Computational Bayesian statistics : an introduction

257 Pages·2019·2.27 MB·English

by  Müller

ComputationalBayesianStatistics
AnIntroduction
MeaningfuluseofadvancedBayesianmethodsrequiresagoodunderstandingofthe
fundamentals.Thisengagingbookexplainstheideasthatunderpintheconstruction
andanalysisofBayesianmodels,withparticularfocusoncomputationalmethodsand
schemes.Theuniquefeaturesofthetextaretheextensivediscussionofavailable
softwarepackagescombinedwithabriefbutcompleteandmathematicallyrigorous
introductiontoBayesianinference.ThetextintroducesMonteCarlomethods,Markov
chainMonteCarlomethods,andBayesiansoftware,withadditionalmaterialonmodel
validationandcomparison,transdimensionalMCMC,andconditionallyGaussian
models.Theinclusionofproblemsmakesthebooksuitableasatextbookforafirst
graduate-levelcourseinBayesiancomputationwithafocusonMonteCarlomethods.
TheextensivediscussionofBayesiansoftware–R/R-INLA,OpenBUGS,JAGS,
STAN,andBayesX–makesitusefulalsoforresearchersandgraduatestudentsfrom
beyondstatistics.
MARIA ANTO´NIA AMARAL TURKMAN was,until2013,afullprofessorinthe
DepartmentofStatisticsandOperationsResearch,FacultyofSciences,Universityof
Lisbon.Thoughretiredfromtheuniversity,sheisstillamemberofitsCentreof
StatisticsandApplications,wheresheheldthepositionofscientificcoordinatoruntil
2017.HerresearchinterestsareBayesianstatistics,medicalandenvironmental
statistics,andspatiotemporalmodeling,withrecentpublicationsoncomputational
methodsinBayesianstatistics,withanemphasisonapplicationsinhealthandforest
fires.ShehasservedasvicepresidentofthePortugueseStatisticalSociety.Shehas
taughtcoursesonBayesianstatisticsandcomputationalstatistics,amongmanyothers.
CARLOS DANIEL PAULINO isasenioracademicresearcherintheCenterof
StatisticsandApplicationsandwasanassociateprofessorwithhabilitationinthe
DepartmentofMathematicsoftheInstitutoSuperiorTe´cnico,bothattheUniversityof
Lisbon.HehaspublishedfrequentlyonBayesianstatisticsandcategoricaldata,with
emphasisonapplicationsinbiostatistics.HehasservedaspresidentofthePortuguese
StatisticalSociety.Hehastaughtmanyundergraduateandgraduate-levelcourses,
notablyinmathematicalstatisticsandBayesianstatistics.
PETER MU¨LLER isaprofessorintheDepartmentofMathematicsandthe
DepartmentofStatistics&DataScienceattheUniversityofTexasatAustin.Hehas
publishedwidelyoncomputationalmethodsinBayesianstatistics,nonparametric
Bayesianstatistics,anddecisionproblems,withemphasisonapplicationsin
biostatisticsandbioinformatics.HehasservedaspresidentoftheInternationalSociety
forBayesianAnalysisandaschairfortheSectiononBayesianStatisticsofthe
AmericanStatisticalAssociation.Besidesmanygraduate-levelcourses,hehastaught
shortcoursesonBayesianbiostatistics,Bayesianclinicaltrialdesign,nonparametric
Bayesianinference,medicaldecision-making,andmore.
INSTITUTE OF MATHEMATICAL STATISTICS
TEXTBOOKS
EditorialBoard
NancyReid(UniversityofToronto)
RamonvanHandel(PrincetonUniversity)
XumingHe(UniversityofMichigan)
SusanHolmes(StanfordUniversity)
ISBAEditorialRepresentative
PeterMu¨ller(UniversityofTexasatAustin)
IMSTextbooksgiveintroductoryaccountsoftopicsofcurrentconcernsuitablefor
advancedcoursesatmaster’slevel,fordoctoralstudentsandforindividualstudy.They
aretypicallyshorterthanafullydevelopedtextbook,oftenarisingfrommaterial
createdforatopicalcourse.Lengthsof100–290pagesareenvisaged.Thebooks
typicallycontainexercises.
IncollaborationwiththeInternationalSocietyforBayesianAnalysis(ISBA),
selectedvolumesintheIMSTextbooksseriescarrythe“withISBA”designationatthe
recommendationoftheISBAeditorialrepresentative.
OtherBooksintheSeries(*withISBA)
1. ProbabilityonGraphs,byGeoffreyGrimmett
2. StochasticNetworks,byFrankKellyandElenaYudovina
3. BayesianFilteringandSmoothing,bySimoSa¨rkka¨
4. TheSurprisingMathematicsofLongestIncreasingSubsequences,byDanRomik
5. NoiseSensitivityofBooleanFunctionsandPercolation,byChristopheGarban
andJeffreyE.Steif
6. CoreStatistics,bySimonN.Wood
7. LecturesonthePoissonProcess,byGu¨nterLastandMathewPenrose
8. ProbabilityonGraphs(SecondEdition),byGeoffreyGrimmett
9. IntroductiontoMalliavinCalculus,byDavidNualartandEula`liaNualart
10. AppliedStochasticDifferentialEquations,bySimoSa¨rkka¨andArnoSolin
11. *ComputationalBayesianStatistics,byM.Anto´niaAmaralTurkman,Carlos
DanielPaulino,andPeterMu¨ller

Computational Bayesian Statistics
An Introduction
MariaAnto´niaAmaralTurkman
UniversityofLisbon
CarlosDanielPaulino
UniversityofLisbon
PeterMu¨ller
UniversityofTexas,Austin
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CambridgeUniversityPressispartoftheUniversityofCambridge.
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levelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108481038
DOI:10.1017/9781108646185
©MariaAnto´niaAmaralTurkman,CarlosDanielPaulino,andPeterMu¨ller2019
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2019
PrintedandboundinGreatBritainbyClaysLtd,ElcografS.p.A.
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Contents
PrefacetotheEnglishVersion viii
Preface ix
1 BayesianInference 1
1.1 TheClassicalParadigm 2
1.2 TheBayesianParadigm 5
1.3 BayesianInference 8
1.3.1 ParametricInference 8
1.3.2 PredictiveInference 12
1.4 Conclusion 13
Problems 14
2 RepresentationofPriorInformation 17
2.1 Non-InformativePriors 18
2.2 NaturalConjugatePriors 23
Problems 26
3 BayesianInferenceinBasicProblems 28
3.1 TheBinomial∧BetaModel 28
3.2 ThePoisson∧GammaModel 30
3.3 Normal(Knownµ)∧InverseGammaModel 31
3.4 Normal(Unknownµ,σ2)∧Jeffreys’Prior 31
3.5 TwoIndependentNormalModels∧MarginalJeffreys’Priors 33
3.6 TwoIndependentBinomials∧BetaDistributions 34
3.7 Multinomial∧DirichletModel 36
3.8 InferenceinFinitePopulations 38
Problems 40
4 InferencebyMonteCarloMethods 43
4.1 SimpleMonteCarlo 43
4.1.1 PosteriorProbabilities 47
4.1.2 CredibleIntervals 47
v
vi Contents
4.1.3 MarginalPosteriorDistributions 48
4.1.4 PredictiveSummaries 50
4.2 MonteCarlowithImportanceSampling 50
4.2.1 CredibleIntervals 54
4.2.2 BayesFactors 56
4.2.3 MarginalPosteriorDensities 57
4.3 SequentialMonteCarlo 59
4.3.1 DynamicStateSpaceModels 59
4.3.2 ParticleFilter 60
4.3.3 AdaptedParticleFilter 62
4.3.4 ParameterLearning 63
Problems 64
5 ModelAssessment 70
5.1 ModelCriticismandAdequacy 70
5.2 ModelSelectionandComparison 76
5.2.1 MeasuresofPredictivePerformance 76
5.2.2 SelectionbyPosteriorPredictivePerformance 81
5.2.3 ModelSelectionUsingBayesFactors 83
5.3 FurtherNotesonSimulationinModelAssessment 85
5.3.1 EvaluatingPosteriorPredictiveDistributions 85
5.3.2 PriorPredictiveDensityEstimation 86
5.3.3 SamplingfromPredictiveDistributions 87
Problems 88
6 MarkovChainMonteCarloMethods 90
6.1 DefinitionsandBasicResultsforMarkovChains 91
6.2 Metropolis–HastingsAlgorithm 94
6.3 GibbsSampler 98
6.4 SliceSampler 105
6.5 HamiltonianMonteCarlo 107
6.5.1 HamiltonianDynamics 107
6.5.2 HamiltonianMonteCarloTransitionProbabilities 111
6.6 ImplementationDetails 113
Problems 116
7 ModelSelectionandTrans-dimensionalMCMC 129
7.1 MCSimulationovertheParameterSpace 129
7.2 MCSimulationovertheModelSpace 131
7.3 MCSimulationoverModelandParameterSpace 135
7.4 ReversibleJumpMCMC 138
Problems 143
Contents vii
8 MethodsBasedonAnalyticApproximations 150
8.1 AnalyticalMethods 151
8.1.1 MultivariateNormalPosteriorApproximation 151
8.1.2 TheClassicalLaplaceMethod 154
8.2 LatentGaussianModels(LGM) 159
8.3 IntegratedNestedLaplaceApproximation 161
8.4 VariationalBayesianInference 164
8.4.1 PosteriorApproximation 164
8.4.2 CoordinateAscentAlgorithm 165
8.4.3 AutomaticDifferentiationVariationalInference 168
Problems 168
9 Software 172
9.1 ApplicationExample 173
9.2 TheBUGSProject:WinBUGSandOpenBUGS 173
9.2.1 ApplicationExample:UsingR2OpenBUGS 175
9.3 JAGS 181
9.3.1 ApplicationExample:UsingR2jags 181
9.4 Stan 185
9.4.1 ApplicationExample:UsingRStan 186
9.5 BayesX 192
9.5.1 ApplicationExample:UsingR2BayesX 194
9.6 ConvergenceDiagnostics:theProgramsCODAandBOA 198
9.6.1 ConvergenceDiagnostics 199
9.6.2 TheCODAandBOAPackages 201
9.6.3 ApplicationExample:CODAandBOA 203
9.7 R-INLAandtheApplicationExample 213
9.7.1 ApplicationExample 215
Problems 222
AppendixA.ProbabilityDistributions 224
AppendixB.ProgrammingNotes 229
References 232
Index 241
Preface to the English Version
This book is based on lecture notes for a short course that was given at
the XXII Congresso da Sociedade Portuguesa de Estatística. In the trans-
lation from the original Portuguese text we have added some additional
material on sequential Monte Carlo, Hamiltonian Monte Carlo, transdi-
mensional Markov chain Monte Carlo (MCMC), and variational Bayes,
and we have introduced problem sets. The inclusion of problems makes
thebooksuitableasatextbookforafirstgraduate-levelclassinBayesian
computationwithafocusonMonteCarlomethods.Theextensivediscus-
sionofBayesiansoftwaremakesitusefulalsoforresearchersandgraduate
studentsfrombeyondstatistics.
The core of the text lies in Chapters 4, 6, and 9 on Monte Carlo meth-
ods,MCMCmethods,andBayesiansoftware.Chapters5,7,and8include
additionalmaterialonmodelvalidationandcomparison,transdimensional
MCMC, and conditionally Gaussian models. Chapters 1 through 3 intro-
ducethebasicsofBayesianinference,andcouldbecoveredfairlyquickly
by way of introduction; these chapters are intended primarily for review
and to introduce notation and terminology. For a more in-depth introduc-
tionwerecommendthetextbooksbyCarlinandLouis(2009),Christensen
etal(2011),Gelmanetal(2014a)orHoff(2009).
viii