately,evenforthesimplerpriorofthetwo,pa;b,theposteriordensityisintractable.Indeed,lettingR+=(0;1),theposteriorexpectationofg(;;2;2e)isgivenby whichisaratiooftwointractableintegrals.ResultsinSection 3.1 showthatitisactuallypossibletointegrateandoutoftheintegralinthedenominatorinclosedform,sothedenominatorisreallya2-dimensionalintractableintegral.However,thenumeratorisusuallyanintractableintegralofdimensionq+3. Intractableintegralsliketheonein( 1{4 )aretypicalinBayesianposterioranalysis.Indeed,posteriorexpectationsthatarisefromusefulstatisticalmodelsareoftenhighdimensionalintegralsthathaveunboundedintegrationranges.Thereforemethodslikeanalyticalapproximationornumericalintegrationarenotalwaysapplicable.Inthenexttwosections,wereviewtheclassicalMonteCarlotechniqueandtheMCMCtechnique 15

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Inpractice,weneedtochoosethesamplesize,N,andthisiswherethecentrallimittheorem(CLT)comesin.Indeed,ifg2L2(),thenthereisaCLTfor

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where Therepresentationin( 1{5 )suggestsestimatingwith^#N= Theprocedureofchoosingalargeenoughsamplesizefor^Nissimilartothatfor Eu2:

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RobertandCasella ( 2004 ,Lemma4.3).Moreover,asimpleconsistentestimatorof2isgivenby^2=1 1.1 .Therefore,toapproximatetheposteriorexpectationin( 1{4 ),weuseMCMCinsteadofclassicalMonteCarlo.TheMCMCmethodappliesthestrategyoutlinedinSection 1.2 withaMarkovchaininplaceoftheiidsample.Andunlikeiidsampling,therearealwaysrecipes,suchasMetropolis-HastingsalgorithmsandGibbssamplers,thatonecanfollowtoconstructaworkableMarkovchainsample.(Byworkable,wemeanthattheMarkovchainisHarrisergodicwithlimitingdistributiontheposteriordistribution.SeeChapter 2 fordetails.)Toexplorea;b,thereareatleasttwoGibbssamplersthatwecanemploy. ThesimpleGibbssamplercyclesthroughtheq+3componentsofthevector(1;:::;q;;2;2e)oneatatimeandsampleseachoneconditionalonthemostcurrentvaluesoftheotherq+2components.Thisalgorithmwasrstintroducedintheseminalpaperby GelfandandSmith ( 1990 )asanapplicationoftheGibbssamplertoaBayesianversionoftheone-waymodelwithproperconjugatepriors. Inthisdissertation,westudyablockGibbssamplerwhoseiterationshavejusttwosteps.Let2=2;2e,=(;)andsupposethatthestateofthechainattimenisXn=2n;n.Oneiterationofoursamplerentailsdrawing2n+1conditionalonn,andthendrawingn+1conditionalon2n+1.\Blocking"variablestogetherinthiswayanddoingmultivariateupdatesoftenleadstoimprovedconvergencepropertiesrelativetothesimple(univariate)versionoftheGibbssampler(see,e.g., Liuetal. 1994 ).Formally,the 19

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1{1 )showsthat,given,2and2eareindependentrandomvariableseachwithinverse-gammadistributions,andgiven2,ismultivariatenormal.(ThespecicformsofthesedistributionsaregiveninSection 3.1 ).HenceitiseasytoprogramtheblockGibbssampler. LetXn1n=0=n2n;no1n=0denotetheblockGibbsMarkovchainandconsiderapplyingthestrategyoutlinedintheprevioussectionwiththeMarkovchaininplaceoftheiidsample.First,theanaloguesof PN1n=0uXn(1{7) wherefunctionsvanduaredenedin( 1{6 ).IfX0issomexedpoint,asitwouldusuallybeinpractice,then MeynandTweedie 1993 ,Chapter17)impliesthattheyarestronglyconsistentestimators.Thus,atthispointinthecomparison,allwehavelostbyusingaMarkovchaininplaceoftheiidsampleisunbiasedness!Inotherwords,theiidsamplecanbereplacedwithaMarkovchainsample(whichismucheasiertoget)andthestrongconsistencyoftheclassicalMonteCarloestimatorstillobtains.Becauseofthis,manyviewMCMCasa\freelunch"relativetoclassicalMonteCarlo.Unfortunately,asweallknow,thereisnofreelunch.Indeed,theroutineuseoftheCLTforchoosinganappropriatesamplesizeintheclassicalMonteCarlocontextisfarfromroutinewhenusingMCMC.Therearetworeasonsforthis.Therstisthat,whentheiidsequenceisreplacedbyaMarkovchain,thesecondmomentconditionsarenolongerenoughtoguaranteethatCLTsexist.Moreover,thestandardmethodofestablishingthatthereisaCLTinvolvesprovingthattheunderlyingMarkovchainisgeometricallyergodic(see 20

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2 forthedenition),andthisoftenrequiresdiculttheoreticalanalysis(see,e.g., JonesandHobert 2001 ; JarnerandHansen 2000 ; MengersenandTweedie 1996 ; MeynandTweedie 1994 ; RobertsandRosenthal 1998 1999 ; RobertsandTweedie 1996 ; RoyandHobert 2007 ).Thesecondreasonisthat,evenwhenaCLTisknowntohold,ndingaconsistentestimatorofthetheasymptoticvarianceisachallengingproblembecausethisvariancehasafairlycomplexformandbecausethedependenceamongthevariablesintheMarkovchaincomplicatestheasymptoticanalysisofestimatorsbasedonthechain(see,e.g., Geyer ,1992; ChanandGeyer ,1994; Jonesetal. ,2006). Inthiswork,weovercometheproblemsdescribedabovefortheblockGibbssamplerthroughaconvergencerateanalysisandthedevelopmentofaregenerativesimulationmethod.Ingeneral,regenerationallowsonetobreakaMarkovchainupintoiidsegments(called\tours")sothatasymptoticanalysiscanproceedusingstandardiidtheory.Whilethetheoreticaldetailsarefairlyinvolved( Myklandetal. 1995 ; Hobertetal. 2002 ),theresultsofthetheoryand,moreimportantly,theapplicationoftheresults,turnsouttobequitestraightforward.Indeed,toapplyourmethod,onesimplyrunstheblockGibbschainasusual,butaftereachiteration,asingleBernoullivariableisdrawn.Specically,supposethatthevalueofthechainattimenisXn.WedrawXn+1asusual,andthendrawaBernoullivariable,callitn,whosesuccessprobabilityisasimplefunctionofXn,Xn+1andafewconstants(seeequation( 3{20 )).Eachtimethatn=1markstheendofatour.(Notethatthetourshaverandomlengths.)NowtoestimateEg,considerrunningtheblockGibbssamplerforRtours;thatis,werunthechainuntiltheRthtimethatan=1.Fort=1;:::;R,letNtbethetthtourlengthandletSt=Pg(Xn)wherethesumrangesoverthevaluesofnthatconstitutethetthtour.ThetotalnumberofiterationsisN=PRt=1Nt.TheobviousestimateofEgbasedonthissimulationis 21

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Furthermore,asimple,consistentestimatorof2isgivenby^2=RPRt=1StegRNt2 1{7 ):geometricergodicityoftheblockGibbschainplusapairofmomentconditionsonthefunctionsuandvwillimplyaCLTfor^N.Insummary,regenerativesimulationoftheMarkovchainallowsustomimicwhatisdoneintheclassicalMonteCarlocontext. Finally,ourconvergencerateanalysisyieldsconditionsunderwhichtheblockGibbschainisgeometricallyergodic.Looselyspeaking,weareabletosaythatthechainisgeometricallyergodicunlessthetotalsamplesizeMisverysmallandthewithingroupsamplesizes,m1;:::;mq,arehighlyunbalanced.Ourconvergencerateresult(Proposition 5 )appliestoallpriorspa;bdenedin( 1{2 ).Hereisaspecialcase. 2;0,theblockGibbsMarkovchainisgeometricallyergodicif

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1-1 displaysallunbalancedcongurationsof(m1;m2;m3)withm12thatsatisfytheconditionsofCorollary 1 Table1-1. Unbalancedcongurations(m1;m2;m3)withm12thatsatisfycondition2ofCorollary 1 7810 71011 6912 81112 455 7910 71111 61012 81212 566 71010 8811 61112 9912 577 8810 8911 61212 91012 667 8910 81011 7712 91112 677 81010 81111 7812 91212 688 9910 9911 7912 101012 778 91010 91011 71012 101112 788 6911 91111 71112 101212 799 61011 101011 71212 111112 889 61111 101111 8812 111212 899 7811 51112 8912 61010 7911 51212 81012 HobertandGeyer ( 1998 )and JonesandHobert (2001,2004).However,ineachofthesestudies,themodelsthatwereconsideredhaveproperpriorsonallparameters.Infact,ourProposition 5 istherstofitskindforrandomeectsmodelswithimproperpriors,which,asweexplainedabove,arethetypeofpriorsrecommendedintheBayesianliterature.ItturnsoutthatusingimproperpriorscomplicatestheanalysisthatisrequiredtostudythecorrespondingMarkovchain.Indeed,Proposition 5 ismuchmorethanastraightforwardextensionoftheexistingresultsforproperpriors. TheBayesianone-wayrandomeectsmodelthatwestudyisasimplestcaseoftheBayesianmixedeectsmodel( vanDykandMeng 2001 ,Sec.8).BlockGibbssamplerscanbedevelopedtoanalyzetheposteriorofthesemodels.However,uniedanalysisfortheconvergencerateoftheassociatedMarkovchainsaredicult.Onecasethathasreceived 23

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JohnsonandJones ( 2008 )derivedasucientconditionforthegeometricallyergodicityoftheblockGibbschainandconstructedaregenerativesimluationschemeforit.Afterall,therearemanyproblemsstilltoberesolvedconcerningcompleteanalysisofBayesianmixedeectsmodelsusingMCMCalgorithms. Anotherrelatedpaperis PapaspiliopoulosandRoberts ( 2008 )whostudiedtheconvergenceratesofGibbssamplersforBayesianhierarchicallinearmodelswithdierentsymmetricerrordistributions.Whatseparatesourresultsfromtheirsisthatthevariancecomponentsinourmodelareconsideredunknownparameters,whileintheirmodelthevariancecomponentsareassumedknown. Therestofthedissertationisorganizedasfollows.InChapter 2 ,wereviewsomeresultsfromgeneralstatespaceMarkovchaintheory.Inparticular,wementionthatgeometricergodicityplaysanimportantroletoensuretheexistenceofMarkovchainCLTs.WealsoprovideanalternativederivationofanexistingCLTbasedonregenerativesimulation.InChapter 3 ,wecarefullystudytheblockGibbssamplerandprovideasucientconditionforitsgeometricconvergence.Further,weidentifyregenerationtimesoftheblockGibbschainbyestablishingaminorizationconditionforitstransitiondensity.TheregenerativemethodisillustratedusingarealdatasetonstyreneexposurewherevalidestimationforthestandarderrorofMCMCestimatorsareprovided.AslightlydierenttopicisdiscussedinChapter 4 .WedescribeanattempttocharacterizegeometricconvergenceofMarkovchainsunderlyingaspecicclassoftwo-variableGibbssamplers.TheseMarkovchainsliveonacommonstatespacethatconstitutespointsonthediagonalandthesubdiagonalofNN,whereNisthesetofnaturalnumbers.Ourresult(Corollary 3 )showsthatageometrictaildecayofthetargetdistributionisalmostnecessaryandsucientfortheassociatedchaintobegeometricallyergodic. 24

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MeynandTweedie ( 1993 )fordenitions.Hererepresentsthemaximalirreducibilitymeasureofthechain,seeAppendix B forthedenition.OnemethodofestablishingHarrisrecurrenceistoshowthateveryboundedharmonicfunctionisconstant( Nummelin 1984 ,Theorem3.8).Here,afunctionh:X!RiscalledharmonicforKifh=Kh,whereKh(x):=RXh(x0)K(x;dx0)forallx2X.Generallyspeaking,Harrisergodicityiseasytoverifyinpractice.Forexample,itsucesthathasastrictlypositivetransitiondensity. Proof. 25

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1 belowstatesoneversionofthedriftcondition.ItisacombinationofLemma15.2.8andTheorem6.0.1from MeynandTweedie ( 1993 ). TheMarkovchainwithtransitionfunctionKiscalledaFellerchainifK(;O)islowersemicontinuousforanyopensetO2(X).Afunctionw:X![0;1)issaidtobeunboundedocompactsetsifthelevelsetfx2X:w(x)dgiscompactforeveryd>0. E[w(X1)jX0=x]w(x)+L;(2{1) 2{1 )iscalledadriftconditionandthefunctionwiscalledadriftfunc-tion. TheconceptsofHarrisergodicityandgeometricergodicityconcernthedistributionofaMarkovchainatstepn.HencetheyarenotdirectlyrelevanttoassessingasingleMarkovchainoutput.However,weshowinthenextsectionthatthesepropertiesaredesirableforMarkovchainsusedinMCMCalgorithomsbecausetheyindicategoodasymptoticbehavioroftheestimatorsdenedin( 1{7 ). 1 thatweconstruct MeynandTweedie 1993 ,p.411).Thetheoremsaysthat,ifg2L1(),thenwithprobability1, 27

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1{6 )belongstoL1(),then^N= uNisstronglyconsistentfor=Ev=Eu.However,theseassumptionsarenotenoughtoguaranteeaCLTfortheseestimators. Here,werefertoTheorem2of ChanandGeyer ( 1994 )forawellknownresultthatrelatesconvergencerateofMarkovchainstotheexistenceofaCLTfor IbragimovandLinnik ( 1971 ), 1 doesnotprovideenoughguidanceforacompleteMCMCanalysisinpractice.Thecomplicatedexpressionfor&2doesnotimmediatelysuggestaconsistentestimator.Thisisnotsurprisinggiventhedependenceinthesamplesaswellastheinuenceofthestartingdistributionofthechain. Tocircumventthisdiculty,weresorttoanalternativeCLTbasedonregenerativesimulation.TheideaistoidentifyregenerationtimesoftheMarkovchainsothatwecanpartitionthedependentsamplesintoiidsegments.Thisallowsustouseiidtheorytoanalyzetheasymptoticbehaviorofestimatorsbasedonthechain. Myklandetal. ( 1995 )and Hobertetal. ( 2002 )derivedCLTsfor Bhattacharya ( 2008 )generalizedtheproofsintheabovepaperstogetaCLTfor^N.Inwhatfollows,weprovideanalternativederivationfortheCLTof Bhattacharya ( 2008 ),whichisourProposition 2 .Ourproofseemstobemoretransparent.Afterthat,weshowthattheCLTof Hobertetal. ( 2002 )canbeviewedasacorollarytoProposition 2 28

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1 ,CLTsbasedonregenerativesimulationclearlyrequireanextraassumption,thatis,wecanindeedidentifyregenerationtimesoftheunderlyingMarkovchain.Thisisanon-issuefordiscretestatespaceMarkovchains,asthechainstartsovereverytimeitreturnstoaxedstate.See,forexample, Asmussen ( 2003 ).ForgeneralstatespaceMarkovchains,theprerequisiteissatisedifonecanestablishthefollowingminorizationconditionfortheMarkovtransitionfunction: wheres:X![0;1]satisesEs>0andisaprobabilitymeasureonX.ThebenetisthatKcanberewrittenasamixtureK(x;dy)=s(x)(dy)+(1s(x))r(x;dy) wherer(x;dy)isdenedtobeK(x;dy)s(x)(dy)1s(x)forx2fx:s(x)<1gandanyarbitraryprobabilitydensityforx2fx:s(x)=1g.ThismixtureprovidesanalternativemethodofsimulatingtheMarkovchain.Giventhecurrentstate,Xn=x,drawingXn+1fromK(x;)canbedonethroughippingacoin,nBer(s(x)),thendrawingXn+1()ifn=1andXn+1r(x;)ifn=0.Thepointis,everytimethatn=1,wehaveXn+1()notdependentofthecurrentvalueXn=x,whichineectstartstheprocessoveratits(n+1)thiteration. SupposeisstartedwithX0().Denoteitsregenerationtimesby0=0<1<2<:::;thatis,t+1=minfn>t;n1=1gfort0.LetthechainrunforRtourswhereRisxed.Thenthetotallengthofthechain(countingX0)isarandomnumber, 29

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WenowstudytheasymptoticbehaviorofeRasRgetslarge. First,byKac'stheorem( MeynandTweedie 1993 ,Thm10.2.2),RR!Eswithprobability1asR!1.Inconjunctionwiththeergodictheorem,thisyields1 30

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where2=E(V1U1)2(EU1)2: ^2=1 Toseewhy^2isindeedaconsistentestimator,notethat,asRgetslarge,~2:=1 2{5 ),weassumedthatEV21andEU21arebothnite.Thesemomentconditionsareactuallyquitediculttocheckdirectly.Indeed,V1andU1aresumsoffunctionsofthestatesoftheMarkovchaincontainingarandomnumberofterms.However, Hobertetal. ( 2002 )showthat,iftheunderlyingMarkovchainisgeometricallyergodic,andthereexistsan">0suchthatEjvj2+"andEjuj2+"arebothnite,thenEV21andEU21arebothniteaswell.Now,wesummarizetheaboveresultsinProposition 2 .ItisthesameastheCLTin Bhattacharya ( 2008 ). 31

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2{2 ).Considerestimatingtheratio=Eg=EvEuwiththeestimatoreRdenedat( 2{4 ).Ifthereexistsan">0suchthatEjvj2+"andEjuj2+"arenite,thenforX0(),R!1withprobability1asn!1andp 2{6 )isastronglyconsistentestimatorof2. supx2Sf(x) Thisconditionbasicallysaysthathasheaviertailsthan.Infact,( 2{7 )isexactlywhatisrequiredtouseasthecandidatedensityinanaccept/rejectalgorithmfor.Ofcourse,thisaccept/rejectalgorithmisnotviableifwecannotmakeexactdrawsfrom.Forathoroughreviewofaccept/rejectmethods,see RobertandCasella ( 2004 ,Chapter2). SupposeweusetheaboveimportancesamplingtechniquetoevaluatetheeectofachangeinthepriordistributionorthelikelihoodfunctioninaBayesiananalysis.Supposethatxandydenoteparametersandobserveddata,respectively.Thinkoftwodierentposteriordensities,and,with(x)/L(x;y)p(x)and(x)/L(x;y)p(x),whereL(x;y)andL(x;y)denotetwodierentlikelihoodfunctions,andp(x)andp(x)denotetwodierentpriordensities.Inthissetting, IfthetwoBayesianmodelsdieronlyinthepriorthatisused,i.e.ifL=L,thenf(x)=f(x)=p(x)=p(x),whichissimplytheratioofthetwopriors.Hence,themomentconditionsinProposition 2 reducetoE(p=p)2+"<1andEjgp=pj2+"<1forsome">0. 32

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2 isadirectgeneralizationoftheresultin Hobertetal. ( 2002 ),whichwestateinCorollary 2 .Thisresultmaybeviewedaspertainingtothespecialcasewhereu1andv=g.Inthiscase,Ut=P1=:NtandVt=Pg(Xn)=:St,wherethesumrangesoverthevaluesofnthatconstitutethetthtour.Clearly,UtisthetthtourlengthandthemomentconditionconcerninguissatisedsinceEjuj2+"=1<1. 2{2 )andjgj2+"2L1()forsomepositive",thenforX0(),R!1withprobability1asn!1andp [EN1]2: Myklandetal. ( 1995 ,Section2),smallvaluesofthecoecientofvariation,CV( N)2.Preferably,cCV( OnenicefeatureofProposition 2 (andCorollary 2 )isthattheconditionsfortheexistenceofaCLTareseparatedintooneconcerningtheconvergencerateoftheMarkovchainandtwoothersthataresimplemomentconditionswithrespecttothetargetdistribution.Hence,ifthechainunderconsiderationisknowntobegeometricallyergodic,thencheckingtheconditionsisquitestraightforward.Incontrast,manyresultsforCLTsforregenerativeprocesseshavesucientconditionsthatarefairlydiculttocheckinpracticebecausetheyinvolveexpectationsofcomplexfunctionsoftheunderlyingprocess. 33

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Myklandetal. ( 1995 )'sCLTrequiresEN21<1andES21<1.OtherexamplesofthiscanbefoundintheoperationsresearchliteraturewhereregenerativesimulationisusedtoassessthevariabilityofMCMCestimatorsintheanalysisofqueueingsystems(see,e.g., Ripley 1987 ; Bratleyetal. 1987 ; CraneandIglehart 1975 ; LavenbergandSlutz 1975 ; GlynnandIglehart 1987 ).TheMarkovprocessesthatunderlytheseanalyseshavecountablestatespaces,whichmakestheidenticationofregenerationtimestrivial.However,unlikeProposition 2 ,theconditionsforCLTsinvolveunwieldymomentconditionswithrespecttotheunderlyingMarkovprocess.Theseconditionsarequitediculttocheckforallbutthesimplestqueueingsystems. 34

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1 .Recallthatundertheconditionalconjugatepriorpa;b;2;2e=2(a+1)2e(b+1); 2expn1 22e(yiji)2o#"qYi=1221 2expn1 22(i)2o#2(a+1)2e(b+1):(3{1) Sincetheobserveddataarealwaysconditionedupon,wehavesuppressedthenotationofdependenceonyintheposteriordensity.Ofcourse,wheneveranimproperpriorisused,onemustcheckthattheresultingposteriorisproper.Aswehavementionedintheintroduction,resultsin HobertandCasella ( 1996 )showthattheposteriorisproperifandonlyifallthreeofthefollowingconditionshold: 2;anda+b>1M whereMisthetotalsamplesize;thatis,M=Pqi=1mi.Notethat( 3{2 )impliesq>12a>1soanecessaryconditionforproprietyisq2.Underthestandarddiuseprior,thatis,when(a;b)=(1 2;0),theposteriorisproperifandonlyifq3. ThefollowingblockGibbssamplerprovidesanecientwaytosimulateaMarkovchainwithinvariantdistribution.Let2=2;2e,=(;)andsupposethatthestateofthechainattimenis2n;n.Oneiterationofoursamplerentailsdrawing2n+1conditionalonn,andthendrawingn+1conditionalon2n+1.Formally,theMarkov 35

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Straightforwardmanipulationof( 3{1 )showsthat,2and2eareconditionallyindependentgiven;thatis,(2j)=(2j)(2ej); 2Xi(i)2)and2ejIGM 2Xi;j(yiji)2: whereV1=264D2211211Tq21375 21+mi2e1.Themean,0,isthesolutionof where 36

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Thisallowsustoobtainthefollowingclosedformexpressions: Vari2;2e=2e Wecanalsouse( 3{5 )towritedowntheclosedformof0,thatis,towritedownE(j;e)andE(kj;e)fork=1;:::;q.WehaveE(j;e)=qXi=1mie

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Since(j2)and(2j)arebothstrictlypositivefor(2;)2R2+Rq+1,itfollowsthattheMarkovtransitiondensitydenedin( 3{3 )isstrictlypositive.Thus,Lemma 1 inSection 2.1 impliesthattheblockGibbsMarkovchain,fXng1n=0=2n;n1n=0,isHarrisergodic.Hence,theMCMCestimatorsdenedin( 1{7 )arestronglyconsistentundersimplerstmomentconditions.Inthefollowingsection,weshowthatthechainisgeometricallyergodic.ThiswillensuretheexistenceofCLTsfortheMCMCestimators. Liuetal. 1994 ).Moreover,theGibbschainanditstwomarginalchainsallconvergeatexactlythesamerate( Diaconisetal. 2008 ; DieboltandRobert 1994 ; RobertsandRosenthal 2001 ).Intermsofourexampleabove,thismeansthatn1n=0isaMarkovchainthatconvergesatthesamerateas(2n;n)1n=0.Therefore,wecanprovethattheblockGibbschainisgeometricbyprovingthatthe-chainisgeometric.The-chainhasaMarkovtransitiondensity(withrespecttoLebesguemeasureonRq+1)givenby 38

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1 showsthatthe-chainisHarrisergodic.WealsoconcludefromLemma 1 thatthemaximalirreducibilitymeasureofthe-chainisequivalenttoLebesguemeasureonRq+1andhenceitssupporthasnon-emptyinterior.Finally,itissimpletoshowthatthe-chainisFellerusingFatou'sLemma. Proof. WearenowinapositiontouseProposition 1 toprovethatthe-chainisgeometric.Therealchallengeisconstructingavaliddriftcondition.Recallthatafunctionw:Rq+1!R+issaidtobeunboundedocompactsetsifthelevelsetf:w()giscompactforevery<1.AccordingtoProposition 1 ,wecanprovethatthe-chainisgeometricbyndingawthatisunboundedocompactsetsandsatisesthedriftcondition Ew(~)jw()+Lforall2Rq+1,(3{8) where<1andL<1.Ourdriftfunctiontakestheformw()=w1()s+w2()s;

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Tokeepthenotationundercontrol,weusewand~wtodenotew()andw(~),respectively.Theleft-handsideof( 3{8 )isEw(~)j=E(~wj)=E~ws1j+E~ws2j: 3{7 showsthatwecangetthenextstate,~,byrstdrawing2(j),andthendrawing~(j2),sographicallywehave!2!~.Thisallowsustocalculatetheexpectationsabovebyconditioningon2.Indeed,fork2f1;2g,wehave E(~wskj)=EE(~wskj2;)j=EE(~wskj2)j(3{9) wherethesecondequalityfollowsfromthefactthat~isconditionallyindependentofgiven2. SincetherearenorestrictionsontheconstantLin( 3{8 ),wedonothavetokeeptrackofanyconstantswhencalculatingE(~wj).Hence,wewillusethenotation\const"torefertoagenericconstant.ThefollowingLemmaprovidesupperandlowerboundsont,whichwillcomeinhandy. 1. max2;2e;(3{10) 40

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3{10 )istruebecauset=qXi=1mi max2;2e: 3{11 ),therstinequalityholdsbecause1 2andt=qXi=1mi 2e: 3 alongwiththeexpressionsin( 3{6 ),wehaveE(~w1j2)=EXi(~i~)22;2e=XihVar(~i~)2;2e+E(~i~)2;2e2i=Xi22e t+constXi22e t+constXi2e t+const

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3{11 ).Wenowboundq=tintwodierentways.Ononehand,byequation( 3{10 ),wegetq tqqXi=1mi 3{11 ),wegetq tqm2 M2e+const:

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M: 3{9 ),notethatfors2(0;1]andanyA;B>0,itfollowseasilythat(A+B)sAs+Bs.TogetherwithJensen'sinequality,thisyields E(~ws1j2)E(~w1j2)s12+22e+constss12s+s22es+const;(3{12) and E(~ws2j2)E(~w2j2)s(q+1)2e+consts(q+1)s2es+const:(3{13) Nowtocompletethecalculationin( 3{9 ),recallthat2jIGq 2; E2sj=(q 2s(q and E2esj=(M 2s(M 2s(M Dene1(s)=(q+1)s(M 2s(M 2s(M 2s(q

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3{9 )and( 3{12 ){( 3{15 ),wehave E(~ws1j)Es12s+s22es+consts1(q 2s(q 2s(M and E(~ws2j)E(q+1)s2es+const(q+1)s(M 2s(M Thefollowingresultexplainshowtheinequalities( 3{16 )and( 3{17 )canbeusedtogethertoformavaliddriftcondition. Proof. 3{16 )and( 3{17 )thatE(~ws1+~ws2j)3(s)ws1+(1(s)+2(s))ws2+const=(;s)(ws1+ws2)+(3(s)(;s))ws1+const Clearly,( 3{18 )requiresthat1(s)<1and3(s)<1.Wenowshowthattheseconditionsarealsosucientfortheexistenceof>0suchthat( 3{18 )issatised. Therearetwocases.Intherstcase,1(s)3(s)<1.Ifwetake=(3(s)1(s))=2(s),then(;s)=3(s)<1and3(s)(;s)=0.Inthesecondcase, 44

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2=1+1(s) 2<1; 2<0: InconjunctionwithProposition 1 ,Proposition 3 showsthatthe-chain(andhencetheblockGibbsMarkovchain)isgeometricallyergodicaslongasthereexistsans2Ssuchthatboth1(s)and3(s)arelessthan1.Thefollowingresultshowswhenthiscanhappen.SeeAppendix C fortheproof. dxlog(x)denotethedigammafunction.IfM+2bq+3and1<2expq 1<2expq 2. 5 showsthatgeometricergodicityholdsunlessthedatasetissmallandunbalancedatthesametime.Indeed,considertherstcondition.Theleft-handsidewillbelarge(andtheconditionwillfail)onlyifbothPqi=1mi Finally,weshowthatCorollary 1 ofChapter 1 ,whichdealswiththespecialcaseof(a;b)=(1 2;0),followsstraightforwardlyfromProposition 5 .Itfollowsfrom( 3{2 )that, 45

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2,theposteriorisimproperifq2.Hence,weonlycareaboutq3.Whenq4,wehave2expq 2=2exp(2(log21)):=2:074: 2,wehave1q 2=2<2expq 2andq4,thecondition1<2expq Now,whenq=3,wehave2expq 2andq=3andthedataarebalanced,thecondition1<2expq 2exp():=2:67orm<2exp() 3M:=0:374M:(3{19) Table 1-1 inChapter 1 providesalistofallunbalanceddatacongurationswithq=3andm12thatsatisfy( 3{19 ). 2 thatregenerativesimulationofaMarkovchainallowustocomputeasymptoticallyvalidstandarderrorsfortheMCMCestimators.Inthissection,weconstructaminorizationconditionthatdrivestheregenerativesimulationofourblockGibbssampler.Theideathatweusetondthisminorizationconditionisoutlinedin Myklandetal. ( 1995 ).RecallthatthetransitiondensityoftheblockGibbschainisgiven 46

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where 2{3 ),whichmustbecalculatedaftereachiterationoftheMarkovchain,involvessandonlythroughtheirproduct.Thus,ccancelsout. Wenowdevelopaclosedformexpressionfors.Since(2j)factorsinto(2j)and(2ej),thebivariateminimizationproblembecomestwoseparateunivariateminimizationproblems.Letwkstandforwkevaluatedatfork=1;2.Then(2j) 2w1)q (1 2w1)q 2w1=2] 2w1=2]=w1 2(w1w1)=2;

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2(w2+SSE)M 2(w2+SSE)M 2(w2+SSE)=2e 2e(M 2(w2+SSE)=2e=w2+SSE 2(w2w2)=2ei: Prn=1(2n;n)=(2;);(2n+1;n+1)=(~2;~)=s()(~2;~) 2(w1w1)= 2(w2w2)= 2(w1w1)=~2iw2+SSE 2(w2w2)=~2eiID(~2)=exp1 2(w1w1)1 ~21 ~2e1 Theoretically,wecoulduseanysetD=[d1;d2][d3;d4]andanydistinguishedpointtoruntheregenerativesimulation.However,theasymptoticsfor^2involveR!1,sowewouldlikeforthechaintoregeneratefairlyoften.Thus,weshouldchooseDandsothattheprobabilityin( 3{20 )isfrequentlyclosetoone.Notsurprisingly,thereistrade-obetweenthesizeofthesetDandthemagnitudeoftheexponentialtermin( 3{20 )(whentheindicatorisunity).OurstrategyforchoosingDandisasfollows.WeruntheblockGibbssamplerforaninitialn0iterations(usingstartingvalue0=( yq; y)forexample).Wetake[d1;d2]tobetheshortestintervalthatcontains60%ofthen0valuesof2,andwecalculate[d3;d4]similarlyusingthen0valuesof2e.Theregenerationprobability( 3{20 )involvesonlythroughw1()andw2().Hence,insteadofsettingequaltothemedian,say,ofthen0valuesofintheinitialrunofthechain,wecalculatew1andw2foreachofthen0valuesofandwesetw1tobethemedianofthew1values 48

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D foraproofofthisresultaswellasguidelinesfortheunbalancedcase. Lylesetal. ( 1997 )usingtheBayesianrandomeectsmodels.Thirteenworkerswererandomlyselectedfromagroupwithinaboatmanufacturingplantandeachone'sstyreneexposurewasmeasuredonthreeseparateoccasions.Sowehaveq=13,mim=3andM=mq=39.ThedataaresummarizedinTables 3-1 and 3-2 Table3-1. Averagestyreneexposurelevelforeachofthe13workers. worker 1234567 worker 8910111213 Table3-2. Summarystatisticsforthestyreneexposuredata SST=3P13i=1( SSE=P13i=1P3j=1(yij Wewouldliketoinvestigatetheeectofusingtwodierentpriorsinthemodel.Theyarethestandarddiuseprior,p=p1 2;0,fromequation( 1{2 ),andthereferenceprior,p=pr,denedin( 1{3 ).Denotetheirassociatedposteriordistributionsby(;;2;2e)and(;;2;2e),respectively.Weknowthatbothposteriorsarewell-dened(proper)sincethedatasetisbalancedandq=133.Wewillfocusontheposteriorexpectationsofthreefunctions:g1(;2;2e)=2,g2(;2;2e)=2e,andg3(;2;2e)=2=(2+2e).Here,g3isthecorrelationbetweenobservationsonthesameworker.Allthreefunctions 49

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6 fromAppendix A impliesthatEjg1j3<1andEjg2j3<1.Therefore,alloftheassumptionsunderlyingtheregenerativesimulationoftheBlockGibbschainaresatised.ThissimulationcanbeusedtoproduceestimatesandstandarderrorsforEgi(;2;2e),i=1;2;3. NextconsiderreusingtheblockGibbsoutputtoapproximateEgi(;2;2e)fori=1;2;3.AccordingtothediscussionfollowingProposition 2 ,ourimportancesamplingresultsareapplicableifwecanndan">0suchthatEjgip=pj2+"<1fori=0;1;2;3,whereg01.First,notethatjgip=pj=gip=psinceallthetermsarepositive.Nowp 22e1"2+2e 2 22e1=21C3 2"2+2e 2p 2; 6 showsthatEjgip=pj3<1fori=0;1;2;3.Hence,ourimportancesamplingtechniqueisapplicable. ImplementationoftheregenerativesimulationrequiresustospecifyR,thetotalnumberofregenerations;thatis,thenumberofiidtoursinthechain.TheproceduretodetermineRhastwosteps.Intherststep,werunthechainforapreliminaryRregenerationsthatisbelievedtoleadtoareasonableestimatoroftheasymptoticvariance,2.Here,weusedR=5;000whichtook87,169iterationsandconsumed20seconds(codedinR).ThesimulationresultsaresummarizedinTable 3-3 .Foreachofthethreefunctionsofinterest,thetableprovidestheestimator,~gR(ortheimportancesamplingestimator,~R),theestimatedasymptoticvariance,^2,theestimatedstandarderrorp 3 fromChapter 2 ,itisnotrecommendedtouse^2toestimate2unlesstheaveragetourlength, 50

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Inthesecondstepoftheprocedure,wedecidehowlargeRneedstobefortheresultingCItobeshorterthanauser-speciedwidthbasedonthepreliminaryanalysisabove.Takethe95%CIofE2forexample.Supposethatwedesireitsmarginoferrortobearound1%ofthemagnitudeofE2.Since~g5000=0:19003,thedesiredwidthofthe95%CI,l,isapproximately20:190031%:=0:0038andwillrequireabout16^2l2=160:03074=0:00382:=38;371regenerations.Totakeintoaccountthepossibilitythattheasymptoticvariancecanbeslightlyunderestimatedbythe^2obtainedinthepreliminaryanalysis,werunthechainalittlelongerthanwhatthecalculationsuggests.Here,wedecidetoproduceachainwithR=40;000regenerations.Actually,only35,000moreiidtoursneedtobesimulated,andthencombinedwiththe5,000iidtourswealreadyhave.Thenalchainwith40,000regenerationsaccountedfor697,869iterationsandtook3minutestogenerate. ThesimulationresultsunderbothpriorsaresummarizedinTable 3-4 andacrudediagnosticwasprovidedusingtheSRQplotinFigure 3-1 .Further,thetraceplotsoftheestimatedasymptotic95%CIsforE2and^2areprovidedinFigure 3-2 .Theseplotssuggestthatthingshavestabilizedquitenicelybythe40,000thregeneration.Finally,weseefromTable 3-4 thattheestimatedposteriormeansunderthetwopriorsareveryclose,sotheissuesregardingchoiceofthepriorraisedby Bernardo ( 1996 )arenotaconcernforthisdataset. 51

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SRQplotsoft=R(verticalaxes)againstt=R(horizontalaxes)oftheblockGibbssamplerforthestyreneexposuredatabasedonrunsoflength5,000and40,000;theassociateddCV( Table3-3. Inthestyreneexposuredataanalysis,therearethreequantitiesofinterest,2,2e,and2=(2+2e),andtwodierentpriors.Foreachcombinationofthese,thetableprovidesestimatesoftheposteriorexpectationandthecorrespondingasymptoticvariance,aswellasthestandarderroranda95%asymptoticCI.TheseresultsarebasedonR=5;000regenerations. Prior Estimate ^2 2 0.19003 0.03463 0.00263 (0.18477,0.19529) Reference 0.18688 0.25732 0.00717 (0.17253,0.20123) 0.61777 0.00883 0.00133 (0.61511,0.62043) Reference 0.61964 0.06613 0.00364 (0.61237,0.62691) 0.21288 0.03532 0.00266 (0.20757,0.21820) Reference 0.21005 0.26278 0.00725 (0.19555,0.22455) Table3-4. ResultsbasedonR=40;000regenerations Prior Estimate ^2 2 0.19023 0.03523 0.00094 (0.18835,0.19210) Reference 0.18720 0.03252 0.00090 (0.18540,0.18900) 0.61849 0.00966 0.00049 (0.61751,0.61947) Reference 0.62031 0.00897 0.00047 (0.61936,0.62126) 0.21304 0.03687 0.00096 (0.21112,0.21496) Reference 0.21033 0.03404 0.00092 (0.20849,0.21218) 52

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ThetopplotshowstheevolutionoftheestimatorofE2andthecorresponding95%asymptoticCIasthenumberofregenerationsgrows.ThesolidlinerepresentstheestimatorandthedashedlinesdenotetheupperandlowerendpointsoftheCI.ThemiddleplotdoesthesameforE2.Thebottomplotdisplaystheevolutionoftheestimateoftheasymptoticvariance,^2,oftheestimatorsofE2(solidline)andE2(dashedline).SincetraceplotsoftheestimatorsunderthetwodierentpriorsarecalculatedusingthesamerunofaMarkovchainwiththesameregenerationtimes,itshouldnotbesurprisingthattheytrackeachotherclosely. 53

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byevaluatingquantitieslikeEXg=RXg(x)X(dx),wheregissomefunctionofinterest.AswementionedinChapter 1 ,itisusuallyimpossibletoperformtheintegrationdirectly.Instead,wecanapproximatetheintegralsusingMonteCarlomethods.ClassicalMonteCarlomethodsarenotapplicablewhenitishardtosimulatefromorXdirectly.Insuchcases,weresorttotheMCMCmethods.Below,wedescribeaclassofusefulMCMCalgorithmsthetransitiondensitiesofwhichhaveacommonstructure. LetY(y)=RX(x;y)X(dx)denotetheY-marginaldensityof.Further,letXjY(xjy)=(x;y)=Y(y)andYjX(yjx)=(x;y)=X(x)denotetheconditionaldensities.ThenthefollowingMarkovtransitiondensitydenesaMarkovchainthathasinvariantdistribution: Itispossibletosimulatethechainifwecansamplefromtheconditionals,XjYandYjX.Supposethatthestateofthechainattimenis(Xn;Yn)=(x;y).Thenthenextstate,(Xn+1;Yn+1),isdrawnasfollows.First,simulateXn+1XjY(jy)andcalltheresultx0;secondly,simulateYn+1YjX(jx0).SincesimulatingeachiterationoftheMarkovchainentailstwosteps,werefertothisMCMCalgorithmasthetwo-variableGibbssampler(TGS). 54

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3 isaTGS.Actually,thereisalargefamilyofveryusefulMCMCalgorithms,calledthedataaugmentationalgorithms( TannerandWong 1987 ),thatfallinthiscategory.ThewideapplicationofTGSsaswellastheirrelativelysimplestructuretemptustostudytheirconvergenceratesinauniedmanner.Asarstsmallsteptowardsthisbiggoal,westudythesimplestpossibleTGS.RecallthataTGSisdeterminedbythejointdensityonXY.ItiswellknownthatifeitherXorYisnite,thenanyirreduciblechainonXYisalsogeometricallyergodic.SothesituationwherebothXandYarecountableprovidesthesimplestcaseforwhichgeometricchainscannotbeeasilycharacterized.Therefore,wedecidetostudyTGSsthatareassociatedwithaspecialclassofdensitiesonXY=NN.First,weusethedriftmethodtondasucientconditionforgeometricergodicityoftheMarkovchainunderlyingtheTGS.Secondly,andmoreinterestingly,weresorttosomeoperatortheorytoderiveanecessaryconditionforgeometricconvergenceofthechain.SincetheinvariantdistributioncompletelydeterminesthecorrespondingTGS,italsodeterminestheconvergencerateofthealgorithm.Asexpected,ourresult(Corollary 3 )showsthatageometrictaildecayofthetargetdistributionisalmostnecessaryandsucientfortheassociatedchaintobegeometricallyergodic. WearenotawareofmuchresultsoncharacterizinggeometricergodicityofGibbssamplersintheliterature.ButtherehavebeenseveralattemptsatthisforMetropolis-Hastingsalgorithms.Forexample, MengersenandTweedie ( 1996 ); RobertsandTweedie ( 1996 ); JarnerandHansen ( 2000 )studiedrandom-walkMetropolisalgorithmswithinvariantdensityonRk.Theirresultsprovidenecessaryandsucientconditionforgeometricergodicityofthealgorithmintermsofthetailbehaviorof. 55

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wherefakg1k=1andfbkg1k=1arestrictlypositivesequencesthatsatisfyP1k=1ak+P1k=1bk=1.Also,letb0=0.DenetheclassPtoconsistofthedensitiessatisfying( 4{2 ).ThemarginaldensitiesassociatedwitharegivenbyX(x)=ax+bx1forx2N; AndtheconditionaldensitiesaregivenbyXjY(xjy)=8>>>><>>>>:ay andYjX(yjx)=8>>>><>>>>:bx1 TheconditionalprobabilitydistributionsnowdetermineaMarkovchainthroughtransitiondensity( 4{1 ).Let=f(Xn;Yn)g1n=0denotetheMarkovchainwithanarbitrarystartingpoint(X0;Y0).Foranystrictlypositivesequencesfakgandfbkg,theassociatedMarkovchain,,isclearlyirreducible,aperiodicandpositiverecurrentwith 56

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MeynandTweedie 1993 ,Prop.8.1.3),isalsoHarrisergodic.Thequestionthatweareinterestedinis,whenisgeometricallyergodic?Inotherwords,forwhatpositivesequences,fakgandfbkg,doestheassociatedMarkovchainconvergeatageometricrate? RecallfromChapter 3 thatthetwomarginalsequencesX=fXngandY=fYngofatwo-variableGibbschainarethemselvesMarkovchains( Liuetal. 1994 ).Moreover,thejointchainandthemarginalchainsallconvergeatexactlythesamerate( Diaconisetal. 2008 ; RobertsandRosenthal 2001 ).Therefore,wecanlearnwhetherisgeometricornotbyanalyzingX.ItiseasytoseethatXisabirthanddeath(BD)chainonNwiththefollowingtransitionprobabilities: Lemma4. 1 ,XisgeometricifwecanndafunctionV:N![0;1)thatisunboundedocompactsetsandsatisesthedriftcondition EV(X1)jX0=xV(x)+Lforsome<1,L<1,andallx2N.(4{5) 57

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2qx(z1)+qx1 21 21 21 2;01 2;0andq2(0;1): 21 4{5 )withL:=maxxx0fE(V(X1)jX0=x)gand<1denedabove.Therefore,( 4{4 )impliesgeometricergodicityofX.

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whichinducesanormonL20(X)denedbykhk=hh;hi1 2.ThenkX(x0jx),thetransitionfunctionofX,denesanoperator,PX,onL20(X).Thatis,foranyh2L20(X),PXh(x)=ZXkX(x0jx)h(x0)X(dx0)=Xx0kX(x0jx)h(x0)forallx: Accordingtotheresultsin RobertsandRosenthal ( 1997 ); RobertsandTweedie ( 2001 )thatapplytoreversibleMarkovchains,XisnotgeometricallyergodicifandonlyifthenormofPX,kPXk,isequalto1.LetX0;X1representthersttworandomvariablesofXwithX0X.Alsolet(X;Y)beapairofrandomvariableswithjointdistribution 59

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wherethesupremumsandinmumrangeoverh2L20;1(X).Therstequalityholdsforallself-adjointoperators( Rudin 1991 ,Thm12.25),thesecondequalityisthedenitionofinnerproduct,thethirdequalityfollowsfrom Liuetal. ( 1994 ,Lemma3.2)andthelastequalityisimpliedbytheformulaVar(h(X))=EVar(h(X)jY)+VarE(h(X)jY): liminfi!1EVar(hi(X)jY)=0;(4{7) thenkPXk=1accordingto( 4{6 )andXisnotgeometric.Weusethisideatoderivethefollowinglemma: 1.

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4{7 ).Recallthatwedenedytobeby=(ay+by)=XjY(y+1jy)inSection 4.2 .ThenEHi(X)jY=y=EH2i(X)jY=y=XjY(yjy)Hi(y)+XjY(y+1jy)Hi(y+1)=8>>>><>>>>:0yi2;i1y=i1;1yi: Therefore,EhVarHi(X)jYi=1Xy=1Y(y)VarHi(X)jY=y=Y(i1)VarHi(X)jY=i1=(ai1+bi1)i1(1i1)=ai1bi1

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wherelimi!1(1i)=limi!1Pix=1(ax+bx1)=1.Hence,equation( 4{7 )holdsifandonlyiflimsupi!1P1x=i(ax+bx) 5 saysthatthechainissubgeometricifthetailmassassociatedwithaiorbiconvergestozeroataslowerratethanaiorbithemselvesasigoestoinnity.Weareabletounderstandthisresultbetterinthenextsection,whereLemma 4 and 5 togetherprovidecharacterizationofgeometricergodicityofaclassofMarkovchains. 4.2 at( 4{2 )).Formost,weareabletotellwhetheritscorrespondingGibbschainisgeometricorsubgeometric.However,thereareafewsthatwecannotcategorize. 62

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5 ,thecorrespondingchainisnotgeometric. 5 ,thecorrespondingchainisnotgeometric. 1+ai 1+bi1 Henceliminfqi1 1+limsupai 1+limsupbi1 Hencelimsuppi 4 .Otherwisewe'llhavetocheckthethreeconditionsinLemma 5 .Ifanyofthemaresatised,thechainisnotgeometric,andifnoneofthemaresatised,itremainsunknownwhattheconvergencerateis. Nevertheless,thereisonecollectionofMarkovchainsforwhichgeometricergodicitycanbefullycharacterized. limi!1ai (ii) (iii)

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:Thisistruebecauseunder(i),r=limi!1pi 4{10 )andq>0by( 4{9 ). (ii)!(iii) :ImmediatebyLemma 4 (iii)!(i) :Ifthechainisgeometric,thenlimi!1ai 5 .Next,ifA=1,thenforanyxedpositiveintegerK,wehavelimi!1ai+1 2;ai+2 2;;ai+K 2: 5 thechainisnotgeometric.Thiscontradictsstatement(iii).SoA6=1.But,Acannotbegreaterthan1either.Otherwise,Pxax=1,whichcontradictsthefactthatPax+Pbx=1.Therefore,A<1. Intheaboveresult,Condition1showsthetypesofdistributionforwhichgoodorbadbehavioroftheassociatedchainmaybeexpected.ThekeypartofthisconditionisA<1,whichmeansthattheprobabilitiesinthediagonaldiedowngeometricallyinthetail.Ontheotherhand,Condition2showsushowtocharacterizegeometricconvergenceofthechainintermsoftransitionprobabilitiesofthemarginalBDchain.Oneneedstobecarefulwheninterpretingthiscondition,becauseanunderlyingassumptiononthesequencesfpigandfqigisthattheredoexistsequencesfaigandfbigsuchthat,together,theysatisfytherelationshipin( 4{3 ).Finally,weendthischapterbyapplyingtheaboveresultstofourconcreteexamples. 64

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3 3 3 wherecisthesolutiontoce1=(1e1)+e2=(1e2)=1.Thenlimi!1ai 3 isnotapplicable.However,wecanresorttothemoregeneralLemma 5 .Notethatlimsupi!1bi 65

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2;0(denedin( 1{2 )),orthereferenceprior,pr(denedin( 1{3 )),isq3.Moreover,ifq3ands;t2R,thenE1 2;02s2et<1 (i) 2
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2;0iswell-dened,andnotethatE1 2;02s2et=1 2;0(y)ZR+ZR+ZRZRq2s1 22et1(j;2;2e)L;;2;2e;yddd2d2e=ms1 2;t(y).m1 2;0(y): 67

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RecallfromSection 2.1 thatdenotesaMarkovchainwithMarkovtransitionfunctionK.Supposethatisanon-trivial,-nitemeasureonX.Theniscalled-irreducibleifforeachx2XandeachsetAwith(A)>0,thereexistsann2f1;2;:::g,whichmaydependonxandA,suchthatKn(x;A)>0.Inwords,thechainis-irreducibleifeverysetwithpositive-measureisaccessiblefromeverypointinthestatespace.Accordingto MeynandTweedie ( 1993 ,Prop.4.2.2),ifis-irreducibleforsome,thenthereexistsaprobabilitymeasure,calledthemaximalirreducibilitymeasure,suchthat 1. is-irreducible,and 2. foranyothermeasure0,thechainis0-irreducibleifandonlyif0isabsolutelycontinuouswithrespectto(denoted0). Itisclearfromabovethatamaximalirreducibilitymeasureisuniqueuptoequivalence;i.e.,if1and2arebothmaximalirreducibilitymeasures,then12and21(denoted12).Hence,theterminology\is-irreducible"iswelldened,ifbythiswemeanthatis-irreducibleforsomeandthatisamaximalirreducibilitymeasurefor.Wecanseethatrepresentsthe\natural"irreducibilitymeasureofachain. 68

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WenowproveProposition 4 byestablishingthat Itiswellknownthat0(x)>0forallx>0andthat(x+1)=(x)+1 2)=2log(2),where:=limp!11+1 2++1 2)=2log(2)+2. Recallthat1(s)isactuallyafunctionofs,qandM 2s: dxhlog((xs))log((x))i=(xs)(x)<0forallx>s>0: 2<1foralls2(0;1)andq2. Proof. (x)(x+s1)sforx>1s: 1.T(x)isstrictlydecreasinginx,and 2.limx!1T(x)=1. 69

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AbramowitzandStegun 1964 ,p.259).Notethat1 Thefractioninthersttermof( C{1 )canbewrittenasthefollowingabsolutelyconvergenttelescopingseries1 Wenowproveclaim2.Fixs2(0;1)anddeneS(x)=xs(x+s)(x).Asx!1,S(x)!1( AbramowitzandStegun 1964 ,p.257).Asaconsequence,limx!1T(x)=limx!1S(x)x x+s1s=1:

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2s>1sand1s;q;q+3 2=q+1 2sq+3 2s 2=Tq+3 2s1: 2s>1andhence1s;q;q+3 2<1. Recallthat3(s)isactuallyafunctionofs,manda.IfwedeneA(s;q;a)=(q 2s(q (q 2A1 Weconcludethatthereexistss2Ssuchthat3(s;m;a)<1if1(m)<2expq C{2 )issharp.Nevertheless,ourproofforProposition 4 doesnotrelyonthisconjecture. 71

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Notethatw1andw2arebothfunctionsof.Henceforanypair,(w1;w2),thatweuseasadistinguishedpointinourminorizationcondition,itisnecessarytocheckthatthereexistsa2Rq+1suchthat(w1;w2)=(w1();w2()).Let~y=( yq)and~m=(m1;;mq).Also,let D{2 )isnotonlysucientbutalsonecessary. Proof. D{1 ),then Ximi( Xi(p and Xi(p withequalityholdingin( D{4 )whenm1==mq=m.Inp-dimensionalspace,letldenotethelinethatpassesthroughtheoriginandthroughthepointp D{3 ),thepointp D{4 ),p D{1 )existsifandonlyifthereexistssuchthatC1andC2()intersect. 72

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Xmi( Finally,becauseof( D{4 ),( D{2 )implies( D{5 ).Thus( D{2 )issucientfortheexistenceofasolution.Now,ifm1==mq,then( D{2 )and( D{5 )areexactlythesame.So,inthiscase,( D{2 )isalsonecessary. 73

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