Bayesian Missing Data Problems: EM, Data Augmentation and Noniterative Computation
Andrea Ming T. Tan
University of Maryland School of Medicine
Baltimore, Maryland, U.S.A.
Guo-Liang Tian
The University of Hong Kong
Hong Kong, China
Kai Wang Ng
*大小：344页。
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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