Decision theory: principles and approaches

Decision Theory......Page 4
Contents......Page 10
Preface......Page 16
Acknowledgments......Page 20
1.1 Controversies......Page 24
1.2 A guided tour of decision theory......Page 29
Part One Foundations......Page 34
2 Coherence......Page 36
2.1.1 Betting odds......Page 38
2.1.2 Coherence and the axioms of probability......Page 40
2.1.3 Coherent conditional probabilities......Page 43
2.1.4 The implications of Dutch Book theorems......Page 44
2.2 Temporal coherence......Page 47
2.3 Scoring rules and the axioms of probabilities......Page 49
2.4 Exercises......Page 50
3 Utility......Page 56
3.1 St. Petersburg paradox......Page 57
3.2.1 Utility and means......Page 60
3.2.2 Associative means......Page 61
3.2.3 Functional means......Page 62
3.3 The expected utility principle......Page 63
3.4.1 Axioms......Page 65
3.4.2 Representation of preferences via expected utility......Page 67
3.5 Allais’ criticism......Page 71
3.7 Exercises......Page 73
4 Utility in action......Page 78
4.1 The “standard gamble”......Page 79
4.2.2 Risk aversion......Page 80
4.2.3 A measure of risk aversion......Page 83
4.3.1 Length and quality of life......Page 86
4.3.3 The time trade-off methods......Page 87
4.3.4 Relation between QALYs and utilities......Page 88
4.3.5 Utilities for time in ill health......Page 89
4.3.6 Difficulties in assessing utility......Page 92
4.4 Exercises......Page 93
5 Ramsey and Savage......Page 98
5.1 Ramsey’s theory......Page 99
5.2.1 Notation and overview......Page 104
5.2.2 The sure thing principle......Page 105
5.2.4 Subjective probability......Page 108
5.2.5 Utility and expected utility......Page 113
5.3 Allais revisited......Page 114
5.4 Ellsberg paradox......Page 115
5.5 Exercises......Page 116
6 State independence......Page 120
6.1 Horse lotteries......Page 121
6.2 State-dependent utilities......Page 123
6.3 State-independent utilities......Page 124
6.4 Anscombe–Aumann representation theorem......Page 126
6.5 Exercises......Page 128
Part Two Statistical Decision Theory......Page 132
7 Decision functions......Page 134
7.1.1 The loss function......Page 135
7.1.2 Minimax......Page 137
7.1.3 Expected utility principle......Page 139
7.1.4 Illustrations......Page 140
7.2.1 Risk......Page 143
7.2.2 Optimality principles......Page 144
7.2.3 Rationality principles and the Likelihood Principle......Page 146
7.2.4 Nuisance parameters......Page 148
7.3 The travel insurance example......Page 149
7.4 Randomized decision rules......Page 154
7.5.1 Hypothesis testing......Page 156
7.5.2 Multiple hypothesis testing......Page 159
7.5.3 Classification......Page 162
7.6.1 Point estimation......Page 163
7.6.2 Interval inference......Page 166
7.7 Minimax–Bayes connections......Page 167
7.8 Exercises......Page 173
8 Admissibility......Page 178
8.1 Admissibility and completeness......Page 179
8.2 Admissibility and minimax......Page 181
8.3.1 Proper Bayes rules......Page 182
8.3.2 Generalized Bayes rules......Page 183
8.4.1 Completeness and Bayes......Page 187
8.4.2 Sufficiency and the Rao–Blackwell inequality......Page 188
8.4.3 The Neyman–Pearson lemma......Page 190
8.5 Using the same α level across studies with different sample sizes is inadmissible......Page 191
8.6 Exercises......Page 194
9 Shrinkage......Page 198
9.1 The Stein effect......Page 199
9.2.1 Is x too big for θ?......Page 202
9.2.2 Empirical Bayes shrinkage......Page 204
9.3.1 Unbiased estimation of the risk of x + g(x)......Page 206
9.3.2 Bayes and minimax shrinkage......Page 208
9.5 Exercises......Page 211
10 Scoring rules......Page 214
10.1 Betting and forecasting......Page 215
10.2.1 Definition......Page 216
10.2.2 Proper scoring rules......Page 217
10.2.3 The quadratic scoring rules......Page 218
10.2.4 Scoring rules that are not proper......Page 219
10.3 Local scoring rules......Page 220
10.4.1 The well-calibrated forecaster......Page 223
10.4.2 Are Bayesians well calibrated?......Page 228
10.5 Exercises......Page 230
11 Choosing models......Page 232
11.1.1 Model probabilities......Page 233
11.1.2 Model selection and Bayes factors......Page 235
11.1.3 Model averaging for prediction and selection......Page 236
11.2 Model elaborations......Page 239
11.3 Exercises......Page 242
Part Three Optimal Design......Page 244
12 Dynamic programming......Page 246
12.1 History......Page 247
12.2 The travel insurance example revisited......Page 249
12.3.1 Two-stage finite decision problems......Page 253
12.3.2 More than two stages......Page 256
12.4.1 The secretary problem......Page 258
12.4.2 The prophet inequality......Page 262
12.5.1 Two-armed bandit problems......Page 264
12.5.2 Adaptive designs for binary outcomes......Page 265
12.6 Variable selection in multiple regression......Page 268
12.7 Computing......Page 271
12.8 Exercises......Page 274
13 Changes in utility as information......Page 278
13.1.1 The value function......Page 279
13.1.2 Information from a perfect experiment......Page 281
13.1.3 Information from a statistical experiment......Page 282
13.1.4 The distribution of information......Page 287
13.2.1 Tasting grapes......Page 288
13.2.2 Medical testing......Page 289
13.2.3 Hypothesis testing......Page 296
13.3.1 Definition......Page 299
13.3.2 Properties......Page 301
13.3.3 Computing......Page 303
13.3.4 Optimal design......Page 304
13.4 Minimax and the value of information......Page 306
13.5 Exercises......Page 308
14 Sample size......Page 312
14.1.2 Sample size as a decision problem......Page 313
14.1.3 Bayes and minimax optimal sample size......Page 315
14.1.4 A minimax paradox......Page 316
14.1.5 Goal sampling......Page 318
14.2 Computing......Page 321
14.3.1 Point estimation with quadratic loss......Page 325
14.3.2 Composite hypothesis testing......Page 327
14.3.3 A two-action problem with linear utility......Page 329
14.3.4 Lindley information for exponential data......Page 332
14.3.5 Multicenter clinical trials......Page 334
14.4 Exercises......Page 339
15 Stopping......Page 346
15.1 Historical note......Page 347
15.2 A motivating example......Page 349
15.3.1 Notation......Page 351
15.3.2 Bayes sequential procedure......Page 352
15.3.3 Bayes truncated procedure......Page 353
15.4.1 Hypotheses testing......Page 355
15.4.2 An example with equivalence between sequential and fixed sample size designs......Page 359
15.5 Sequential sampling to reduce uncertainty......Page 360
15.6.1 Stopping rules and the Likelihood Principle......Page 362
15.6.2 Sampling to a foregone conclusion......Page 363
15.7 Exercises......Page 365
A.1 Notation......Page 368
A.2 Relations......Page 372
A.4 Conjugate updating......Page 373
References......Page 376
Index......Page 390
Series......Page 396