Decision Theory - Parmigiani, Giovanni

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Résumé :

Preface xiii
Acknowledgments xvii

1 Introduction 1
1.1 Controversies 1
1.2 A guided tour of decision theory 6

Part One Foundations 11

2 Coherence 13
2.1 The Dutch Book theorem 15
2.2 Temporal coherence 24
2.3 Scoring rules and the axioms of probabilities 26
2.4 Exercises 27

3 Utility 33
3.1 St. Petersburg paradox 34
3.2 Expected utility theory and the theory of means 37
3.3 The expected utility principle 40
3.4 The von Neumann-Morgenstern representation theorem 42
3.5 Allais' criticism 48
3.6 Extensions 50
3.7 Exercises 50

4 Utility in action 55
4.1 The standard gamble 56
4.2 Utility of money 57
4.3 Utility functions for medical decisions 63
4.4 Exercises 70

5 Ramsey and Savage 75
5.1 Ramsey's theory 76
5.2 Savage's theory 81
5.3 Allais revisited 91
5.4 Ellsberg paradox 92
5.5 Exercises 93

6 State independence 97
6.1 Horse lotteries 98
6.2 State-dependent utilities 100
6.3 State-independent utilities 101
6.4 Anscombe-Aumann representation theorem 103
6.5 Exercises 105

Part Two Statistical Decision Theory 109

7 Decision functions 111
7.1 Basic concepts 112
7.2 Data-based decisions 120
7.3 The travel insurance example 126
7.4 Randomized decision rules 131
7.5 Classification and hypothesis tests 133
7.6 Estimation 140
7.7 Minimax-Bayes connections 144
7.8 Exercises 150

8 Admissibility 155
8.1 Admissibility and completeness 156
8.2 Admissibility and minimax 158
8.3 Admissibility and Bayes 159
8.4 Complete classes 164
8.5 Using the same ? level across studies with different sample sizes is inadmissible 168
8.6 Exercises 171

9 Shrinkage 175
9.1 The Stein effect 176
9.2 Geometric and empirical Bayes heuristics 179
9.3 General shrinkage functions 183
9.4 Shrinkage with different likelihood and losses 188
9.5 Exercises 188

10 Scoring rules 191
10.1 Betting and forecasting 192
10.2 Scoring rules 193
10.3 Local scoring rules 197
10.4 Calibration and refinement 200
10.5 Exercises 207

11 Choosing models 209
11.1 The true model perspective 210
11.2 Model elaborations 216
11.3 Exercises 219

Part Three Optimal Design 221

12 Dynamic programming 223
12.1 History 224
12.2 The travel insurance example revisited 226
12.3 Dynamic programming 230
12.4 Trading off immediate gains and information 235
12.5 Sequential clinical trials 241
12.6 Variable selection in multiple regression 245
12.7 Computing 248
12.8 Exercises 251

13 Changes in utility as information 255
13.1 Measuring the value of information 256
13.2 Examples 265
13.3 Lindley information 276
13.4 Minimax and the value of information 283
13.5 Exercises 285

14 Sample size 289
14.1 Decision-theoretic approaches to sample size 290
14.2 Computing 298
14.3 Examples 302
14.4 Exercises 316

15 Stopping 323
15.1 Historical note 324
15.2 A motivating example 326
15.3 Bayesian optimal stopping 328
15.4 Examples 332
15.5 Sequential sampling to reduce uncertainty 337
15.6 The stopping rule principle 339
15.7 Exercises 342

Appendix 345
A.1 Notation 345
A.2 Relations 349
A.3 Probability (density) functions of some distributions 350
A.4 Conjugate updating 350

References 353
Index 367