ISBN 978-0-387-71598-8 e-ISBN 978-0-387-71599-5<br><br>1 Introduction 1<br>1.1 Statistical problems and statistical models 1<br>1.2 The Bayesian paradigm as a duality principle 8<br>1.3 Likelihood Principle and Sufficiency Principle 13<br>1.3.1 Sufficiency 13<br>1.3.2 The Likelihood Principle 15<br>1.3.3 Derivation of the Likelihood Principle 18<br>1.3.4 Implementation of the Likelihood Principle 19<br>1.3.5 Maximum likelihood estimation 20<br>1.4 Prior and posterior distributions 22<br>1.5 Improper prior distributions 26<br>1.6 The Bayesian choice 31<br>1.7 Exercises 31<br>1.8 Notes 45<br>2 Decision-Theoretic Foundations 51<br>2.1 Evaluating estimators 51<br>2.2 Existence of a utility function 54<br>2.3 Utility and loss 60<br>2.4 Two optimalities: minimaxity and admissibility 65<br>2.4.1 Randomized estimators 65<br>2.4.2 Minimaxity 66<br>2.4.3 Existence of minimax rules and maximin strategy 69<br>2.4.4 Admissibility 74<br>2.5 Usual loss functions 77<br>2.5.1 The quadratic loss 77<br>2.5.2 The absolute error loss 79<br>2.5.3 The 0 − 1 loss 80<br>2.5.4 Intrinsic losses 81<br>2.6 Criticisms and alternatives 83<br>2.7 Exercises 85<br>2.8 Notes 96<br>3 From Prior Information to Prior Distributions 105<br>3.1 The difficulty in selecting a prior distribution 105<br>3.2 Subjective determination and approximations 106<br>3.2.1 Existence 106<br>3.2.2 Approximations to the prior distribution 108<br>3.2.3 Maximum entropy priors 109<br>3.2.4 Parametric approximations 111<br>3.2.5 Other techniques 113<br>3.3 Conjugate priors 113<br>3.3.1 Introduction 113<br>3.3.2 Justifications 114<br>3.3.3 Exponential families 115<br>3.3.4 Conjugate distributions for exponential families 120<br>3.4 Criticisms and extensions 123<br>3.5 Noninformative prior distributions 127<br>3.5.1 Laplace’s prior 127<br>3.5.2 Invariant priors 128<br>3.5.3 The Jeffreys prior 129<br>3.5.4 Reference priors 133<br>3.5.5 Matching priors 137<br>3.5.6 Other approaches 140<br>3.6 Posterior validation and rob<br>3.7 Exercises 144<br>3.8 Notes 158<br>4 Bayesian Point Estimation 165<br>4.1 Bayesian inference 165<br>4.1.1 Introduction 165<br>4.1.2 MAP estimator 166<br>4.1.3 Likelihood Principle 167<br>4.1.4 Restricted parameter space 168<br>4.1.5 Precision of the Bayes estimators 170<br>4.1.6 Prediction 171<br>4.1.7 Back to Decision Theory 173<br>4.2 Bayesian Decision Theory 173<br>4.2.1 Bayes estimators 173<br>4.2.2 Conjugate priors 175<br>4.2.3 Loss estimation 178<br>4.3 Sampling models 180<br>4.3.1 Laplace succession rule 180<br>Contents xix<br>4.3.2 The tramcar problem 181<br>4.3.3 Capture-recapture models 182<br>4.4 The particular case of the normal model 186<br>4.4.1 Introduction 186<br>4.4.2 Estimation of variance 187<br>4.4.3 Linear models and G–priors 190<br>4.5 Dynamic models 193<br>4.5.1 Introduction 193<br>4.5.2 The AR model 196<br>4.5.3 The MA model 198<br>4.5.4 The ARMA model 201<br>4.6 Exercises 201<br>4.7 Notes 216<br>5 Tests and Confidence Regions 223<br>5.1 Introduction 223<br>5.2 A first approach to testing theory 224<br>5.2.1 Decision-theoretic testing 224<br>5.2.2 The Bayes factor 227<br>5.2.3 Modification of the prior 229<br>5.2.4 Point-null hypotheses 230<br>5.2.5 Improper priors 232<br>5.2.6 Pseudo-Bayes factors 236<br>5.3 Comparisons with the classical approach 242<br>5.3.1 UMP and UMPU tests 242<br>5.3.2 Least favorable prior distributions 245<br>5.3.3 Criticisms 247<br>5.3.4 The p-values 249<br>5.3.5 Least favorable Bayesian answers 250<br>5.3.6 The one-sided case 254<br>5.4 A second decision-theoretic approach 256<br>5.5 Confidence regions 259<br>5.5.1 Credible intervals 260<br>5.5.2 Classical confidence intervals 263<br>5.5.3 Decision-theoretic evaluation of confidence sets 264<br>5.6 Exercises 267<br>5.7 Notes 279<br>6 Bayesian Calculations 285<br>6.1 Implementation difficulties 285<br>6.2 Classical approximation methods 293<br>6.2.1 Numerical integration 293<br>6.2.2 Monte Carlo methods 294<br>6.2.3 Laplace analytic approximation 298<br>6.3 Markov chain Monte Carlo<br><br><br><br><br>

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