1st Edition
Markov Chain Monte Carlo in Practice
In a family study of breast cancer, epidemiologists in Southern California increase the power for detecting a gene-environment interaction. In Gambia, a study helps a vaccination program reduce the incidence of Hepatitis B carriage. Archaeologists in Austria place a Bronze Age site in its true temporal location on the calendar scale. And in France, researchers map a rare disease with relatively little variation.
Each of these studies applied Markov chain Monte Carlo methods to produce more accurate and inclusive results. General state-space Markov chain theory has seen several developments that have made it both more accessible and more powerful to the general statistician. Markov Chain Monte Carlo in Practice introduces MCMC methods and their applications, providing some theoretical background as well. The authors are researchers who have made key contributions in the recent development of MCMC methodology and its application.
Considering the broad audience, the editors emphasize practice rather than theory, keeping the technical content to a minimum. The examples range from the simplest application, Gibbs sampling, to more complex applications. The first chapter contains enough information to allow the reader to start applying MCMC in a basic way. The following chapters cover main issues, important concepts and results, techniques for implementing MCMC, improving its performance, assessing model adequacy, choosing between models, and applications and their domains.
Markov Chain Monte Carlo in Practice is a thorough, clear introduction to the methodology and applications of this simple idea with enormous potential. It shows the importance of MCMC in real applications, such as archaeology, astronomy, biostatistics, genetics, epidemiology, and image analysis, and provides an excellent base for MCMC to be applied to other fields as well.
Introduction
The Problem
Markov Chain Monte Carlo
Implementation
Discussion
HEPATITIS B: A CASE STUDY IN MCMC METHODS
Introduction
Hepatitis B Immunization
Modelling
Fitting a Model Using Gibbs Sampling
Model Elaboration
Conclusion
MARKOV CHAIN CONCEPTS RELATED TO SAMPLING ALGORITHMS
Markov Chains
Rates of Convergence
Estimation
The Gibbs Sampler and Metropolis-Hastings Algorithm
INTRODUCTION TO GENERAL STATE-SPACE MARKOV CHAIN THEORY
Introduction
Notation and Definitions
Irreducibility, Recurrence, and Convergence
Harris Recurrence
Mixing Rates and Central Limit Theorems
Regeneration
Discussion
FULL CONDITIONAL DISTRIBUTIONS
Introduction
Deriving Full Conditional Distributions
Sampling from Full Conditional Distributions
Discussion
STRATEGIES FOR IMPROVING MCMC
Introduction
Reparameterization
Random and Adaptive Direction Sampling
Modifying the Stationary Distribution
Methods Based on Continuous-Time Processes
Discussion
IMPLEMENTING MCMC
Introduction
Determining the Number of Iterations
Software and Implementation
Output Analysis
Generic Metropolis Algorithms
Discussion
INFERENCE AND MONITORING CONVERGENCE
Difficulties in Inference from Markov Chain Simulation
The Risk of Undiagnosed Slow Convergence
Multiple Sequences and Overdispersed Starting Points
Monitoring Convergence Using Simulation Output
Output Analysis for Inference
Output Analysis for Improving Efficiency
MODEL DETERMINATION USING SAMPLING-BASED METHODS
Introduction
Classical Approaches
The Bayesian Perspective and the Bayes Factor
Alternative Predictive Distributions
How to Use Predictive Distributions
Computational Issues
An Example
Discussion
HYPOTHESIS TESTING AND MODEL SELECTION
Introduction
Uses of Bayes Factors
Marginal Likelihood Estimation by Importance Sampling
Marginal Likelihood Estimation Using Maximum Likelihood
Application: How Many Components in a Mixture?
Discussion
Appendix: S-PLUS Code for the Laplace-Metropolis Estimator
MODEL CHECKING AND MODEL IMPROVEMENT
Introduction
Model Checking Using Posterior Predictive Simulation
Model Improvement via Expansion
Example: Hierarchical Mixture Modelling of Reaction Times
STOCHASTIC SEARCH VARIABLE SELECTION
Introduction
A Hierarchical Bayesian Model for Variable Selection
Searching the Posterior by Gibbs Sampling
Extensions
Constructing Stock Portfolios With SSVS
Discussion
BAYESIAN MODEL COMPARISON VIA JUMP DIFFUSIONS
Introduction
Model Choice
Jump-Diffusion Sampling
Mixture Deconvolution
Object Recognition
Variable Selection
Change-Point Identification
Conclusions
ESTIMATION AND OPTIMIZATION OF FUNCTIONS
Non-Bayesian Applications of MCMC
Monte Carlo Optimization
Monte Carlo Likelihood Analysis
Normalizing-Constant Families
Missing Data
Decision Theory
Which Sampling Distribution?
Importance Sampling
Discussion
STOCHASTIC EM: METHOD AND APPLICATION
Introduction
The EM Algorithm
The Stochastic EM Algorithm
Examples
GENERALIZED LINEAR MIXED MODELS
Introduction
Generalized Linear Models (GLMs)
Bayesian Estimation of GLMs
Gibbs Sampling for GLMs
Generalized Linear Mixed Models (GLMMs)
Specification of Random-Effect Distributions
Hyperpriors and the Estimation of Hyperparameters
Some Examples
Discussion
HIERARCHICAL LONGITUDINAL MODELLING
Introduction
Clinical Background
Model Detail and MCMC Implementation
Results
Summary and Discussion
MEDICAL MONITORING
Introduction
Modelling Medical Monitoring
Computing Posterior Distributions
Forecasting
Model Criticism
Illustrative Application
Discussion
MCMC FOR NONLINEAR HIERARCHICAL MODELS
Introduction
Implementing MCMC
Comparison of Strategies
A Case Study from Pharmacokinetics-Pharmacodynamics
Extensions and Discussion
BAYESIAN MAPPING OF DISEASE
Introduction
Hypotheses and Notation
Maximum Likelihood Estimation of Relative Risks
Hierarchical Bayesian Model of Relative Risks
Empirical Bayes Estimation of Relative Risks
Fully Bayesian Estimation of Relative Risks
Discussion
MCMC IN IMAGE ANALYSIS
Introduction
The Relevance of MCMC to Image Analysis
Image Models at Different Levels
Methodological Innovations in MCMC Stimulated by Imaging
Discussion
MEASUREMENT ERROR
Introduction
Conditional-Independence Modelling
Illustrative examples
Discussion
GIBBS SAMPLING METHODS IN GENETICS
Introduction
Standard Methods in Genetics
Gibbs Sampling Approaches
MCMC Maximum Likelihood
Application to a Family Study of Breast Cancer
Conclusions
MIXTURES OF DISTRIBUTIONS: INFERENCE AND ESTIMATION
Introduction
The Missing Data Structure
Gibbs Sampling Implementation
Convergence of the Algorithm
Testing for Mixtures
Infinite Mixtures and Other Extensions
AN ARCHAEOLOGICAL EXAMPLE: RADIOCARBON DATING
Introduction
Background to Radiocarbon Dating
Archaeological Problems and Questions
Illustrative Examples
Discussion
Index
Biography
W.R. Gilks Institute of Public Health, Cambridge, UK; S. Richardson Imperial College, London, UK; David Spiegelhalter MRC Biostatistics Unit, Cambridge, UK.
