Case Studies in Bayesian Statistical Modelling and Analysis E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Probability and Statistics
• Sprache: Englisch

Seitenzahl: 808

Veröffentlichungsjahr: 2012

Beschreibung

Provides an accessible foundation to Bayesian analysis using real world models

This book aims to present an introduction to Bayesian modelling and computation, by considering real case studies drawn from diverse fields spanning ecology, health, genetics and finance. Each chapter comprises a description of the problem, the corresponding model, the computational method, results and inferences as well as the issues that arise in the implementation of these approaches.

Case Studies in Bayesian Statistical Modelling and Analysis:

• Illustrates how to do Bayesian analysis in a clear and concise manner using real-world problems.
• Each chapter focuses on a real-world problem and describes the way in which the problem may be analysed using Bayesian methods.
• Features approaches that can be used in a wide area of application, such as, health, the environment, genetics, information science, medicine, biology, industry and remote sensing.

Case Studies in Bayesian Statistical Modelling and Analysis is aimed at statisticians, researchers and practitioners who have some expertise in statistical modelling and analysis, and some understanding of the basics of Bayesian statistics, but little experience in its application. Graduate students of statistics and biostatistics will also find this book beneficial.

Leseprobe

Contents

Cover

Wiley Series in Probability and Statistics

Title Page

Copyright

Preface

List of contributors

Chapter 1: Introduction

1.1 Introduction

1.2 Overview

1.3 Further Reading

References

Chapter 2: Introduction to MCMC

2.1 Introduction

2.2 Gibbs Sampling

2.3 Metropolis–Hastings Algorithms

2.4 Approximate Bayesian Computation

2.5 Reversible Jump MCMC

2.6 MCMC for some Further Applications

References

Chapter 3: Priors: Silent or active Partners of Bayesian Inference?

3.1 Priors in the very Beginning

3.2 Methodology I: Priors Defined by Mathematical Criteria

3.3 Methodology II: Modelling Informative Priors

3.4 Case Studies

3.5 Discussion

Acknowledgements

References

Chapter 4: Bayesian Analysis of the Normal Linear Regression Model

4.1 Introduction

4.2 Case Studies

4.3 Matrix Notation and the Likelihood

4.4 Posterior Inference

4.5 Analysis

References

Chapter 5: Adapting ICU Mortality Models for Local Data: A Bayesian Approach

5.1 Introduction

5.2 Case Study: Updating a known Risk-Adjustment Model for Local Use

5.3 Models and Methods

5.4 Data Analysis and Results

5.5 Discussion

References

Chapter 6: A Bayesian Regression Model with Variable Selection for Genome-Wide Association Studies

6.1 Introduction

6.2 Case Study: Case–Control of Type 1 Diabetes

6.3 Case Study: GENICA

6.4 Models and Methods

6.5 Data Analysis and Results

6.6 Discussion

Acknowledgements

References

6.A Appendix: SNP IDs

Chapter 7: Bayesian Meta-Analysis

7.1 Introduction

7.2 Case Study 1: Association between Red Meat Consumption and Breast Cancer

7.3 Case study 2: Trends in Fish Growth Rate and Size

Acknowledgements

References

Chapter 8: Bayesian Mixed Effects Models

8.1 Introduction

8.2 Case Studies

8.3 Models and Methods

8.4 Data Analysis and Results

8.5 Discussion

References

Chapter 9: Ordering of Hierarchies in Hierarchical Models: Bone Mineral Density Estimation

9.1 Introduction

9.2 Case Study

9.3 Models

9.4 Data Analysis and Results

9.5 Discussion

References

9.A Appendix: Likelihoods

Chapter 10: Bayesian Weibull Survival Model for Gene Expression Data

10.1 Introduction

10.2 Survival Analysis

10.3 Bayesian Inference for the Weibull Survival Model

10.4 Case Study

10.5 Discussion

References

Chapter 11: Bayesian Change Point Detection in Monitoring Clinical Outcomes

11.1 Introduction

11.2 Case Study: Monitoring Intensive Care Unit Outcomes

11.3 Risk-Adjusted Control Charts

11.4 Change Point Model

11.5 Evaluation

11.6 Performance Analysis

11.7 Comparison of Bayesian Estimator with Other Methods

11.8 Conclusion

References

Chapter 12: Bayesian Splines

12.1 Introduction

12.2 Models and Methods

12.3 Case Studies

12.4 Conclusion

References

Chapter 13: Disease Mapping using Bayesian Hierarchical Models

13.1 Introduction

13.2 Case Studies

13.3 Models and Methods

13.4 Data Analysis and Results

13.5 Discussion

References

Chapter 14: Moisture, Crops and Salination: An Analysis of a Three-Dimensional Agricultural Data Set

14.1 Introduction

14.2 Case Study

14.3 Review

14.4 Case Study Modelling

14.5 Model Implementation: Coding Considerations

14.6 Case Study Results

14.7 Conclusions

References

Chapter 15: A Bayesian Approach to Multivariate State Space Modelling: A Study of a Fama–French Asset-Pricing Model with Time-Varying Regressors

15.1 Introduction

15.2 Case Study: Asset Pricing in Financial Markets

15.3 Time-varying Fama–French Model

15.4 Bayesian Estimation

15.5 Analysis

15.6 Conclusion

References

Chapter 16: Bayesian Mixture Models: When the Thing you need to know is the Thing you cannot Measure

16.1 Introduction

16.2 Case Study: CT Scan Images of Sheep

16.3 Models and Methods

16.4 Data Analysis and Results

16.5 Discussion

References

Chapter 17: Latent Class Models in Medicine

17.1 Introduction

17.2 Case Studies

17.3 Models and Methods

17.4 Data Analysis and Results

17.5 Discussion

References

Chapter 18: Hidden Markov Models for Complex Stochastic Processes: A Case Study in Electrophysiology

18.1 Introduction

18.2 Case Study: Spike Identification and Sorting of Extracellular Recordings

18.3 Models and Methods

18.4 Data Analysis and Results

18.5 Discussion

References

Chapter 19: Bayesian Classification and Regression Trees

19.1 Introduction

19.2 Case Studies

19.3 Models and Methods

19.4 Computation

19.5 Case Studies – Results

19.6 Discussion

References

Chapter 20: Tangled Webs: Using Bayesian Networks in the Fight Against Infection

20.1 Introduction to Bayesian Network Modelling

20.2 Introduction to Case Study

20.3 Model

20.4 Methods

20.5 Results

20.6 Discussion

References

Chapter 21: Implementing Adaptive Dose Finding Studies using Sequential Monte Carlo

21.1 Introduction

21.2 Model and Priors

21.3 SMC for Dose Finding Studies

21.4 Example

21.5 Discussion

References

21.A Appendix: Extra Example

Chapter 22: Likelihood-Free Inference for Transmission Rates of Nosocomial Pathogens

22.1 Introduction

22.2 Case Study: Estimating Transmission Rates of Nosocomial Pathogens

22.3 Models and Methods

22.4 Data Analysis and Results

22.5 Discussion

References

Chapter 23: Variational Bayesian Inference for Mixture Models

23.1 Introduction

23.2 Case Study: Computed Tomography (CT) Scanning of a Loin Portion of a Pork Carcase

23.3 Models and Methods

23.4 Data Analysis and Results

23.5 Discussion

References

23.A Appendix: Form of the Variational Posterior for a Mixture of Multivariate Normal Densities

Chapter 24: Issues in Designing Hybrid Algorithms

24.1 Introduction

24.2 Algorithms and Hybrid Approaches

24.3 Illustration of Hybrid Algorithms

24.4 Discussion

References

Chapter 25: A Python Package for Bayesian Estimation using Markov Chain Monte Carlo

25.1 Introduction

25.2 Bayesian Analysis

25.3 Empirical Illustrations

25.4 Using PyMCMC Efficiently

25.5 PyMCMC Interacting with R

25.6 Conclusions

25.7 Obtaining PyMCMC

References

Index

Wiley Series in Probability and Statistics

Wiley Series in Probability and Statistics

Established by WALTER A. SHEWHART and SAMUEL S. WILKS

Editors

David J. Balding, Noel A.C. Cressie, Garrett M. Fitzmaurice, Harvey Goldstein,

Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F.M. Smith,

Ruey S. Tsay, Sanford Weisberg

Editors Emeriti

Vic Barnett, Ralph A. Bradley, J. Stuart Hunter, J.B. Kadane, David G. Kendall,

Jozef L. Teugels

A complete list of the titles in this series appears at the end of this volume.

This edition first published 2013

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Library of Congress Cataloging-in-Publication Data

Case studies in Bayesian statistical modelling and analysis / edited by Clair Alston, Kerrie Mengersen, and Anthony Pettitt.

pages cm

Includes bibliographical references and index.

ISBN 978-1-119-94182-8 (cloth)

1. Bayesian statistical decision theory. I. Alston, Clair. II. Mengersen, Kerrie L. III. Pettitt, Anthony (Anthony N.)

QA279.5.C367 2013

519.5’42–dc23

2012024683

A catalogue record for this book is available from the British Library.

ISBN: 978-1-119-94182-8

Preface

Bayesian statistics is now an established statistical methodology in almost all research disciplines and is being applied to a very wide range of problems. These approaches are endemic in areas of health, the environment, genetics, information science, medicine, biology, industry, remote sensing, and so on. Despite this, most statisticians, researchers and practitioners will not have encountered Bayesian statistics as part of their formal training and often find it difficult to start understanding and employing these methods. As a result of the growing popularity of Bayesian statistics and the concomitant demand for learning about these methods, there is an emerging body of literature on Bayesian theory, methodology, computation and application. Some of this is generic and some is specific to particular fields. While some of this material is introductory, much is at a level that is too complex to be replicated or extrapolated to other problems by an informed Bayesian beginner.

As a result, there is still a need for books that show how to do Bayesian analysis, using real-world problems, at an accessible level.

This book aims to meet this need. Each chapter of this text focuses on a real-world problem that has been addressed by members of our research group, and describes the way in which the problem may be analysed using Bayesian methods. The chapters generally comprise a description of the problem, the corresponding model, the computational method, results and inferences, as well as the issues arising in the implementation of these approaches. In order to meet the objective of making the approaches accessible to the informed Bayesian beginner, the material presented in these chapters is sometimes a simplification of that used in the full projects. However, references are typically given to published literature that provides further details about the projects and/or methods.

This book is targeted at those statisticians, researchers and practitioners who have some expertise in statistical modelling and analysis, and some understanding of the basics of Bayesian statistics, but little experience in its application. As a result, we provide only a brief introduction to the basics of Bayesian statistics and an overview of existing texts and major published reviews of the subject in Chapter 2, along with references for further reading. Moreover, this basic background in statistics and Bayesian concepts is assumed in the chapters themselves.

Of course, there are many ways to analyse a problem. In these chapters, we describe how we approached these problems, and acknowledge that there may be alternatives or improvements. Moreover, there are very many models and a vast number of applications that are not addressed in this book. However, we hope that the material presented here provides a foundation for the informed Bayesian beginner to engage with Bayesian modelling and analysis. At the least, we hope that beginners will become better acquainted with Bayesian concepts, models and computation, Bayesian ways of thinking about a problem, and Bayesian inferences. We hope that this will provide them with confidence in reading Bayesian material in their own discipline or for their own project. At the most, we hope that they will be better equipped to extend this learning to do Bayesian statistics. As we all learn about, implement and extend Bayesian statistics, we all contribute to ongoing improvement in the philosophy, methodology and inferential capability of this powerful approach.

This book includes an accompanying website. Please visit www.wiley.com/go/statistical_modelling

Clair L. AlstonKerrie L. MengersenAnthony N. Pettitt

List of contributors

Clair L. Alston

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Hassan Assareh

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Carla Chen

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Samuel Clifford

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

David A. Cook

Princess Alexandra Hospital

Brisbane, Australia

Susanna M. Cramb

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

and

Viertel Centre for Research in

Cancer Control

Cancer Council Queensland

Australia

Robert J. Denham

Department of Environment and Resource Management

Brisbane, Australia

Margaret Donald

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Christopher C. Drovandi

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Arul Earnest

Tan Tock Seng Hospital, Singapore & Duke–NUS Graduate Medical School

Singapore

Graham E. Gardner

School of Veterinary and Biomedical Sciences

Murdoch University

Perth, Australia

Philip Gharghori

Department of Accounting and Finance

Monash University

Melbourne, Australia

Petra L. Graham

Department of Statistics

Macquarie University

North Ryde, Australia

Candice M. Hincksman

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Wenbiao Hu

School of Population Health and Institute of Health and Biomedical Innovation

University of Queensland

Brisbane, Australia

Katja Ickstadt

Faculty of Statistics

TU Dortmund University

Germany

Helen Johnson

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Sandra Johnson

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Jonathan M. Keith

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

and

Monash University

Melbourne, Australia

Jeong E. Lee

School of Computing and Mathematical Sciences

Auckland University of Technology

New Zealand

Samantha Low Choy

Cooperative Research Centre for

National Plant Biosecurity, Australia

and

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

James M. McGree

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Clare A. McGrory

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

and

School of Mathematics

University of Queensland

St. Lucia, Australia

Kerrie L. Mengersen

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Rebecca A. O'Leary

Department of Agriculture and Food

Western Australia, Australia

Anthony N. Pettitt

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Jegar O. Pitchforth

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Christian P. Robert

Université Paris-Dauphine

Paris, France

and

Centre de Recherche

en Économie et Statistique

(CREST), Paris, France

Margaret Rolfe

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Judith Rousseau

Université Paris-Dauphine

Paris, France

and

Centre de Recherche

en Économie et Statistique

(CREST), Paris, France

Peter Silburn

St. Andrew's War Memorial

Hospital and Medical Institute

Brisbane, Australia

Ian Smith

St. Andrew's War Memorial

Hospital and Medical Institute

Brisbane, Australia

Christopher M. Strickland

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

Sri Astuti Thamrin

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

and

Hasanuddin University, Indonesia

Cathal D. Walsh

Department of Statistics

Trinity College Dublin

Ireland

Mary Waterhouse

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

and

Wesley Research Institute

Brisbane, Australia

Nicole M. White

School of Mathematical Sciences

Queensland University of Technology

Brisbane, Australia

and

CRC for Spatial Information, Australia

Rick Young

Tamworth Agricultural Institute

Department of Primary Industries

Tamworth, Australia

1

Introduction

Clair L. Alston, Margaret Donald, Kerrie L. Mengersen and Anthony N. Pettitt

Queensland University of Technology, Brisbane, Australia

1.1 Introduction

This book aims to present an introduction to Bayesian modelling and computation, by considering real case studies drawn from diverse fields spanning ecology, health, genetics and finance. As discussed in the Preface, the chapters are intended to be introductory and it is openly acknowledged that there may be many other ways to address the case studies presented here. However, the intention is to provide the Bayesian beginner with a practical and accessible foundation on which to build their own Bayesian solutions to problems encountered in research and practice.

In the following, we first provide an overview of the chapters in the book and then present a list of texts for further reading. This book does not seek to teach the novice about Bayesian statistics per se, nor does it seek to cover the whole field. However, there is now a substantial literature on Bayesian theory, methodology, computation and application that can be used as support and extension. While we cannot hope to cover all of the relevant publications, we provide a selected review of texts now available on Bayesian statistics, in the hope that this will guide the reader to other reference material of interest.

1.2 Overview

In this section we give an overview of the chapters in this book. Given that the models are developed and described in the context of the particular case studies, the first two chapters focus on the other two primary cornerstones of Bayesian modelling: computational methods and prior distributions. Building on this foundation, Chapters 4–9 describe canonical examples of Bayesian normal linear and hierarchical models. The following five chapters then focus on extensions to the regression models for the analysis of survival, change points, nonlinearity (via splines) and spatial data. The wide class of latent variables models is then illustrated in Chapters 15–19 by considering multivariate linear state space models, mixtures, latent class analysis, hidden Markov models and structural equation models. Chapters 20 and 21 then describe other model structures, namely Bayesian classification and regression trees, and Bayesian networks. The next four chapters of the book focus on different computational methods for solving diverse problems, including approximate Bayesian computation for modelling the transmission of infection, variational Bayes methods for the analysis of remotely sensed data and sequential Monte Carlo to facilitate experimental design. Finally, the last chapter describes a software package, PyMCMC, that has been developed by researchers in our group to provide accessible, efficient Markov chain Monte Carlo algorithms for solving some of the problems addressed in the book.

The chapters are now described in more detail.

Modern Bayesian computation has been hailed as a ‘model-liberating’ revolution in Bayesian modelling, since it facilitates the analysis of a very wide range of models, diverse and complex data sets, and practically relevant estimation and inference. One of the fundamental computational algorithms used in Bayesian analysis is the Markov chain Monte Carlo (MCMC) algorithm. In order to set the stage for the computational approaches described in subsequent chapters, Chapter 2 provides an overview of the Gibbs and Metropolis–Hastings algorithms, followed by extensions such as adaptive MCMC, approximate Bayesian computation (ABC) and reversible jump MCMC (RJMCMC).

One of the distinguishing features of Bayesian methodology is the use of prior distributions. In Chapter 3 the range of methodology for constructing priors for a Bayesian analysis is described. The approach can broadly be categorized as one of the following two: (i) priors are based on mathematical criteria, such as conjugacy; or (ii) priors model the existing information about the unknown quantity. The chapter shows that in practice a balance must be struck between these two categories. This is illustrated by case studies from the author's experience. The case studies employ methodology for formulating prior models for different types of likelihood models: binomial, logistic regression, normal and a finite mixture of multivariate normal distributions. The case studies involve the following: time to submit research dissertations; surveillance for exotic plant pests; species distribution models; and delineating ecoregions. There is a review of practical issues. One aim of this chapter is to alert the reader to the important and multi-faceted role of priors in Bayesian inference. The author argues that, in practice, the prior often assumes a silent presence in many Bayesian analyses. Many practitioners or researchers often passively select an ‘inoffensive prior'. This chapter provides practical approaches towards more active selection and evaluation of priors.

Chapter 4 presents the ubiquitous and important normal linear regression model, firstly under the usual assumption of independent, homoscedastic, normal residuals, and secondly for the situation in which the error covariance matrix is not necessarily diagonal and has unknown parameters. For the latter case, a first-order serial correlation model is considered in detail. In line with the introductory nature of this chapter, two well-known case studies are considered, one involving house prices from a cross-sectional study and the other a time series of monthly vehicle production data from Australia. The theory is extended to the situation where the error covariance matrix is not necessarily diagonal and has unknown parameters, and a first-order serial correlation model is considered in detail. The problem of covariate selection is considered from two perspectives: the stochastic search variable selection approach and a Bayesian lasso. MCMC algorithms are given for the various models. Results are obtained for the two case studies for the fixed model and the variable selection methods.

The application of Bayesian linear regression with informed priors is described in Chapter 5 in the context of modelling patient risk. Risk stratification models are typically constructed via ‘gold-standard’ logistic regressions of health outcomes of interest, often based on a population that has different characteristics to the patient group to which the model is applied. A Bayesian model can augment the local data with priors based on the gold-standard models, resulting in a locally calibrated model that better reflects the target patient group.

A further illustration of linear regression and variable selection is presented in Chapter 6. This concerns a case study involving a genome-wide association (GWA) study. This involves regressing the trait or disease status of interest (a continuous or binary variable) against all the single nucleotide polymorphisms (SNPs) available in order to find the significant SNPs or effects and identify important genes. The case studies involve investigations of genes associated with Type 1 diabetes and breast cancer. Typical SNP studies involve a large number of SNPs and the diabetes study has over 26 000 SNPs while the number of cases is relatively small. A main effects model and an interaction model are described. Bayesian stochastic search algorithms can be used to find the significant effects and the search algorithm to find the important SNPs is described, which uses Gibbs sampling and MCMC. There is an extensive discussion of the results from both case studies, relating the findings to those of other studies of the genetics of these diseases.

The ease with which hierarchical models are constructed in a Bayesian framework is illustrated in Chapter 7 by considering the problem of Bayesian meta-analysis. Meta-analysis involves a systematic review of the relevant literature on the topic of interest and quantitative synthesis of available estimates of the associated effect. For one of the case studies in the chapter this is the association between red meat consumption and the incidence of breast cancer. Formal studies of the association have reported conflicting results, from no association between any level of red meat consumption to a significantly raised relative risk of breast cancer. The second case study is illustrative of a range of problems requiring the synthesis of results from time series or repeated measures studies and involves the growth rate and size of fish. A multivariate analysis is used to capture the dependence between parameters of interest. The chapter illustrates the use of the WinBUGS software to carry out the computations.

Mixed models are a popular statistical model and are used in a range of disciplines to model complex data structures. Chapter 8 presents an exposition of the theory and computation of Bayesian mixed models.

Considering the various models presented to date, Chapter 9 reflects on the need to carefully consider the way in which a Bayesian hierarchical model is constructed. Two different hierarchical models are fitted to data concerning the reduction in bone mineral density (BMD) seen in a sample of patients attending a hospital. In the sample, one of three distinct methods of measuring BMD is used with a patient and patients can be in one of two study groups, either outpatient or inpatient. Hence there are six combinations of data, the three BMD measurement methods and in-or outpatient. The data can be represented by covariates in a linear model, as described in Chapter 2, or can be represented by a nested structure. For the latter, there is a choice of two structures, either method measurement within study group or vice versa, both of which provide estimates of the overall population mean BMD level. The resulting posterior distributions, obtained using WinBUGS, are shown to depend substantially on the model construction.

Returning to regression models, Chapter 10 focuses on a Bayesian formulation of a Weibull model for the analysis of survival data. The problem is motivated by the current interest in using genetic data to inform the probability of patient survival. Issues of model fit, variable selection and sensitivity to specification of the priors are considered.

Chapter 11 considers a regression model tailored to detect change points. The standard model in the Bayesian context provides inferences for a change point and is relatively straightforward to implement in MCMC. The motivation of this study arose from a monitoring programme of mortality of patients admitted to an intensive care unit (ICU) in a hospital in Brisbane, Australia. A scoring system is used to quantify patient mortality based on a logistic regression and the score is assumed to be correct before the change point and changed after by a fixed amount on the odds ratio scale. The problem is set within the context of the application of process control to health care. Calculations were again carried out using WinBUGS software.

The parametric regression models considered so far are extended in Chapter 12 to smoothing splines. Thin-plate splines are discussed in a regression context and a Bayesian hierarchical model is described along with an MCMC algorithm to estimate the parameters. B-splines are described along with an MCMC algorithm and extensions to generalized additive models. The ideas are illustrated with an adaptation to data on the circle (averaged 24 hour temperatures) and other data sets. MATLAB code is provided on the book's website.

Extending the regression model to the analysis of spatial data, Chapter 13 concerns disease mapping which generally involves modelling the observed and expected counts of morbidity or mortality and expressing each as a ratio, a standardized mortality/morbidity rate (SMR), for an area in a given region. Crude SMRs can have large variances for sparsely populated areas or rare diseases. Models that have spatial correlation are used to smooth area estimates of disease risk and the chapter shows how appropriate Bayesian hierarchical models can be formulated. One case study involves the incidence of birth defects in New South Wales, Australia. A conditional autoregressive (CAR) model is used for modelling the observed number of defects in an area and various neighbour weightings considered and compared. WinBUGS is used for computation. A second case study involves survival from breast cancer in Queensland and excess mortality, a count, is modelled using a CAR model. Various priors are used and sensitivity analyses carried out. Again WinBUGS is used to estimate the relative excess risk. The approach is particularly useful when there are sparsely populated areas, as is the situation in the two case studies.

The focus on spatial data is continued in Chapter 14 with a description of the analysis carried out to investigate the effects of different cropping systems on the moisture of soil at varying depths up to 300 cm below the surface at 108 different sites, set out in a row by column design. The experiment involved collecting daily data on about 60 occasions over 5 years but here only one day's data are analysed. The approach uses a Gaussian Markov random field model defined using the CAR formulation to model the spatial dependence for each horizontal level and linear splines to model the smooth change in moisture with depth. The analysis was carried out using the WinBUGS software and the code on the book's website is described.

Complex data structures can be readily modelled in a Bayesian framework by extending the models considered to data to include latent structures. This concept is illustrated in Chapter 15 by describing a Bayesian analysis for multivariate linear state space modelling. The theory is developed for the Fama–French model of excess return for asset portfolios. For each portfolio the excess return is explained by a linear model with time-varying regression coefficients described by a linear state space model. Three different models are described which allow for different degrees of dependence between the portfolios and across time. A Gibbs algorithm is described for the unknown parameters while an efficient algorithm for simulating from the smoothing distribution for the system parameters is provided. Discrimination between the three possible models is carried out using a likelihood criterion. Efficient computation of the likelihood is also considered. Some results for the regression models for different contrasting types of portfolios are given which confirm the characteristics of these portfolios.

The interest in latent structure models is continued in Chapter 16 with an exposition of mixture distributions, in particular finite normal mixture models. Mixture models can be used as non-parametric density estimates, for cluster analysis and for identifying specific components in a data set. The latent structure in this model indicates mixture components and component membership. A Gibbs algorithm is described for obtaining samples from the posterior distribution. A case study describes the application of mixtures to image analysis for computer tomography (CT) for scans taken from a sheep's carcase in order to determine the quantities of bone, muscle and fat. The basic model is extended so that the spatial smoothness of the image can be taken into account and a Potts model is used to spatially cluster the different components. A brief description of how the method can be extended to estimate the volume of bone, muscle and fat in a carcase is given. Some practical hints on how to set up the models are also given.

Chapter 17 again involves latent structures, this time through latent class models for clustering subgroups of patients or subjects, leading to identification of meaningful clinical phenotypes. Between-subject variability can be large and these differences can be modelled by an unobservable, or latent, process. The first case study involves the identification of subgroups for patients suffering from Parkinson's disease using symptom information. The second case study involves breast cancer patients and their cognitive impairment possibly as a result of therapy. The latent class models involving finite mixture models and trajectory mixture models are reviewed, and various aspects of MCMC implementation discussed. The finite mixture model is used to analyse the Parkinson's disease data using binary and multinomial models in the mixture. The trajectory mixture model is used with regression models to analyse the cognitive impairment of breast cancer patients. The methods indicate two or three latent classes in the case studies. Some WinBUGS code is provided for the trajectory mixture model on the book's website.

A related form of latent structure representation, described in Chapter 18, is hidden Markov models (HMMs) which have been extensively developed and used for the analysis of speech data and DNA sequences. Here a case study involves electrophysiology and the application of HMMs to the identification and sorting of action potentials in extracellular recordings involving firing neurons in the brain. Data have been collected during deep brain stimulation, a popular treatment for advanced Parkinson's disease. The HMM is described in general and in the context of a single neuron firing. An extension to a factorial HMM is considered to model several neurons firing, essentially each neuron having its own HMM. A Gibbs algorithm for posterior simulation is described and applied to simulated data as well as the deep brain stimulation data.

Bayesian models can extend to other constructs to describe complex data structures. Chapter 19 concerns classification and regression trees (CARTs) and, in particular, the Bayesian version, BCART. The BCART model has been found to be highly rated in terms of interpretability. Classification and regression trees give sets of binary rules, repeatedly splitting the predictor variables, to finally end at the predicted value. The case studies here are from epidemiology, concerning a parasite living in the human gut (cryptosporidium), and from medical science, concerning disease of the spine (kyphosis), and extensive analyses of the data sets are given. The CART approach is described and then the BCART is detailed. The BCART approach employs a stochastic search over possible regression trees with different structures and parameters. The original BART employed reversible jump MCMC and is compared with a recent implementation. MATLAB code is available on the book's website and a discussion on implementation is provided. The kyphosis data set involves a binary indicator for disease for subjects after surgery and a small number of predictor variables. The cryptosporidiosis case study involves predicting incidence rates of the disease. The results of the BCART analyses are described and details of implementation provided.

As another example of alternative model constructs, the idea of a Bayesian network (BN) for modelling the relationship between variables is introduced in Chapter 20. A BN can also be considered as a directed graphical model. Some details about software for fitting BNs are given. A case study concerns MRSA transmission in hospitals (see also Chapter 19). The mechanisms behind MRSA transmission and containment have many confounding factors and control strategies may only be effective when used in combination. The BN is developed to investigate the possible role of high bed occupancy on transmission of MRSA while simultaneously taking into account other risk factors. The case study illustrates the use of the iterative BN development cycle approach and then can be used to identify the most influential factors on MRSA transmission and to investigate different scenarios.

In Chapter 21 the ideas of design from a Bayesian perspective are considered in particular in the context of adaptively designing phase I clinical trials which are aimed at determining a maximum tolerated dose (MTD) of a drug. There are only two possible outcomes after the administration of a drug dosage: that is, whether or not a toxic event (or adverse reaction) was observed for the subject and that each response is available before the next subject is treated. The chapter describes how sequential designs which choose the next dose level can be found using SMC (Sequential Monte Carlo). Details of models and priors are given along with the SMC procedure. Results of simulation studies are given. The design criteria considered are based on the posterior distribution of the MTD, and also ways of formally taking into account the safety of subjects in the design are discussed. This chapter initiates the consideration of other computational algorithms that is the focus of the remaining chapters of the book.

Chapter 22 concerns the area of inference known as approximate Bayesian computation (ABC) or likelihood-free inference. Bayesian statistics is reliant on the availability of the likelihood function and the ABC approach is available when the likelihood function is not computationally tractable but simulation of data from it is relatively easy. The case study involves the application of infectious disease models to estimate the transmission rates of nosocomial pathogens within a hospital ward and in particular the case of Methicillin-resistant Staphylococcus aureus (MRSA). A Markov process is used to model the data and simulations from the model are straightforward, but computation of the likelihood is computationally intensive. The ABC inference methods are briefly reviewed and an adaptive SMC algorithm is described and used. Results are given showing the accuracy of the ABC approach.

Chapter 23 describes a computational method, variational Bayes (VB), for Bayesian inference which provides a deterministic solution to finding the posterior instead of one based on simulation, such as MCMC. In certain circumstances VB provides an alternative to simulation which is relatively fast. The chapter gives an overview of some of the properties of VB and application to a case study involving levels of chlorophyll in the waters of the Great Barrier Reef. The data are analysed using a VB approximation for the finite normal mixture models described in Chapter 14 and details of the iterative process are given. The data set is relatively large with over 16 000 observations but results are obtained for fitting the mixture model in a few minutes. Some advice on implementing the VB approach for mixtures, such as initiating the algorithm, is given.

The final investigation into computational Bayesian algorithms is presented in Chapter 24. The focus of this chapter is on ways of developing different MCMC algorithms which combine various features in order to improve performance. The approaches include a delayed rejection algorithm (DRA), a Metropolis adjusted Langevin algorithm (MALA), a repulsive proposal incorporated into a Metropolis–Hastings algorithm, and particle Monte Carlo (PMC). In the regular Metropolis–Hastings algorithm (MHA) a single proposal is made and either accepted or rejected, whereas in this algorithm the possibility of a second proposal is considered if the first proposal is rejected. The MALA uses the derivative of the log posterior to direct proposals in the MHA. In PMC there are parallel chains and the iteration values are known as particles. The particles usually interact in some way. The repulsive proposal (RP) modifies the target distribution to have holes around the particles and so induces a repulsion away from other values. The PMC avoids degeneracy of the particles by using an importance distribution which incorporates repulsion. So here two features are combined to give a hybrid algorithm. Other hybrids include DRA in MALA, MHA with RP. The various hybrid algorithms are compared in terms of statistical efficiency, computation and applicability. The algorithms are compared on a simulated data set and a data set concerning aerosol particle size. Some advantages are given and some caution provided.

The book closes with a chapter that describes PyMCMC, a new software package for Bayesian computation. The package aims to provide a suite of efficient MCMC algorithms, thus alleviating some of the programming load on Bayesian analysts while still providing flexibility of choice and application. PyMCMC is written in Python and takes advantage of Python libraries Numpy, Scipy. It is straightforward to optimize, extensible to C or Fortran, and parallelizable. PyMCMC also provides wrappers for a range of common models, including linear models (with stochastic search), linear and generalized linear mixed models, logit and probit models, independent and spatial mixtures, and a time series suite. As a result, it can be used to address many of the problems considered throughout the book.

1.3 Further Reading

We divide this discussion into parts, dealing with books that focus on theory and methodology, those focused on computation, those providing an exposition of Bayesian methods through a software package, and those written for particular disciplines.

1.3.1 Bayesian Theory and Methodology

Foundations

There are many books that can be considered as foundations of Bayesian thinking. While we focus almost exclusively on reviews of books in this chapter, we acknowledge that there are excellent articles that provide a review of Bayesian statistics. For example, Fienberg (2006) in an article ‘When did Bayesian inference become “Bayesian?”’ charts the history of how the proposition published posthumously in the Transactions of the Royal Society of London (Bayes 1763) became so important for statistics, so that now it has become perhaps the dominant paradigm for doing statistics.

Foundational authors who have influenced modern Bayesian thinking include De Finetti (1974, 1975), who developed the ideas of subjective probability, exchangeability and predictive inference; Lindley (1965, 1980, 1972) and Jeffreys and Zellner (1980), who set the foundations of Bayesian inference; and Jaynes (2003), who developed the field of objective priors. Modern Bayesian foundational texts that have eloquently and clearly embedded Bayesian theory in a decision theory framework include those by Bernardo and Smith (1994, 2000), Berger (2010) and Robert (1994, 2001) which all provide a wide coverage of Bayesian theory, methods and models.

Other texts that may appeal to the reader are the very readable account of Bayesian epistemology provided by Bovens and Hartmann (2003) and the seminal discussion of the theory and practice of probability and statistics from both classical and Bayesian perspectives by DeGroot et al. (1986).

Introductory Texts

The number of introductory books on Bayesian statistics is increasing exponentially. Early texts include those by Schmitt (1969), who gives an introduction to the field through the focal lens of uncertainty analysis, and by Martin (1967), who addresses Bayesian decision problems and Markov chains.

Box and Tiao (1973, 1992) give an early exposition of the use of Bayes' theorem, showing how it relates to more classical statistics with a concern to see in what way the assumed prior distributions may be influencing the conclusions. A more modern exposition of Bayesian statistics is given by Gelman et al. (1995, 2004). This book is currently used as an Honours text for our students in Mathematical Sciences.

Other texts that provide an overview of Bayesian statistical inference, models and applications include those by Meyer (1970), Iversen (1984), Press (1989, 2002) and Leonard and Hsu (1999). The last of these explicitly focuses on interdisciplinary research. The books by Lee (2004b) and Bolstad (2004) also provide informative introductions to this field, particularly for the less mathematically trained.

Two texts by Congdon (2006, 2010) provide a comprehensive coverage of modern Bayesian statistics, and include chapters on such topics as hierarchical models, latent trait models, structural equation models, mixture models and nonlinear regression models. The books also discuss applications in the health and social sciences. The chapters typically form a brief introduction to the salient theory, together with the many references for further reading. In both these books a very short appendix is provided about software (‘Using WinBUGS and BayesX').

Compilations

The maturity of the field of Bayesian statistics is reflected by the emergence of texts that comprise reviews and compilations. One of the most well-known series of such texts is the Proceedings of the Valencia Conferences, held every 4 years in Spain. Edited by Bernardo and co-authors (Bernardo et al. 2003, 2007, 2011, 1992, 1996, 1999, 1980, 1985, 1988), these books showcase frontier methodology and application over the course of the past 30 years.

Edited volumes addressing general Bayesian statistics include The Oxford Handbook of Applied Bayesian Data Analysis by O'Hagan (2010). Edited volumes within specialist areas of statistics are also available. For example, Gelfand et al. (2010) 's Handbook of Spatial Statistics is a collection of chapters from prominent researchers in the field of spatial statistics, and forms a coherent whole while at the same time pointing to the latest research in each contributor's field. Mengersen et al. (2011) have recently edited a series of contributions on methods and applications of Bayesian mixtures. Edited volumes in specialist discipline areas are discussed below.

1.3.2 Bayesian Methodology

Texts on specific areas of Bayesian methodology are also now quite common, as given in Table 1.1.

Table 1.1 Bayesian methodology books.

Author and YearTopicBroemeling (1985)Bayesian analysis of linear modelsSpall (1988)Bayesian analysis of time series and dynamic modelsWest and Harrison (1989, 1997)Bayesian forecasting and dynamic modelsBerry and Stangl (1996)Bayesian biostatisticsNeal (1996)Bayesian learning for neural networksKopparapu and Desai (2001)Bayesian approach to image interpretationDenison (2002)Bayesian methods for nonlinear classification and regressionGhosh and Ramamoorthi (2003)Bayesian non-parametricsBanerjee et al. (2004)Hierarchical modelling and analysis for spatial dataLee (2004a)Bayesian non-parametrics via neural networksCongdon (2005)Bayesian models for categorical dataO'Hagan et al. (2006)Uncertain judgements: eliciting expert probabilitiesLee et al. (2008)Semi-parametric Bayesian analysis of structural equation modelsBroemeling (2009)Bayesian methods for measures of agreementAndo (2010)Bayesian model selection and statistical modellingFox (2010)Bayesian item response modelling (free e-book)Hjort et al. (2010)Bayesian non-parametricsIbrahim (2010)Bayesian survival analysis

1.3.3 Bayesian Computation

There is a wide literature on Monte Carlo methods in general, from different perspectives of statistics, computer science, physics, and so on. There are also many books that contain sections on Bayesian computation as part of a wider scope, and similarly books that focus on narrow sets of algorithms. Finally, the books and documentation associated with Bayesian software most often contain descriptions of the underlying computational approaches. In light of this, here we review a selected set of books targeted at the Bayesian community by Christian Robert, who is a leading authority on modern Bayesian computation and analysis.

Three books by Robert and co-authors provide a comprehensive overview of Monte Carlo methods applicable to Bayesian analysis. The earliest, Discretization and MCMC Convergence Assessment (Robert 1998), describes common MCMC algorithms as well as less well-known ones such as perfect simulation and Langevin Metropolis–Hastings. The text then focuses on convergence diagnostics, largely grouped into those based on graphical plots, stopping rules and confidence bounds. The approaches are illustrated through benchmark examples and case studies.

The second book, by Robert and Casella, Monte Carlo Statistical Methods (Robert and Casella 1999 2004), commences with an introduction (statistical models, likelihood methods, Bayesian methods, deterministic numerical methods, prior distributions and bootstrap methods), then covers random variable generation, Monte Carlo approaches (integration, variance, optimization), Markov chains, popular algorithms (Metropolis–Hastings, slice sampler, two-stage and multi-stage Gibbs, variable selection, reversible jump, perfect sampling, iterated and sequential importance sampling) and convergence.

The more recent text by Robert and Casella, Introducing Monte Carlo Methods in R (Robert and Casella 2009), presents updated ideas about this topic and comprehensive R code. The code is available as freestanding algorithms as well as via an R package, mcsm. This book covers basic R programs, Monte Carlo integration, Metropolis–Hastings and Gibbs algorithms, and issues such as convergence, optimization, monitoring and adaptation.

1.3.4 Bayesian Software

There is now a range of software for Bayesian computation. In the following, we focus on books that describe general purpose software, with accompanying descriptions about Bayesian methods, models and application. These texts can therefore act as introductory (and often sophisticated) texts in their own right. We also acknowledge that there are other texts and papers, both hard copy and online, that describe software built for more specific applications.

WinBUGS at http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/contents.shtml, a free program whose aim is to ‘make practical MCMC methods available to applied statisticians', comes with two manuals, one for WinBUGS (Spiegelhalter et al. 2003) (under the Help button) and the other for GeoBUGS (Thomas et al. 2004) (under the Map button), which together with the examples (also under the Help and Map buttons) explain the software and show how to get started. Ntzoufras (2009) is a useful introductory text which looks at modelling via WinBUGS and includes chapters on generalized linear models and also hierarchical models.

In Albert (2009), a paragraph suffices to introduce us to Bayesian priors, and on the next page we are modelling in R using the LearnBayes R package. This deceptive start disguises an excellent introductory undergraduate text, or ‘teach yourself’ text, with generally minimal theory and a restricted list of references. It is a book to add to the shelf if you are unfamiliar with R and even integrates complex integrals using the Laplace approximation for which there is a function in LearnBayes.

1.3.5 Applications

The number of books on Bayesian statistics for particular disciplines has grown enormously in the past 20 years. In this section we do not attempt a serious review of this literature. Instead, in Table 1.2 we have listed a selection of books on a selection of subjects, indicating the focal topic of each book. Note that there is some inevitable overlap with texts described above, where these describe methodology applicable across disciplines, but are strongly adopted in a particular discipline. The aim is thus to illustrate the breadth of fields covered and to give some pointers to literature within these fields.

Table 1.2 Applied Bayesian books.

Discipline/Author and yearTitleEconomics    Jeffreys and Zellner (1980)Bayesian Analysis in Econometrics and Statistics    Dorfman (1997, 2007)Bayesian Economics through Numerical Methods    Bauwens et al. (1999)Bayesian Inference in Dynamic Econometric Models    Koop (2003)Bayesian EconometricsBusiness    Neapolitan (2003)Learning Bayesian Networks    Rossi et al. (2006)Bayesian Statistics and Marketing    Neapolitan and Jiang (2007)Probabilistic Methods for Financial & Marketing InformaticsHealth    Spiegelhalter (2004)Bayesian Approaches to Clinical Trials and Health-Care Evaluation    Berry (2011)Bayesian Adaptive Methods for Clinical TrialsEarth sciences    Koch (1990)Bayesian Inference with Geodetic ApplicationsEcology    McCarthy (2007)Bayesian Methods for Ecology    King (2009)Bayesian Analysis for Population Ecology    Link and Barker (2009)Bayesian Inference with Ecological ApplicationsSpace    Hobson et al. (2009)Bayesian Methods in CosmologySocial sciences    Jackman (2009)Bayesian Analysis for the Social SciencesBioinformatics    Do et al. (2006)Bayesian Inference for Gene Expression and Proteomics    Mallick et al. (2009)Bayesian Analysis of Gene Expression Data    Dey (2010)Bayesian Modeling in BioinformaticsEngineering    Candy (2009)Bayesian Signal Processing    Yuen (2010)Bayesian Methods for Structural Dynamics and Civil EngineeringArchaeology    Buck et al. (1996)The Bayesian Approach to Interpreting Archaeological Data    Buck and Millard (2004)Tools for Constructing Chronologies

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2

Introduction to MCMC

Anthony N. Pettitt and Candice M. Hincksman

Queensland University of Technology, Brisbane, Australia

2.1 Introduction

Although Markov chain Monte Carlo (MCMC) techniques have been available since Metropolis and Ulam (1949), which is almost as long as the invention of computational Monte Carlo techniques in the 1940s by the Los Alamos physicists working on the atomic bomb, they have only been popular in mainstream statistics since the pioneering paper of Gelfand and Smith (1990) and the subsequent papers in the early 1990s. Gelfand and Smith (1990) introduced Gibbs sampling to the statistics community. It is no coincidence that the BUGS project started in 1989 in Cambridge, UK, and was led by David Spiegelhalter, who had been a PhD student of Adrian Smith's at Oxford. Both share a passion for Bayesian statistics. Recent accounts of MCMC techniques can be found in the book by Gamerman and Lopes (2006) or in Robert and Casella (2011).

Hastings (1970) generalized the Metropolis algorithm but the idea had remained unused in the statistics literature. It was soon realized that Metropolis–Hastings could be used within Gibbs for those situations where it was difficult to implement so-called pure Gibbs. With a clear connection between the expectation–maximization (EM) algorithm, for obtaining modal values of likelihoods or posteriors where there are missing values or latent values, and Gibbs sampling, MCMC approaches were developed for models where there are latent variables used in the likelihood, such as mixed models or mixture models, and models for stochastic processes such as those involving infectious diseases with various unobserved times. Almost synonymous with MCMC is the notion of a hierarchical model where the probability model, likelihood times prior, is defined in terms of conditional distributions and the model can be described by a directed acyclic graph (DAG), a key component of generic Gibbs sampling computation such as BUGS. WinBUGS has the facility to define a model through defining an appropriate DAG and the specification of explicit MCMC algorithms is not required from the user. The important ingredients of MCMC are the following. There is a target distribution, π, of several variables x1, ..., xk. The target distribution in Bayesian statistics is defined as the posterior, p(θ|y