Bayesian Missing Data Problems

Bayesian Missing Data Problems: EM, Data Augmentation and Noniterative Computation

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Features

  • Presents a wide range of practical missing data problems that are solvable using a Bayesian approach
  • Provides worked out noniterative sampling calculations of posteriors for the problems
  • Uses inverse Bayesian formulae, the EM algorithm, and data augmentation algorithms for computation
  • Illustrates methods with a variety of biostatistical models and real-world applications, including mixed effects and hierarchical models, nonresponse and contingency tables, and the constrained parameter problem reformulated as a missing data problem
  • Includes S-PLUS and R computer codes

Summary

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.

Table of Contents

Introduction

Background

Scope, Aim and Outline

Inverse Bayes Formulae (IBF)

The Bayesian Methodology

The Missing Data Problems

Entropy

Optimization, Monte Carlo Simulation and Numerical Integration

Optimization

Monte Carlo Simulation

Numerical Integration

Exact Solutions

Sample Surveys with Nonresponse

Misclassified Multinomial Model

Genetic Linkage Model

Weibull Process with Missing Data

Prediction Problem with Missing Data

Binormal Model with Missing Data

The 2 × 2 Crossover Trial with Missing Data

Hierarchical Models

Nonproduct Measurable Space (NPMS)

Discrete Missing Data Problems

The Exact IBF Sampling

Genetic Linkage Model

Contingency Tables with One Supplemental Margin

Contingency Tables with Two Supplemental Margins

The Hidden Sensitivity Model for Surveys with Two Sensitive Questions

Zero-Inflated Poisson Model

Changepoint Problems

Capture-Recapture Model

Computing Posteriors in the EM-Type Structures

The IBF Method

Incomplete Pro-Post Test Problems

Right Censored Regression Model

Linear Mixed Models for Longitudinal Data

Probit Regression Models for Independent Binary Data

A Probit-Normal GLMM for Repeated Binary Data

Hierarchical Models for Correlated Binary Data

Hybrid Algorithms: Combining the IBF Sampler with the Gibbs Sampler

Assessing Convergence ofMCMC Methods

Remarks

Constrained Parameter Problems

Linear Inequality Constraints

Constrained Normal Models

Constrained Poisson Models

Constrained Binomial Models

Checking Compatibility and Uniqueness

Introduction

Two Continuous Conditional Distributions: Product Measurable Space (PMS)

Finite Discrete Conditional Distributions: PMS

Two Conditional Distributions: NPMS

One Marginal and Another Conditional Distribution

Appendix: Basic Statistical Distributions and Stochastic Processes

Discrete Distributions

Continuous Distributions

Mixture Distributions

Stochastic Processes

References

Author Index

Subject Index

Problems appear at the end of each chapter.

Author Bio(s)

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.

Editorial Reviews

In Bayesian Missing Data Problems, the authors provide a new and appealing approach to handle missing data problems (MDPs), based on noniterative methods. … the examples and real applications following key theorems and concepts are useful for readers to further understand the results and pinpoint major advantages or drawbacks about the proposed methodology. … I recommend this book as a valuable reference for researchers interested in MDPs, and I believe that the methodology described in the book should be included in the up-to-date literature on missing data. … the book stimulated my interest, suggesting an alternative way to think about MDPs. …
Biometrics, June 2011

… [this book] sits nicely alongside Tanner’s Tools for Statistical Inference. … For those interested in Bayesian computational methods, this book will be of great interest. …
International Statistical Review (2010), 78, 3

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