목차 일부
CONTENTS
1. INTRODUCTION ... 1
1.1. Thomas Bayes ... 1
1.2. The subjectivist view of probability ... 2
1.3. Bayesian Statistics in perspective ... 3
1.4. An overview of Bayesian Theory .....
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목차 전체
CONTENTS
1. INTRODUCTION ... 1
1.1. Thomas Bayes ... 1
1.2. The subjectivist view of probability ... 2
1.3. Bayesian Statistics in perspective ... 3
1.4. An overview of Bayesian Theory ... 5
1.4.1. Scope ... 5
1.4.2. Foundations ... 5
1.4.3. Generalisations ... 6
1.4.4. Modelling ... 7
1.4.5. Inference ... 7
1.4.6. Remodelling ... 8
1.4.7. Basic formulae ... 8
1.4.8. Non-Bayesian theories ... 9
1.5. A Bayesian reading list ... 9
2. FOUNDATIONS ... 13
2.1. Beliefs and actions ... 13
2.2. Decision problems ... 16
2.2.1. Basic elements ... 16
2.2.2. Formal representation ... 18
2.3. Coherence and quantification ... 23
2.3.1. Events, options and preferences ... 23
2.3.2. Coherent preferences ... 23
2.3.3. Quantification ... 28
2.4. Beliefs and probabilities ... 33
2.4.1. Representation of beliefs ... 33
2.4.2. Revision of beliefs and Bayes theorem ... 38
2.4.3. Conditional independence ... 45
2.4.4. Sequential revision of beliefs ... 47
2.5. Actions and utilities ... 49
2.5.1. Bounded sets of consequences ... 49
2.5.2. Bounded decision problems ... 50
2.5.3. General decision problems ... 54
2.6. Sequential decision problems ... 56
2.6.1. Complex decision problems ... 56
2.6.2. Backward induction ... 59
2.6.3. Design of experiments ... 63
2.7. Inference and information ... 67
2.7.1. Reporting beliefs as a decision problem ... 67
2.7.2. The utility of a probability distribution ... 69
2.7.3. Approximation and discrepancy ... 75
2.7.4. Information ... 77
2.8. Discussion and further references ... 81
2.8. 1. Operational definitions ... 81
2.8.2. Quantitative coherence theories ... 83
2.8.3. Related theories ... 85
2.8.4. Critical issues ... 92
3. GENERALISATIONS ... 105
3.1. Generalised representation of beliefs ... 105
3.1.1. Motivation ... 105
3.1.2. Countable additivity ... 106
3.2. Review of probability theory ... 109
3.2.1. Random quantities and distributions ... 109
3.2.2. Some particular univariate distributions ... 114
3.2.3. Convergence and limit theorems ... 125
3.2.4. Random vectors, Bayes theorem ... 127
3.2.5. Some particular multivariate distributions ... 133
3.3. Generalised options and utilities ... 141
3.3.1. Motivation and preliminaries ... 141
3.3.2. Generalised preferences ... 145
3.3.3. The value of information ... 147
3.4. Generalised information measures ... 150
3.4.1. The general problem of reporting beliefs ... 150
3.4.2. The utility of a general probability distribution ... 151
3.4.3. Generalised approximation and discrepancy ... 154
3.4.4. Generalised information ... 157
3.5. Discussion and further references ... 160
3.5.1. The role of mathematics ... 160
3.5.2. Critical issues ... 161
4. MODELLING ... 165
4.1. Statistical models ... 165
4.1.1. Beliefs and models ... 165
4.2. Exchangeability and related concepts ... 167
4.2.1. Dependence and independence ... 167
4.2.2. Exchangeability and partial exchangeability ... 168
4.3. Models via exchangeability ... 172
4.3.1. The Bernoulli and binomial models ... 172
4.3.2. The multinomial model ... 176
4.3.3. The general model ... 177
4.4. Models via invariance ... 181
4.4.1. The normal model ... 181
4.4.2. The multivariate normal model ... 185
4.4.3. The exponential model ... 187
4.4.4. The geometric model ... 189
4.5. Models via sufficient statistics ... 190
4.5.1. Summary statistics ... 190
4.5.2. Predictive sufficiency and parametric sufficiency ... 191
4.5.3. Sufficiency and the exponential family ... 197
4.5.4. Information measures and the exponential family ... 207
4.6. Models via partial exchangeability ... 209
4.6.1. Models for extended data structures ... 209
4.6.2. Several samples ... 211
4.6.3. Structured layouts ... 217
4.6.4. Covariates ... 219
4.6.5. Hierarchical models ... 222
4.7. Pragmatic aspects ... 226
4.7.1. Finite and infinite exchangeability ... 226
4.7.2. Parametric and nonparametric models ... 228
4.7.3. Model elaboration ... 229
4.7.4. Model simplification ... 233
4.7.5. Prior distributions ... 234
4.8. Discussion and further references ... 235
4.8.1. Representation theorems ... 235
4.8.2. Subjectivity and objectivity ... 236
4.8.3. Critical issues ... 237
5. INFERENCE ... 241
5.1. The Bayesian paradigm ... 241
5.1.1. Observables, beliefs and models ... 241
5.1.2. The role of Bayes theorem ... 242
5.1.3. Predictive and parametric inference ... 243
5.1.4. Sufficiency, ancillarity and stopping rules ... 247
5.1.5. Decisions and inference summaries ... 255
5.1.6. Implementation issues ... 263
5.2. Conjugate analysis ... 265
5.2.1. Conjugate families ... 265
5.2.2. Canonical conjugate analysis ... 269
5.2.3. Approximations with conjugate families ... 279
5.3. Asymptotic analysis ... 285
5.3.1. Discrete asymptotics ... 286
5.3.2. Continuous asymptotics ... 287
5.3.3. Asymptotics under transformations ... 295
5.4. Reference analysis ... 298
5.4.1. Reference decisions ... 299
5.4.2. One-dimensional reference distributions ... 302
5.4.3. Restricted reference distributions ... 316
5.4.4. Nuisance parameters ... 320
5.4.5. Multiparameter problems ... 333
5.5. Numerical approximations ... 339
5.5.1. Laplace approximation ... 340
5.5.2. Iterative quadrature ... 346
5.5.3. Importance sampling ... 348
5.5.4. Sampling-importance-resampling ... 350
5.5.5. Markov chain Monte Carlo ... 353
5.6. Discussion and further references ... 356
5.6.1. An historical footnote ... 356
5.6.2. Prior ignorance ... 357
5.6.3. Robustness ... 367
5.6.4. Hierarchical and empirical Bayes ... 371
5.6.5. Further methodological developments ... 373
5.6.6. Critical issues ... 374
6. REMODELLING ... 377
6.1. Model comparison ... 377
6.1.1. Ranges of models ... 377
6.1.2. Perspectives on model comparison ... 383
6.1.3. Model comparison as a decision problem ... 386
6.1.4. Zero-one utilities and Bayes factors ... 389
6.1.5. General utilities ... 395
6.1.6. Approximation by cross-validation ... 403
6.1.7. Covariate selection ... 407
6.2. Model rejection ... 409
6.2.1. Model rejection through model comparison ... 409
6.2.2. Discrepancy measures for model rejection ... 412
6.2.3. Zero-one discrepancies ... 413
6.2.4. General discrepancies ... 415
6.3. Discussion and further references ... 417
6.3.1. Overview ... 417
6.3.2. Modelling and remodelling ... 418
6.3.3. Critical issues ... 418
A. SUMMARY OF BASIC FORMULAE ... 427
A.1. Probability distributions ... 427
A.2. Inferential processes ... 436
B. NON-BAYESIAN THEORIES ... 443
B.1. Overview ... 443
B.2. Alternative approaches ... 445
B.2.1. Classical decision theory ... 445
B.2.2. Frequentist procedures ... 449
B.2.3. Likelihood inference ... 454
B.2.4. Fiducial, and related theories ... 456
B.3. Stylised inference problems ... 460
B.3.1. Point estimation ... 460
B.3.2. Interval estimation ... 465
B.3.3. Hypothesis testing ... 469
B.3.4. Significance testing ... 475
B.4. Comparative issues ... 478
B.4.1. Conditional and unconditional inference ... 478
B.4.2. Nuisance parameters and marginalisation ... 479
B.4.3. Approaches to prediction ... 482
B.4.4. Aspects of asymptotics ... 485
B.4.5. Model choice criteria ... 486
REFERENCES ... 489
SUBJECT INDEX ... 555
AUTHOR INDEX ... 573
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