Contents
Preface to the Paperback Edition vii
Preface to the Second Edition ix
Preface to the First Edition xiii
List of Tables xxiii
List of Figures xxv
1 Introduction 1
1.1 Statistical problems and statistical models 1
1.2 The Bayesian paradigm as a duality principle 8
1.3 Likelihood Principle and Sufficiency Principle 13
1.4 Prior and posterior distributions 22
1.5 Improper prior distributions 26
1.6 The Bayesian choice 31
1.7 Exercises 31
1.8 Notes 45
2 Decision-Theoretic Foundations 51
2.1 Evaluating estimators 51
2.2 Existence of a utility function 54
2.3 Utility and loss 60
2.4 Two optimalities: minimaxity and admissibility 65
2.5 Usual loss functions 77
2.6 Criticisms and alternatives 83
2.7 Exercises 85
2.8 Notes 96
3 From Prior Information to Prior Distributions 105
3.1 The difficulty in selecting a prior distribution 105
3.2 Subjective determination and approximations 106
3.3 Conjugate priors 113
3.4 Criticisms and extensions 123
3.5 Noninformative prior distributions 127
3.6 Posterior validation and robustness 141
3.7 Exercises 144
3.8 Notes 158
4 Bayesian Point Estimation 165
4.1 Bayesian inference 165
4.2 Bayesian Decision Theory 173
4.3 Sampling models 180
4.4 The particular case of the normal model 186
4.5 Dynamic models 193
4.6 Exercises 201
4.7 Notes 216
5 Tests and Confidence Regions 223
5.1 Introduction 223
5.2 A first approach to testing theory 224
5.3 Comparisons with the classical approach 242
5.4 A second decision-theoretic approach 256
5.5 Confidence regions 259
5.6 Exercises 267
5.7 Notes 279
6 Bayesian Calculations 285
6.1 Implementation difficulties 285
6.2 Classical approximation methods 293
6.3 Markov chain Monte Carlo methods 301
6.4 An application to mixture estimation 318
6.5 Exercises 321
6.6 Notes 334
7 Model Choice 343
7.1 Introduction 343
7.2 Standard framework 348
7.3 Monte Carlo and MCMC approximations 356
7.4 Model averaging 366
7.5 Model projections 369
7.6 Goodness-of-fit 374
7.7 Exercises 377
7.8 Notes 386
8 Admissibility and Complete Classes 391
8.1 Introduction 391
8.2 Admissibility of Bayes estimators 391
8.3 Necessary and sufficient admissibility conditions 400
8.4 Complete classes 409
8.5 Necessary admissibility conditions 412
8.6 Exercises 416
8.7 Notes 425
9 Invariance, Haar Measures, and Equivariant Estimators 427
9.1 Invariance principles 427
9.2 The particular case of location parameters 429
9.3 Invariant decision problems 431
9.4 Best equivariant noninformative distributions 436
9.5 The Hunt–Stein theorem 441
9.6 The role of invariance in Bayesian Statistics 445
9.7 Exercises 446
9.8 Notes 454
10 Hierarchical and Empirical Bayes Extensions 457
10.1 Incompletely Specified Priors 457
10.2 Hierarchical Bayes analysis 460
10.3 Optimality of hierarchical Bayes estimators 474
10.4 The empirical Bayes alternative 478
10.5 Empirical Bayes justifications of the Stein effect 484
10.6 Exercises 490
10.7 Notes 502
11 A Defense of the Bayesian Choice 507
A Probability Distributions 519
B Usual Pseudo-random Generators 523
C Notations 527
References 531
Author Index 579
Subject Index 587