Bayesian Computation with R / Edition 1

Bayesian Computation with R / Edition 1

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by Jim Albert
     
 

There has been a dramatic growth in the development and application of Bayesian inferential methods. Some of this growth is due to the availability of powerful simulation-based algorithms to summarize posterior distributions. There has been also a growing interest in the use of the system R for statistical analyses. R's open source nature, free availability, and

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Overview

There has been a dramatic growth in the development and application of Bayesian inferential methods. Some of this growth is due to the availability of powerful simulation-based algorithms to summarize posterior distributions. There has been also a growing interest in the use of the system R for statistical analyses. R's open source nature, free availability, and large number of contributor packages have made R the software of choice for many statisticians in education and industry.

Bayesian Computation with R introduces Bayesian modeling by the use of computation using the R language. The early chapters present the basic tenets of Bayesian thinking by use of familiar one and two-parameter inferential problems. Bayesian computational methods such as Laplace's method, rejection sampling, and the SIR algorithm are illustrated in the context of a random effects model. The construction and implementation of Markov Chain Monte Carlo (MCMC) methods is introduced. These simulation-based algorithms are implemented for a variety of Bayesian applications such as normal and binary response regression, hierarchical modeling, order-restricted inference, and robust modeling, Algorithms written in R are used to develop Bayesian tests and assess Bayesian models by use of the posterior predictive distribution. The use of R to interface with WinBUGS, a popular MCMC computing language, is described with several illustrative examples.

This book is a suitable companion book for an introductory course on Bayesian methods and is valuable to the statistical practitioner who wishes to learn more about the R language and Bayesian methodology. The LearnBayes package, written by the author and availablefrom the CRAN website, contains all of the R functions described in the book.

The second edition contains several new topics such as the use of mixtures of conjugate priors and the use of Zellner's priors to choose between models in linear regression. There are more illustrations of the construction of informative prior distributions, such as the use of conditional means priors and multivariate normal priors in binary regressions. The new edition contains changes in the R code illustrations according to the latest edition of the LearnBayes package.

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Product Details

ISBN-13:
9780387713847
Publisher:
Springer-Verlag New York, LLC
Publication date:
04/18/2008
Series:
Use R Series
Edition description:
1st ed. 2007. Corr. 2nd printing
Pages:
270
Product dimensions:
6.00(w) x 9.10(h) x 0.60(d)

Table of Contents

1 An Introduction to R 1

1.1 Overview 1

1.2 Exploring a Student Dataset 1

1.2.1 Introduction to the Dataset 1

1.2.2 Reading the Data into R 2

1.2.3 R Commands to Summarize and Graph a Single Batch 2

1.2.4 R Commands to Compare Batches 5

1.2.5 R Commands for Studying Relationships 6

1.3 Exploring the Robustness of the t Statistic 8

1.3.1 Introduction 8

1.3.2 Writing a Function to Compute the t Statistic 9

1.3.3 Programming a Monte Carlo Simulation 10

1.3.4 The Behavior of the True Significance Level Under Different Assumptions 11

1.4 Further Reading 13

1.5 Summary of R Functions 14

1.6 Exercises 15

2 Introduction to Bayesian Thinking 19

2.1 Introduction 19

2.2 Learning About the Proportion of Heavy Sleepers 19

2.3 Using a Discrete Prior 20

2.4 Using a Beta Prior 22

2.5 Using a Histogram Prior 26

2.6 Prediction 28

2.7 Further Reading 34

2.8 Summary of R Functions 34

2.9 Exercises 35

3 Single-Parameter Models 39

3.1 Introduction 39

3.2 Normal Distribution with Known Mean but Unknown Variance 39

3.3 Estimating a Heart Transplant Mortality Rate 41

3.4 An Illustration of Bayesian Robustness 44

3.5 Mixtures of Conjugate Priors 49

3.6 A Bayesian Test of the Fairness of a Coin 52

3.7 Further Reading 57

3.8 Summary of R Functions 57

3.9 Exercises 58

4 Multiparameter Models 63

4.1 Introduction 63

4.2 Normal Data with Both Parameters Unknown 63

4.3 A Multinomial Model 66

4.4 A Bioassay Experiment 69

4.5 Comparing Two Proportions 75

4.6 Further Reading 80

4.7 Summary of R Functions 80

4.8 Exercises 81

5 Introduction to Bayesian Computation 87

5.1 Introduction 87

5.2 Computing Integrals 88

5.3 Setting Up aProblem in R 89

5.4 A Beta-Binomial Model for Overdispersion 90

5.5 Approximations Based on Posterior Modes 94

5.6 The Example 95

5.7 Monte Carlo Method for Computing Integrals 97

5.8 Rejection Sampling 98

5.9 Importance Sampling 101

5.9.1 Introduction 101

5.9.2 Using a Multivariate t as a Proposal Density 103

5.10 Sampling Importance Resampling 105

5.11 Further Reading 105

5.12 Summary of R Functions 109

5.13 Exercises 110

6 Markov Chain Monte Carlo Methods 117

6.1 Introduction 117

6.2 Introduction to discrete Markov Chains 117

6.3 Metropolis-Hastings Algorithms 120

6.4 Gibbs Sampling 122

6.5 MCMC Output Analysis 122

6.6 A Strategy in Bayesian Computing 124

6.7 Learning About a Normal Population from Grouped Data 124

6.8 Example of Output Analysis 129

6.9 Modeling Data with Cauchy Errors 131

6.10 Analysis of the Stanford Heart Transplant Data 140

6.11 Further Reading 145

6.12 Summary of R Functions 146

6.13 Exercises 147

7 Hierarchical Modeling 153

7.1 Introduction 153

7.2 Three Examples 153

7.3 Individual and Combined Estimates 155

7.4 Equal Mortality Rates? 157

7.5 Modeling a Prior Belief of Exchangeability 161

7.6 Posterior Distribution 163

7.7 Simulating from the Posterior 163

7.8 Posterior Inferences 168

7.8.1 Shrinkage 168

7.8.2 Comparing Hospitals 169

7.9 Bayesian Sensitivity Analysis 171

7.10 Posterior Predictive Model Checking 173

7.11 Further Reading 175

7.12 Summary of R Functions 175

7.13 Exercises 176

8 Model Comparison 181

8.1 Introduction 181

8.2 Comparison of Hypotheses 181

8.3 A One-Sided Test of a Normal Mean 182

8.4 A Two-Sided Test of a Normal Mean 185

8.5 Comparing Two Models 186

8.6 Models for Soccer Goals 187

8.7 Is a Baseball Hitter Really Streaky? 190

8.8 A Test of Independence in a Two-Way Contingency Table 194

8.9 Further Reading 199

8.10 Summary of R Functions 199

8.11 Exercises 201

9 Regression Models 205

9.1 Introduction 205

9.2 Normal Linear Regression 205

9.2.1 The Model 205

9.2.2 The Posterior Distribution 206

9.2.3 Prediction of Future Observations 206

9.2.4 Computation 207

9.2.5 Model Checking 207

9.2.6 An Example 208

9.3 Model Selection Using Zellner's Prior 217

9.4 Survival Modeling 222

9.5 Further Reading 227

9.6 Summary of R Functions 227

9.7 Exercises 229

10 Gibbs Sampling 235

10.1 Introduction 235

10.2 Robust Modeling 236

10.3 Binary Response Regression with a Probit Link 240

10.3.1 Missing Data and Gibbs Sampling 240

10.3.2 Proper Priors and Model Selection 243

10.4 Estimating a Table of Means 248

10.4.1 Introduction 248

10.4.2 A Flat Prior Over the Restricted Space 250

10.4.3 A Hierarchical Regression Prior 254

10.4.4 Predicting the Success of Future Students 259

10.5 Further Reading 260

10.6 Summary of R Functions 260

10.7 Exercises 261

11 Using R to Interface with WinBUGS 265

11.1 Introduction to WinBUGS 265

11.2 An R Interface to WinBUGS 266

11.3 MCMC Diagnostics Using the coda Package 267

11.4 A Change-Point Model 268

11.5 A Robust Regression Model 272

11.6 Estimating Career Trajectories 276

11.7 Further Reading 281

11.8 Exercises 282

References 287

Index 293

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