1st Edition

# Markov Chain Monte Carlo in Practice

**Also available as eBook on:**

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.