ABSTRACT
CONTENTS 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 8.2 A General Framework for Bayesian Modeling . . . . . . . . . . . . . . . . . . . . . . . . 212
8.2.1 Discrete Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.2.2 Continuous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.2.3 Prior and Posterior Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8.2.4 Predictive Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 8.2.5 Model Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.2.6 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.3 Conjugate or Classical Bayesian Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.3.1 Linear Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.3.2 Dynamic Linear Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 8.3.3 Beyond Linear Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.4 Computational Bayesian Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 8.4.1 Normal Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 8.4.2 Integral Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.4.2.1 Quadrature Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8.4.2.2 Laplace Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8.4.2.3 Monte Carlo Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.4.3 Monte Carlo-Based Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 8.4.3.1 Rejection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 8.4.3.2 Weighted Resampling Method . . . . . . . . . . . . . . . . . . . . . . . 227 8.4.3.3 Metropolis-Hastings Algorithm . . . . . . . . . . . . . . . . . . . . . . 227 8.4.3.4 Gibbs Sampler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.4.3.5 MCMC Over Model Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.4.3.6 Public Domain Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.5 Bayesian Survival Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 8.5.1 Models for Univariate Survival Times . . . . . . . . . . . . . . . . . . . . . . . . . . 230 8.5.2 Shared Frailty Models for Multivariate Survival Times . . . . . 231
8.5.2.1 Baseline Hazard Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2.3 Positive Stable Shared Frailty Model . . . . . . . . . . . . . . . . 233
8.5.3 Extensions of Frailty Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 8.6 Disease Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8.7 Bayesian Clinical Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
8.7.1 Principles of Bayesian Clinical Trial Design . . . . . . . . . . . . . . . . . . . 240 8.7.2 Operating Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.7.3 A Two-Agent Dose-Finding Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
8.8 Microarray Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 8.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 8.8.2 The Gamma/Gamma Hierarchical Model . . . . . . . . . . . . . . . . . . . . . 244 8.8.3 A Nonparametric Bayesian Model for Differential Gene
Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 8.8.4 The Probability of Expression Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 8.8.5 Multiplicity Correction: Controlling False Discovery Rate . . 247
8.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
This articlegives a surveyofBayesian techniquesuseful forbiomedical applications. Given the extensive use of Bayesian methods, especially with the recent advent of Markov Chain Monte Carlo (MCMC), we can never do exhaustive justice. Nevertheless, we have made an attempt to present different Bayesian applications from different viewpoints and differing levels of complexity. We start with a brief introduction to the Bayesian paradigm in Section 8.2, and give some basic formulas. In Section 8.3, we review conjugate Bayesian analysis in both the static and dynamic inferential frameworks, and give references to some biomedical applications. The conjugate Bayesian approach is often insufficient for handling complex problems that arise in several applications. The advent of sampling-based Bayesian methods has opened the door to carrying out inference in a variety of settings. They must of course be used with care, and with sufficient understanding of the underlying stochastics. In Section 8.4, we present details on the algorithms most commonly used in Bayesian computing and provide exhaustive references. Sections 8.5-8.8 show illustrations of Bayesian computing in biomedical applications that are of current interest.
