This highly acclaimed text, now available in paperback, provides a thorough account of key concepts and theoretical results, with particular emphasis on viewing statistical inference as a special case of decision theory. Information-theoretic concepts play a central role in the development of the theory, which provides, in particular, a detailed discussion of the problem of specification of so-called ‘prior ignorance. The work is written from the authorss committed Bayesian perspective, but an overview of non-Bayesian theories is also provided, and each chapter contains a wide-ranging critical re-examination of controversial issues. The level of mathematics used is such that most material is accessible to readers with knowledge of advanced calculus. In particular, no knowledge of abstract measure theory is assumed, and the emphasis throughout is on statistical concepts rather than rigorous mathematics. The book will be an ideal source for all students and researchers in statistics, mathematics, decision analysis, economic and business studies, and all branches of science and engineering, who wish to further their understanding of Bayesian statistics
Author(s): José M. Bernardo, Adrian F. M. Smith
Series: Wiley Series in Probability and Statistics
Edition: 3
Publisher: John Wiley & Sons
Year: 2000
Language: English
Pages: 611
Tags: Математика;Теория вероятностей и математическая статистика;Математическая статистика;
BAYESIAN THEORY......Page 5
Contents......Page 11
1.1. Thomas Bayes......Page 17
1.2. The subjectivist view of probability......Page 18
1.3. Bayesian Statistics in perspective......Page 19
1.4.2. Foundations......Page 21
1.4.3. Generalisations......Page 22
1.4.5. Inference......Page 23
1.4.7. Basic formulae......Page 24
1.5. A Bayesian reading list......Page 25
2.1. Beliefs and actions......Page 29
2.2.1. Basic elements......Page 32
2.2.2. Formal representation......Page 34
2.3.2. Coherent preferences......Page 39
2.3.3. Quantification......Page 44
2.4.1. Representation of beliefs......Page 49
2.4.2. Revision of beliefs and Bayes’ theorem......Page 54
2.4.3. Conditional independence......Page 61
2.4.4. Sequential revision of beliefs......Page 63
2.5.1. Bounded sets of consequences......Page 65
2.5.2. Bounded decision problems......Page 66
2.5.3. General decision problems......Page 70
2.6.1. Complex decision problems......Page 72
2.6.2. Backward induction......Page 75
2.6.3. Design of experiments......Page 79
2.7.1. Reporting beliefs as a decision problem......Page 83
2.7.2. The utility of a probability distribution......Page 85
2.7.3. Approximation and discrepancy......Page 91
2.7.4. Information......Page 93
2.8.1. Operational definitions......Page 97
2.8.2. Quantitative coherence theories......Page 99
2.8.3. Related theories......Page 101
2.8.4. Critical issues......Page 108
3.1.1. Motivation......Page 121
3.1.2. Countable additivity......Page 122
3.2.1. Random quantities and distributions......Page 125
3.2.2. Some particular univariate distributions......Page 130
3.2.3. Convergence and limit theorems......Page 141
3.2.4. Random vectors, Bayes' theorem......Page 143
3.2.5. Some particular multivariate distributions......Page 149
3.3.1. Motivation and preliminaries......Page 157
3.3.2. Generalised preferences......Page 161
3.3.3. The value of information......Page 163
3.4.1. The general problem of reporting beliefs......Page 166
3.4.2. The utility of a general probability distribution......Page 167
3.4.3. Generalised approximation and discrepancy......Page 170
3.4.4. Generalised information......Page 173
3.5.1. The role of mathematics......Page 176
3.5.2. Critical issues......Page 177
4.1.1. Beliefs and models......Page 181
4.2.1. Dependence and independence......Page 183
4.2.2. Exchangeability and partial exchangeability......Page 184
4.3.1. The Bernoulli and binomial models......Page 188
4.3.2. The multinomial model......Page 192
4.3.3. The general model......Page 193
4.4.1. The normal model......Page 197
4.4.2. The multivariate normal model......Page 201
4.4.3. The exponential model......Page 203
4.4.4. The geometric model......Page 205
4.5.1. Summary statistics......Page 206
4.5.2. Predictive sufficiency and parametric sufficiency......Page 207
4.5.3. Sufficiency and the exponential family......Page 213
4.5.4. Information measures and the exponential family......Page 223
4.6.1. Models for extended data structures......Page 225
4.6.2. Several samples......Page 227
4.6.3. Structured layouts......Page 233
4.6.4. Covariates......Page 235
4.6.5. Hierarchical models......Page 238
4.7.1. Finite and infinite exchangeability......Page 242
4.7.2. Parametric and nonparametric models......Page 244
4.7.3. Model elaboration......Page 245
4.7.4. Model simplification......Page 249
4.7.5. Prior distributions......Page 250
4.8.1 Representation theorems......Page 251
4.8.2. Subjectivity and objectivity......Page 252
4.8.3. Critical issues......Page 253
5.1.1. Observables, beliefs and models......Page 257
5.1.2. The role of Bayes' theorem......Page 258
5.1.3. Predictive and parametric inference......Page 259
5.1.4. Sufficiency. ancillarity and stopping rules......Page 263
5.1.5. Decisions and inference summaries......Page 271
5.1.6. Implementation issues......Page 279
5.2.1. Conjugate families......Page 281
5.2.2. Canonical conjugate analysis......Page 285
5.2.3. Approximations with conjugate families......Page 295
5.3. Asymptotic analysis......Page 301
5.3.1. Discrete asymptotics......Page 302
5.3.2. Continuous asymptotics......Page 303
5.3.3. Asymptotics under transformations......Page 311
5.4. Reference analysis......Page 314
5.4.1. Reference decisions......Page 315
5.4.2. One-dimensional reference distributions......Page 318
5.4.3. Restricted reference distributions......Page 332
5.4.4. Nuisance parameters......Page 336
5.4.5. Multiparameter problems......Page 349
5.5. Numerical approximations......Page 355
5.5.1. Laplace approximation......Page 356
5.5.2. Iterative quadrature......Page 362
5.5.3. Importance sampling......Page 364
5.5.4. Sampling-importance-resampling......Page 366
5.5.5. Markov chain Monte Carlo......Page 369
5.6.1. An historical footnote......Page 372
5.6.2. Prior ignorance......Page 373
5.6.3. Robustness......Page 383
5.6.4. Hierarchical and empirical Bayes......Page 387
5.6.5. Further methodological developments......Page 389
5.6.6. Critical issues......Page 390
6.1.1. Ranges of models......Page 393
6.1.2. Perspectives on model comparison......Page 399
6.1.3. Model comparison as a decision problem......Page 402
6.1.4. Zero-one utilities and Bayes factors......Page 405
6.1.5. General utilities......Page 411
6.1.6. Approximation by cross-validation......Page 419
6.1.7. Covariate selection......Page 423
6.2.1. Model rejection through model comparison......Page 425
6.2.2. Discrepancy measures for model rejection......Page 428
6.2.3. Zero-one discrepancies......Page 429
6.2.4. General discrepancies......Page 431
6.3.1. Overview......Page 433
6.3.3. Critical issues......Page 434
A.1. Probability distributions......Page 443
A.2. Inferential processes......Page 452
B.1. Overview......Page 459
B.2.1. Classical decision theory......Page 461
B.2.2. Frequentist procedures......Page 465
B.2.3. Likelihood inference......Page 470
B.2.4. Fiducial and related theories......Page 472
B.3.1. Point estimation......Page 476
B.3.2. Interval estimation......Page 481
B.3.3. Hypothesis testing......Page 485
8.3.4. Significance testing......Page 491
8.4.1. Conditional and unconditional inference......Page 494
8.4.2. Nuisance paramctcrs and marginalisation......Page 495
B.4.3. Approaches to prediction......Page 498
B.4.4. Aspects of asymptotics......Page 501
B.4.5. Model choice criteria......Page 502
REFERENCES......Page 505
SUBJECT INDEX......Page 571
AUTHOR INDEX......Page 589
