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Introduction and examplesp. 1
Introductionp. 1
Why Bayes?p. 2
Estimating the probability of a rare eventp. 3
Building a predictive modelp. 8
Where we are goingp. 11
Discussion and further referencesp. 12
Belief, probability and exchangeabilityp. 13
Belief functions and probabilitiesp. 13
Events, partitions and Bayes' rulep. 14
Independencep. 17
Random variablesp. 17
Discrete random variablesp. 18
Continuous random variablesp. 19
Descriptions of distributionsp. 21
Joint distributionsp. 23
Independent random variablesp. 26
Exchangeabilityp. 27
de Finetti's theoremp. 29
Discussion and further referencesp. 30
One-parameter modelsp. 31
The binomial modelp. 31
Inference for exchangeable binary datap. 35
Confidence regionsp. 41
The Poisson modelp. 43
Posterior inferencep. 45
Example: Birth ratesp. 48
Exponential families and conjugate priorsp. 51
Discussion and further referencesp. 52
Monte Carlo approximationp. 53
The Monte Carlo methodp. 53
Posterior inference for arbitrary functionsp. 57
Sampling from predictive distributionsp. 60
Posterior predictive model checkingp. 62
Discussion and further referencesp. 65
The normal modelp. 67
The normal modelp. 67
Inference for the mean, conditional on the variancep. 69
Joint inference for the mean and variancep. 73
Bias, variance and mean squared errorp. 79
Prior specification based on expectationsp. 83
The normal model for non-normal datap. 84
Discussion and further referencesp. 86
Posterior approximation with the Gibbs samplerp. 89
A semiconjugate prior distributionp. 89
Discrete approximationsp. 90
Sampling from the conditional distributionsp. 92
Gibbs samplingp. 93
General properties of the Gibbs samplerp. 96
Introduction to MCMC diagnosticsp. 98
Discussion and further referencesp. 104
The multivariate normal modelp. 105
The multivariate normal densityp. 105
A semiconjugate prior distribution for the meanp. 107
The inverse-Wishart distributionp. 109
Gibbs sampling of the mean and covariancep. 112
Missing data and imputationp. 115
Discussion and further referencesp. 123
Group comparisons and hierarchical modelingp. 125
Comparing two groupsp. 125
Comparing multiple groupsp. 130
Exchangeability and hierarchical modelsp. 131
The hierarchical normal modelp. 132
Posterior inferencep. 133
Example: Math scores in U.S. public schoolsp. 135
Prior distributions and posterior approximationp. 137
Posterior summaries and shrinkagep. 140
Hierarchical modeling of means and variancesp. 143
Analysis of math score datap. 145
Discussion and further referencesp. 146
Linear regressionp. 149
The linear regression modelp. 149
Least squares estimation for the oxygen uptake datap. 153
Bayesian estimation for a regression modelp. 154
A semiconjugate prior distributionp. 154
Default and weakly informative prior distributionsp. 155
Model selectionp. 160
Bayesian model comparisonp. 163
Gibbs sampling and model averagingp. 167
Discussion and further referencesp. 170
Nonconjugate priors and Metropolis-Hastings algorithmsp. 171
Generalized linear modelsp. 171
The Metropolis algorithmp. 173
The Metropolis algorithm for Poisson regressionp. 179
Metropolis, Metropolis-Hastings and Gibbsp. 181
The Metropolis-Hastings algorithmp. 182
Why does the Metropolis-Hastings algorithm work?p. 184
Combining the Metropolis and Gibbs algorithmsp. 187
A regression model with correlated errorsp. 188
Analysis of the ice core datap. 191
Discussion and further referencesp. 192
Linear and generalized linear mixed effects modelsp. 195
A hierarchical regression modelp. 195
Full conditional distributionsp. 198
Posterior analysis of the math score datap. 200
Generalized linear mixed effects modelsp. 201
A Metropolis-Gibbs algorithm for posterior approximationp. 202
Analysis of tumor location datap. 203
Discussion and further referencesp. 207
Latent variable methods for ordinal datap. 209
Ordered probit regression and the rank likelihoodp. 209
Probit regressionp. 211
Transformation models and the rank likelihoodp. 214
The Gaussian copula modelp. 217
Rank likelihood for copula estimationp. 218
Discussion and further referencesp. 223
Exercisesp. 225
Common distributionsp. 253
Referencesp. 259
Indexp. 267
Table of Contents provided by Ingram. All Rights Reserved.

This book provides a compact self-contained introduction to the theory and application of Bayesian statistical methods. The book is accessible to readers with only a basic familiarity with probability, yet allows more advanced readers to quickly grasp the principles underlying Bayesian theory and methods. The examples and computer code allow the reader to understand and implement basic Bayesian data analyses using standard statistical models and to extend the standard models to specialized data analysis situations. The book begins with fundamental notions such as probability, exchangeability and Bayes' rule, and ends with modern topics such as variable selection in regression, generalized linear mixed effects models and semiparametric copula estimation. Numerous examples from the social, biological and physical sciences show how to implement these methodologies in practice.Monte Carlo summaries of posterior distributions play an important role in Bayesian data analysis. The open-source R statistical computing environment provides sufficient functionality to make Monte Carlo estimation very easy for a large number of statistical models, and example R-code is provided throughout the text. Much of the example code can be run ``as is'' in R, and essentially all of it can be run after downloading the relevant datasets from the companion website for this book.