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Bayesian Missing Data Problems: EM, Data Augmentation and Noniterative Computation (Chapman & Hall/CRC Biostatistics Series) (Hardcover)
Ming T. Tan (Author), Guo-Liang Tian (Author), Kai Wang Ng (Author)

Editorial Reviews
Product Description
Bayesian Missing Data Problems: EM, Data Augmentation and Noniterative Computation presents solutions to missing data problems through explicit or noniterative sampling calculation of Bayesian posteriors. The methods are based on the inverse Bayes formulae discovered by one of the author in 1995. Applying the Bayesian approach to important real-world problems, the authors focus on exact numerical solutions, a conditional sampling approach via data augmentation, and a noniterative sampling approach via EM-type algorithms.
After introducing the missing data problems, Bayesian approach, and posterior computation, the book succinctly describes EM-type algorithms, Monte Carlo simulation, numerical techniques, and optimization methods. It then gives exact posterior solutions for problems, such as nonresponses in surveys and cross-over trials with missing values. It also provides noniterative posterior sampling solutions for problems, such as contingency tables with supplemental margins, aggregated responses in surveys, zero-inflated Poisson, capture-recapture models, mixed effects models, right-censored regression model, and constrained parameter models. The text concludes with a discussion on compatibility, a fundamental issue in Bayesian inference.
This book offers a unified treatment of an array of statistical problems that involve missing data and constrained parameters. It shows how Bayesian procedures can be useful in solving these problems.
About the Author
Ming T. Tan is Professor of Biostatistics in the Department of Epidemiology and Preventive Medicine at the University of Maryland School of Medicine and Director of the Division of Biostatistics at the University of Maryland Greenebaum Cancer Center.
Guo-Liang Tian is Associate Professor in the Department of Statistics and Actuarial Science at the University of Hong Kong.
Kai Wang Ng is Professor and Head of the Department of Statistics and Actuarial Science at the University of Hong Kong.

Product Details

· Hardcover: 344 pages

· Publisher: Chapman & Hall; 1 edition (August 26, 2009)

· Language: English

· ISBN-10: 142007749X

· ISBN-13: 978-1420077490

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关键词：Augmentation Bayesian Problems missing problem problems methods through Series

Contents

Preface xv

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Scope, Aim and Outline . . . . . . . . . . . . . . . . . 6

1.3 Inverse Bayes Formulae (IBF) . . . . . . . . . . . . . . 9

1.3.1 The point-wise, function-wise and sampling IBF 10

1.3.2 Monte Carlo versions of the IBF . . . . . . . . 12

1.3.3 Generalization to the case of three vectors . . . 14

1.4 The Bayesian Methodology . . . . . . . . . . . . . . . 15

1.4.1 The posterior distribution . . . . . . . . . . . . 15

1.4.2 Nuisance parameters . . . . . . . . . . . . . . . 17

1.4.3 Posterior predictive distribution . . . . . . . . 18

1.4.4 Bayes factor . . . . . . . . . . . . . . . . . . . . 20

1.4.5 Marginal likelihood . . . . . . . . . . . . . . . . 21

1.5 The Missing Data Problems . . . . . . . . . . . . . . . 22

1.5.1 Missing data mechanism . . . . . . . . . . . . . 23

1.5.2 Data augmentation (DA) . . . . . . . . . . . . 23

1.5.3 The original DA algorithm . . . . . . . . . . . 24

1.5.4 Connection with the Gibbs sampler . . . . . . 26

1.5.5 Connection with the IBF . . . . . . . . . . . . 28

1.6 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.6.1 Shannon entropy . . . . . . . . . . . . . . . . . 29

1.6.2 Kullback{Leibler divergence . . . . . . . . . . . 30

Problems . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 Optimization, Monte Carlo Simulation and

Numerical Integration 35

2.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . 36

2.1.1 The Newton{Raphson (NR) algorithm . . . . . 36

2.1.2 The expectation{maximization (EM) algorithm 40

2.1.3 The ECM algorithm . . . . . . . . . . . . . . . 47

2.1.4 Minorization{maximization (MM) algorithms . 49

2.2 Monte Carlo Simulation . . . . . . . . . . . . . . . . . 56

2.2.1 The inversion method . . . . . . . . . . . . . . 56

2.2.2 The rejection method . . . . . . . . . . . . . . 58

2.2.3 The sampling/importance resampling method . 62

2.2.4 The stochastic representation method . . . . . 66

2.2.5 The conditional sampling method . . . . . . . . 70

2.2.6 The vertical density representation method . . 72

2.3 Numerical Integration . . . . . . . . . . . . . . . . . . 75

2.3.1 Laplace approximations . . . . . . . . . . . . . 75

2.3.2 Riemannian simulation . . . . . . . . . . . . . . 77

2.3.3 The importance sampling method . . . . . . . 80

2.3.4 The cross{entropy method . . . . . . . . . . . . 84

Problems . . . . . . . . . . . . . . . . . . . . . . . . . 89

3 Exact Solutions 93

3.1 Sample Surveys with Nonresponse . . . . . . . . . . . 93

3.2 Misclassi¯ed Multinomial Model . . . . . . . . . . . . 95

3.3 Genetic Linkage Model . . . . . . . . . . . . . . . . . . 97

3.4 Weibull Process with Missing Data . . . . . . . . . . . 99

3.5 Prediction Problem with Missing Data . . . . . . . . . 101

3.6 Binormal Model with Missing Data . . . . . . . . . . . 103

3.7 The 2 £ 2 Crossover Trial with Missing Data . . . . . 105

3.8 Hierarchical Models . . . . . . . . . . . . . . . . . . . 108

3.9 Nonproduct Measurable Space (NPMS) . . . . . . . . 109

Problems . . . . . . . . . . . . . . . . . . . . . . . . . 112

4 Discrete Missing Data Problems 117

4.1 The Exact IBF Sampling . . . . . . . . . . . . . . . . 118

4.2 Genetic Linkage Model . . . . . . . . . . . . . . . . . . 119

4.3 Contingency Tables with One Supplemental Margin . 121

4.4 Contingency Tables with Two Supplemental Margins . 123

4.4.1 Neurological complication data . . . . . . . . . 123

4.4.2 MLEs via the EM algorithm . . . . . . . . . . 123

4.4.3 Generation of i.i.d. posterior samples . . . . . . 125

4.5 The Hidden Sensitivity (HS) Model for Surveys with

Two Sensitive Questions . . . . . . . . . . . . . . . . . 126

4.5.1 Randomized response models . . . . . . . . . . 126

4.5.2 Nonrandomized response models . . . . . . . . 127

4.5.3 The nonrandomized hidden sensitivity model . 128

4.6 Zero{In°ated Poisson Model . . . . . . . . . . . . . . . 132

4.7 Changepoint Problems . . . . . . . . . . . . . . . . . . 133

4.7.1 Bayesian formulation . . . . . . . . . . . . . . . 134

4.7.2 Binomial changepoint models . . . . . . . . . . 137

4.7.3 Poisson changepoint models . . . . . . . . . . . 139

4.8 Capture{Recapture Model . . . . . . . . . . . . . . . . 145

Problems . . . . . . . . . . . . . . . . . . . . . . . . . 148

5 Computing Posteriors in the EM-Type Structures 155

5.1 The IBF Method . . . . . . . . . . . . . . . . . . . . . 156

5.1.1 The IBF sampling in the EM structure . . . . 156

5.1.2 The IBF sampling in the ECM structure . . . . 163

5.1.3 The IBF sampling in the MCEM structure . . 164

5.2 Incomplete Pro-Post Test Problems . . . . . . . . . . . 165

5.2.1 Motivating example: Sickle cell disease study . 166

5.2.2 Binormal model with missing data and known

variance . . . . . . . . . . . . . . . . . . . . . . 167

5.2.3 Binormal model with missing data and

unknown mean and variance . . . . . . . . . . 168

5.3 Right Censored Regression Model . . . . . . . . . . . . 173

5.4 Linear Mixed Models for Longitudinal Data . . . . . . 176

5.5 Probit Regression Models for Independent

Binary Data . . . . . . . . . . . . . . . . . . . . . . . . 181

5.6 A Probit-Normal GLMM for Repeated Binary Data . 185

5.6.1 Model formulation . . . . . . . . . . . . . . . . 186

5.6.2 An MCEM algorithm without using

the Gibbs sampler at E-step . . . . . . . . . . . 187

5.7 Hierarchical Models for Correlated Binary Data . . . . 195

5.8 Hybrid Algorithms: Combining the IBF Sampler

with the Gibbs Sampler . . . . . . . . . . . . . . . . . 197

5.8.1 Nonlinear regression models . . . . . . . . . . . 198

5.8.2 Binary regression models with t link . . . . . . 199

5.9 Assessing Convergence of MCMC Methods . . . . . . 201

5.9.1 Gelman and Rubin's PSR statistic . . . . . . . 202

5.9.2 The di®erence and ratio criteria . . . . . . . . . 203

5.9.3 The Kullback{Leibler divergence criterion . . . 204

5.10 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Problems . . . . . . . . . . . . . . . . . . . . . . . . . 206

6 Constrained Parameter Problems 211

6.1 Linear Inequality Constraints . . . . . . . . . . . . . . 211

6.1.1 Motivating examples . . . . . . . . . . . . . . . 211

6.1.2 Linear transformation . . . . . . . . . . . . . . 212

6.2 Constrained Normal Models . . . . . . . . . . . . . . . 214

6.2.1 Estimation when variances are known . . . . . 214

6.2.2 Estimation when variances are unknown . . . . 219

6.2.3 Two examples . . . . . . . . . . . . . . . . . . 222

6.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . 227

6.3 Constrained Poisson Models . . . . . . . . . . . . . . . 228

6.3.1 Simplex restrictions on Poisson rates . . . . . . 228

6.3.2 Data augmentation . . . . . . . . . . . . . . . . 228

6.3.3 MLE via the EM algorithm . . . . . . . . . . . 229

6.3.4 Bayes estimation via the DA algorithm . . . . 230

6.3.5 Life insurance data analysis . . . . . . . . . . . 231

6.4 Constrained Binomial Models . . . . . . . . . . . . . . 233

6.4.1 Statistical model . . . . . . . . . . . . . . . . . 233

6.4.2 A physical particle model . . . . . . . . . . . . 234

6.4.3 MLE via the EM algorithm . . . . . . . . . . . 236

6.4.4 Bayes estimation via the DA algorithm . . . . 239

Problems . . . . . . . . . . . . . . . . . . . . . . . . . 240

7 Checking Compatibility and Uniqueness 241

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 241

7.2 Two Continuous Conditional Distributions:

Product Measurable Space (PMS) . . . . . . . . . . . 243

7.2.1 Several basic notions . . . . . . . . . . . . . . . 243

7.2.2 A review on existing methods . . . . . . . . . . 244

7.2.3 Two examples . . . . . . . . . . . . . . . . . . 246

7.3 Finite Discrete Conditional Distributions: PMS . . . . 247

7.3.1 The formulation of the problems . . . . . . . . 248

7.3.2 The connection with quadratic optimization

under box constraints . . . . . . . . . . . . . . 248

7.3.3 Numerical examples . . . . . . . . . . . . . . . 250

7.3.4 Extension to more than two dimensions . . . . 253

7.3.5 The compatibility of regression function and

conditional distribution . . . . . . . . . . . . . 255

7.3.6 Appendix: S-plus function (lseb) . . . . . . . . 258

7.3.7 Discussion . . . . . . . . . . . . . . . . . . . . . 258

7.4 Two Conditional Distributions: NPMS . . . . . . . . . 259

7.5 One Marginal and Another Conditional Distribution . 262

7.5.1 A su±cient condition for uniqueness . . . . . . 262

7.5.2 The continuous case . . . . . . . . . . . . . . . 265

7.5.3 The ¯nite discrete case . . . . . . . . . . . . . . 266

7.5.4 The connection with quadratic optimization

under box constraints . . . . . . . . . . . . . . 269

Problems . . . . . . . . . . . . . . . . . . . . . . . . . 271

A Basic Statistical Distributions and Stochastic

Processes 273

A.1 Discrete Distributions . . . . . . . . . . . . . . . . . . 273

A.2 Continuous Distributions . . . . . . . . . . . . . . . . 275

A.3 Mixture Distributions . . . . . . . . . . . . . . . . . . 283

A.4 Stochastic Processes . . . . . . . . . . . . . . . . . . . 285

List of Figures 287

List of Tables 290

List of Acronyms 292

List of Symbols 294

References 298

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