ISBN 978-0-387-71598-8 e-ISBN 978-0-387-71599-5

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
1.3.1 Sufficiency 13
1.3.2 The Likelihood Principle 15
1.3.3 Derivation of the Likelihood Principle 18
1.3.4 Implementation of the Likelihood Principle 19
1.3.5 Maximum likelihood estimation 20
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
2.4.1 Randomized estimators 65
2.4.2 Minimaxity 66
2.4.3 Existence of minimax rules and maximin strategy 69
2.4.4 Admissibility 74
2.5 Usual loss functions 77
2.5.1 The quadratic loss 77
2.5.2 The absolute error loss 79
2.5.3 The 0 − 1 loss 80
2.5.4 Intrinsic losses 81
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
3.2.1 Existence 106
3.2.2 Approximations to the prior distribution 108
3.2.3 Maximum entropy priors 109
3.2.4 Parametric approximations 111
3.2.5 Other techniques 113
3.3 Conjugate priors 113
3.3.1 Introduction 113
3.3.2 Justifications 114
3.3.3 Exponential families 115
3.3.4 Conjugate distributions for exponential families 120
3.4 Criticisms and extensions 123
3.5 Noninformative prior distributions 127
3.5.1 Laplace’s prior 127
3.5.2 Invariant priors 128
3.5.3 The Jeffreys prior 129
3.5.4 Reference priors 133
3.5.5 Matching priors 137
3.5.6 Other approaches 140
3.6 Posterior validation and rob
3.7 Exercises 144
3.8 Notes 158
4 Bayesian Point Estimation 165
4.1 Bayesian inference 165
4.1.1 Introduction 165
4.1.2 MAP estimator 166
4.1.3 Likelihood Principle 167
4.1.4 Restricted parameter space 168
4.1.5 Precision of the Bayes estimators 170
4.1.6 Prediction 171
4.1.7 Back to Decision Theory 173
4.2 Bayesian Decision Theory 173
4.2.1 Bayes estimators 173
4.2.2 Conjugate priors 175
4.2.3 Loss estimation 178
4.3 Sampling models 180
4.3.1 Laplace succession rule 180
Contents xix
4.3.2 The tramcar problem 181
4.3.3 Capture-recapture models 182
4.4 The particular case of the normal model 186
4.4.1 Introduction 186
4.4.2 Estimation of variance 187
4.4.3 Linear models and G–priors 190
4.5 Dynamic models 193
4.5.1 Introduction 193
4.5.2 The AR model 196
4.5.3 The MA model 198
4.5.4 The ARMA model 201
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
5.2.1 Decision-theoretic testing 224
5.2.2 The Bayes factor 227
5.2.3 Modification of the prior 229
5.2.4 Point-null hypotheses 230
5.2.5 Improper priors 232
5.2.6 Pseudo-Bayes factors 236
5.3 Comparisons with the classical approach 242
5.3.1 UMP and UMPU tests 242
5.3.2 Least favorable prior distributions 245
5.3.3 Criticisms 247
5.3.4 The p-values 249
5.3.5 Least favorable Bayesian answers 250
5.3.6 The one-sided case 254
5.4 A second decision-theoretic approach 256
5.5 Confidence regions 259
5.5.1 Credible intervals 260
5.5.2 Classical confidence intervals 263
5.5.3 Decision-theoretic evaluation of confidence sets 264
5.6 Exercises 267
5.7 Notes 279
6 Bayesian Calculations 285
6.1 Implementation difficulties 285
6.2 Classical approximation methods 293
6.2.1 Numerical integration 293
6.2.2 Monte Carlo methods 294
6.2.3 Laplace analytic approximation 298
6.3 Markov chain Monte Carlo