Bayesian Essentials with R (Springer Texts in Statistics)

This Bayesian modeling book provides a self-contained entry to computational Bayesian statistics. Focusing on the most standard statistical models and backed up by real datasets and an all-inclusive R (CRAN) package called bayess, the book provides an operational methodology for conducting Bayesian inference, rather than focusing on its theoretical and philosophical justifications. Readers are empowered to participate in the real-life data analysis situations depicted here from the beginning. The stakes are high and the reader determines the outcome. Special attention is paid to the derivation of prior distributions in each case and specific reference solutions are given for each of the models. Similarly, computational details are worked out to lead the reader towards an effective programming of the methods given in the book. In particular, all R codes are discussed with enough detail to make them readily understandable and expandable. This works in conjunction with the bayess package. Bayesian Essentials with R can be used as a textbook at both undergraduate and graduate levels, as exemplified by courses given at Université Paris Dauphine (France), University of Canterbury (New Zealand), and University of British Columbia (Canada). It is particularly useful with students in professional degree programs and scientists to analyze data the Bayesian way. The text will also enhance introductory courses on Bayesian statistics. Prerequisites for the book are an undergraduate background in probability and statistics, if not in Bayesian statistics. A strength of the text is the noteworthy emphasis on the role of models in statistical analysis. This is the new, fully-revised edition to the book Bayesian Core: A Practical Approach to Computational Bayesian Statistics. **Jean-Michel Marin** is Professor of Statistics at Université Montpellier 2, France, and Head of the Mathematics and Modelling research unit. He has written over 40 papers on Bayesian methodology and computing, as well as worked closely with population geneticists over the past ten years. **Christian Robert** is Professor of Statistics at Université Paris-Dauphine, France. He has written over 150 papers on Bayesian Statistics and computational methods and is the author or co-author of seven books on those topics, including The Bayesian Choice (Springer, 2001), winner of the ISBA DeGroot Prize in 2004. He is a Fellow of the Institute of Mathematical Statistics, the Royal Statistical Society and the American Statistical Society. He has been co-editor of the Journal of the Royal Statistical Society, Series B, and in the editorial boards of the Journal of the American Statistical Society, the Annals of Statistics, Statistical Science, and Bayesian Analysis. He is also a recipient of an Erskine Fellowship from the University of Canterbury (NZ) in 2006 and a senior member of the Institut Universitaire de France (2010-2015). Preface 8 Contents 12 1 User's Manual 16 1.1 Expectations 17 1.2 Prerequisites and Further Reading 18 1.3 Styles and Fonts 19 1.4 An Introduction to R 20 1.4.1 Getting Started 21 1.4.2 R Objects 23 1.4.3 Probability Distributions in R 30 1.4.4 Graphical Facilities 31 1.4.5 Writing New R Functions 34 1.4.6 Input and Output in R 36 1.4.7 Administration of R Objects 36 1.5 The bayess Package 37 2 Normal Models 39 2.1 Normal Modeling 40 2.2 The Bayesian Toolkit 42 2.2.1 Posterior Distribution 42 2.2.2 Bayesian Estimates 47 2.2.3 Conjugate Prior Distributions 48 2.2.4 Noninformative Priors 49 2.2.5 Bayesian Credible Intervals 51 2.3 Bayesian Model Choice 52 2.3.1 The Model Index as a Parameter 53 2.3.2 The Bayes Factor 55 2.3.3 The Ban on Improper Priors 57 2.4 Monte Carlo Methods 60 2.4.1 An Approximation Based on Simulations 61 2.4.2 Importance Sampling 63 2.4.3 Approximation of Bayes Factors 66 2.5 Outlier Detection 72 2.6 Exercises 75 3 Regression and Variable Selection 79 3.1 Linear Models 80 3.2 Classical Least Squares Estimator 83 3.3 The Jeffreys Prior Analysis 87 3.4 Zellner's G-Prior Analysis 88 3.4.1 A Semi-noninformative Solution 89 3.4.2 The BayesReg R Function 94 3.4.3 Bayes Factors and Model Comparison 95 3.4.4 Prediction 98 3.5 Markov Chain Monte Carlo Methods 99 3.5.1 Conditionals 100 3.5.2 Two-Stage Gibbs Sampler 101 3.5.3 The General Gibbs Sampler 104 3.6 Variable Selection 105 3.6.1 Deciding on Explanatory Variables 105 3.6.2 G-Prior Distributions for Model Choice 107 3.6.3 A Stochastic Search for the Most Likely Model 110 3.7 Exercises 112 4 Generalized Linear Models 116 4.1 A Generalization of the Linear Model 117 4.1.1 Motivation 117 4.1.2 Link Functions 119 4.2 Metropolis–Hastings Algorithms 121 4.2.1 Definition 122 4.2.2 The Independence Sampler 123 4.2.3 The Random Walk Sampler 124 4.2.4 Output Analysis and Proposal Design 124 4.3 The Probit Model 128 4.3.1 Flat Prior 128 4.3.2 Noninformative G-Priors 130 4.3.3 About Informative Prior Analyses 135 4.4 The Logit Model 137 4.5 Log-Linear Models 140 4.5.1 Contingency Tables 140 4.5.2 Inference Under a Flat Prior 144 4.5.3 Model Choice and Significance of the Parameters 146 4.6 Exercises 150 5 Capture–Recapture Experiments 152 5.1 Inference in a Finite Population 153 5.2 Sampling Models 155 5.2.1 The Binomial Capture Model 155 5.2.2 The Two-Stage Capture–Recapture Model 156 5.2.3 The T-Stage Capture–Recapture Model 161 5.3 Open Populations 165 5.4 Accept–Reject Algorithms 169 5.5 The Arnason–Schwarz Capture–Recapture Model 173 5.5.1 Modeling 174 5.5.2 Gibbs Sampler 178 5.6 Exercises 181 6 Mixture Models 185 6.1 Missing Variable Models 186 6.2 Finite Mixture Models 188 6.3 Mixture Likelihoods and Posteriors 189 6.4 MCMC Solutions 194 6.5 Label Switching Difficulty 204 6.6 Prior Selection 210 6.7 Tempering 211 6.8 Mixtures with an Unknown Number of Components 213 6.9 Exercises 218 7 Time Series 220 7.1 Time-Indexed Data 221 7.1.1 Setting 221 7.1.2 Stability of Time Series 223 7.2 Autoregressive (AR) Models 225 7.2.1 The Models 226 7.2.2 Exploring the Parameter Space by MCMCAlgorithms 230 7.3 Moving Average (MA) Models 237 7.4 ARMA Models and Other Extensions 243 7.5 Hidden Markov Models 247 7.5.1 Basics 248 7.5.2 Forward–Backward Representation 252 7.6 Exercises 259 8 Image Analysis 262 8.1 Image Analysis as a Statistical Problem 263 8.2 Spatial Dependence 263 8.2.1 Grids and Lattices 263 8.2.2 Markov Random Fields 265 8.2.3 The Ising Model 267 8.2.4 The Potts Model 271 8.3 Handling the Normalizing Constant 273 8.3.1 Path Sampling 275 8.3.2 The ABC Method 278 8.3.3 Inference on Potts Models 281 8.4 Image Segmentation 284 8.5 Exercises 292 About the Authors 295 References 296 Index 300 This Bayesian modeling book provides a self-contained entry to computational Bayesian statistics. Focusing on the most standard statistical models and backed up by real datasets and an all-inclusive R (CRAN) package called bayess, the book provides an operational methodology for conducting Bayesian inference, rather than focusing on its theoretical and philosophical justifications. Readers are empowered to participate in the real-life data analysis situations depicted here from the beginning. The stakes are high and the reader determines the outcome. Special attention is paid to the derivation of prior distributions in each case and specific reference solutions are given for each of the models. Similarly, computational details are worked out to lead the reader towards an effective programming of the methods given in the book. In particular, all R codes are discussed with enough detail to make them readily understandable and expandable. This works in conjunction with the bayess package. Bayesian Essentials with R can be used as a textbook at both undergraduate and graduate levels, as exemplified by courses given at Université Paris Dauphine (France), University of Canterbury (New Zealand), and University of British Columbia (Canada). It is particularly useful with students in professional degree programs and scientists to analyze data the Bayesian way. The text will also enhance introductory courses on Bayesian statistics. Prerequisites for the book are an undergraduate background in probability and statistics, if not in Bayesian statistics. A strength of the text is the noteworthy emphasis on the role of models in statistical analysis. This is the new, fully-revised edition to the book Bayesian Core: A Practical Approach to Computational Bayesian Statistics. (Quelle: buch.ch) This Bayesian Modeling Book Provides A Self-contained Entry To Computational Bayesian Statistics. Focusing On The Most Standard Statistical Models And Backed Up By Real Datasets And An All-inclusive R (cran) Package Called Bayess, The Book Provides An Operational Methodology For Conducting Bayesian Inference, Rather Than Focusing On Its Theoretical And Philosophical Justifications. Readers Are Empowered To Participate In The Real-life Data Analysis Situations Depicted Here From The Beginning. The Stakes Are High And The Reader Determines The Outcome. Special Attention Is Paid To The Derivation Of Prior Distributions In Each Case And Specific Reference Solutions Are Given For Each Of The Models. Similarly, Computational Details Are Worked Out To Lead The Reader Towards An Effective Programming Of The Methods Given In The Book. In Particular, All R Codes Are Discussed With Enough Detail To Make Them Readily Understandable And Expandable. User's Manual -- Normal Models -- Regression And Variable Selection -- Generalized Linear Models -- Capture-recapture Experiments -- Mixture Models -- Time Series -- Image Analysis. Jean-michel Marin, Christian P. Robert. Includes Bibliographical References (pages 287-290) And Index. Also Available Online. Front Matter....Pages i-xiv User’s Manual....Pages 1-23 Normal Models....Pages 25-64 Regression and Variable Selection....Pages 65-101 Generalized Linear Models....Pages 103-138 Capture–Recapture Experiments....Pages 139-171 Mixture Models....Pages 173-207 Time Series....Pages 209-250 Image Analysis....Pages 251-283 Back Matter....Pages 285-296