Understanding Computational Bayesian StatisticsISBN: 9780470046098
336 pages
December 2009

Description
Providing a solid grounding in statistics while uniquely covering the topics from a Bayesian perspective, Understanding Computational Bayesian Statistics successfully guides readers through this new, cuttingedge approach. With its handson treatment of the topic, the book shows how samples can be drawn from the posterior distribution when the formula giving its shape is all that is known, and how Bayesian inferences can be based on these samples from the posterior. These ideas are illustrated on common statistical models, including the multiple linear regression model, the hierarchical mean model, the logistic regression model, and the proportional hazards model.
The book begins with an outline of the similarities and differences between Bayesian and the likelihood approaches to statistics. Subsequent chapters present key techniques for using computer software to draw Monte Carlo samples from the incompletely known posterior distribution and performing the Bayesian inference calculated from these samples. Topics of coverage include:
 Direct ways to draw a random sample from the posterior by reshaping a random sample drawn from an easily sampled starting distribution
 The distributions from the onedimensional exponential family
 Markov chains and their longrun behavior
 The MetropolisHastings algorithm
 Gibbs sampling algorithm and methods for speeding up convergence
 Markov chain Monte Carlo sampling
Using numerous graphs and diagrams, the author emphasizes a stepbystep approach to computational Bayesian statistics. At each step, important aspects of application are detailed, such as how to choose a prior for logistic regression model, the Poisson regression model, and the proportional hazards model. A related Web site houses R functions and Minitab macros for Bayesian analysis and Monte Carlo simulations, and detailed appendices in the book guide readers through the use of these software packages.
Understanding Computational Bayesian Statistics is an excellent book for courses on computational statistics at the upperlevel undergraduate and graduate levels. It is also a valuable reference for researchers and practitioners who use computer programs to conduct statistical analyses of data and solve problems in their everyday work.
Table of Contents
Preface xi
1 Introduction to Bayesian Statistics I
1.1 The Frequentist Approach to Statistics 1
1.2 The Bayesian Approach to Statistics 3
1.3 Comparing Likelihood and Bayesian Approaches to Statistics 6
1.4 Computational Bayesian Statistics 19
1.5 Purpose and Organization of This Book 20
2 Monte Carlo Sampling from the Posterior 25
2.1 AcceptanceRejectionSampling 27
2.2 SamplingImportanceResampling 33
2.3 AdaptiveRejectionSampling from a LogConcave Distribution 35
2.4 Why Direct Methods Are Inefficient for HighDimension Parameter Space 42
3. Bayesian Inference 47
3.1 Bayesian Inference from the Numerical Posterior 47
3.2 Bayesian Inference from Posterior Random Sample 54
4. Bayesian Statistics Using Conjugate Priors 61
4.1 OneDimensional Exponential Family of Densities 61
4.2 Distributions for Count Data 62
4.3 Distributions for Waiting Times 69
4.4 Normally Distributed Observations with Known Variance 75
4.5 Normally Distributed Observations with Known Mean 78
4.6 Normally Distributed Observations with Unknown Mean and Variance 80
4.7 Multivariate Normal Observations with Known Covariance Matrix 85
4.8 Observations from Normal Linear Regression Model 87
Appendix: Proof of Poisson Process Theorem 97
5. Markov Chains 101
5.1 Stochastic Processes 102
5.2 Markov Chains 103
5.3 TimeInvariant Markov Chains with Finite State Space 104
5.4 Classification of States of a Markov Chain 109
5.5 Sampling from a Markov Chain 114
5.6 TimeReversible Markov Chains and Detailed Balance 117
5.7 Markov Chains with Continuous State Space 120
6. Markov Chain Monte Carlo Sampling from Posterior 127
6.1 MetropolisHastings Algorithm for a Single Parameter 130
6.2 MetropolisHastings Algorithm for Multiple Parameters 137
6.3 Blockwise MetropolisHastings Algorithm 144
6.4 Gibbs Sampling 149
6.5 Summary 150
7 Statistical Inference from a Markov Chain Monte Carlo Sample 159
7.1 Mixing Properties of the Chain 160
7.2 Finding a HeavyTailed Matched Curvature Candidate Density 162
7.3 Obtaining An Approximate Random Sample For Inference 168
Appendix: Procedure for Finding the Matched
Curvature Candidate Density for a Multivariate Parameter 176
8 Logistic Regression 179
8.1 Logistic Regression Model 180
8.2 Computational Bayesian Approach to the Logistic Regression Model 184
8.3 Modelling with the Multiple Logistic Regression Model 192
9 Poisson Regression and Proportional Hazards Model 203
9.1 Poisson Regression Model 204
9.2 Computational Approach to Poisson Regression Model 207
9.3 The Proportional Hazards Model 214
9.4 Computational Bayesian Approach to Proportional Hazards Model 218
10 Gibbs Sampling and Hierarchical Models 235
10.1 Gibbs Sampling Procedure 236
10.2 The Gibbs Sampler for the Normal Distribution 237
10.3 Hierarchical Models and Gibbs Sampling 242
10.4 Modelling Related Populations with Hierarchical Models 244
Appendix: Proof That Improper Jeffrey's Prior Distribution for the Hypervariance Can Lead to an
Improper Postenor 261
11 Going Forward with Markov Chain Monte Carlo 265
A Using the Included Minitab Macros 271
B Using the Included R Functions 289
References 307
Topic Index 313
Author Information
The Wiley Advantage
 The coverage of topics is similar in range to a traditional statistics coursebook.

Exercises are provided throughout the book, along with selected answers.

Each summer concludes with a summary of main points, helping readers understand the key principles related to each discussion.

Bayesian statistics is now more appropriate than ever because computer programs enable the practitioner to work on problems that contain many parameters

The author supplies a calculus refresher anda summary on the use of statistical tables to provide the necessary background information.

A related Web site provides R functions and Minitab macros for Bayesian analysis and Monte Carlo simulations.
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