Decision theory : principles and approaches

Decision theory provides a formal framework for making logical choices in the face of uncertainty. Given a set of alternatives, a set of consequences, and a correspondence between those sets, decision theory offers conceptually simple procedures for choice. This book presents an overview of the fundamental concepts and outcomes of rational decision making under uncertainty, highlighting the implications for statistical practice.

The authors have developed a series of self contained chapters focusing on bridging the gaps between the different fields that have contributed to rational decision making and presenting ideas in a unified framework and notation while respecting and highlighting the different and sometimes conflicting perspectives.

This book:

* Provides a rich collection of techniques and procedures.
* Discusses the foundational aspects and modern day practice.
* Links foundations to practical applications in biostatistics, computer science, engineering and economics.
* Presents different perspectives and controversies to encourage readers to form their own opinion of decision making and statistics.

Decision Theory is fundamental to all scientific disciplines, including biostatistics, computer science, economics and engineering. Anyone interested in the whys and wherefores of statistical science will find much to enjoy in this book.

Author Notes

Giovanni Parmigiani is the author of Decision Theory: Principles and Approaches, published by Wiley.

Lurdes Yoshiko Tani Inoue is a Brazilian-born statistician of Japanese descent, who specializes in Bayesian inference. She works as a professor of biostatistics in the University of Washington School of Public Health.

Reviews 1

Choice Review

This book is designed for graduate students in statistics and biostatistics at both the master's and PhD levels. The work's novel feature, as Parmigiani (Johns Hopkins) and Inoue (Univ. of Washington) point out in the preface, is that instead of using a standard textbook format, they have "selected a set of exciting papers and book chapters, and developed a self-contained lecture around each one." These selections fall into three broad categories that constitute the three parts of the book. The first part, "Foundations," discusses utility in two separate chapters, Ramsey and Savage theories, and more. Part 2, "Statistical Decision Theory," includes a chapter titled "Decision Functions." "Optimal Design," the final section, contains chapters titled "Dynamic Programming" and "Sample Size." The authors are to be commended on their attractive approach, which seems to work very well. The extensive list of references at the end, in addition to the key articles in each chapter, makes the book a valuable reference that should be in all libraries supporting advanced students in statistics and its applications. It can also serve as a good textbook in these areas. Summing Up: Recommended. Graduate students and above. R. Bharath emeritus, Northern Michigan University

Preface	p. xiii
Acknowledgments	p. xvii
1 Introduction	p. 1
1.1 Controversies	p. 1
1.2 A guided tour of decision theory	p. 6
Part 1 Foundations	p. 11
2 Coherence	p. 13
2.1 The "Dutch Book" theorem	p. 15
2.1.1 Betting odds	p. 15
2.1.2 Coherence and the axioms of probability	p. 17
2.1.3 Coherent conditional probabilities	p. 20
2.1.4 The implications of Dutch Book theorems	p. 21
2.2 Temporal coherence	p. 24
2.3 Scoring rules and the axioms of probabilities	p. 26
2.4 Exercises	p. 27
3 Utility	p. 33
3.1 St. Petersburg paradox	p. 34
3.2 Expected utility theory and the theory of means	p. 37
3.2.1 Utility and means	p. 37
3.2.2 Associative means	p. 38
3.2.3 Functional means	p. 39
3.3 The expected utility principle	p. 40
3.4 The von Neumann-Morgenstern representation theorem	p. 42
3.4.1 Axioms	p. 42
3.4.2 Representation of preferences via expected utility	p. 44
3.5 Allais' criticism	p. 48
3.6 Extensions	p. 50
3.7 Exercises	p. 50
4 Utility in action	p. 55
4.1 The "standard gamble"	p. 56
4.2 Utility of money	p. 57
4.2.1 Certainty equivalents	p. 57
4.2.2 Risk aversion	p. 57
4.2.3 A measure of risk aversion	p. 60
4.3 Utility functions for medical decisions	p. 63
4.3.1 Length and quality of life	p. 63
4.3.2 Standard gamble for health states	p. 64
4.3.3 The time trade-off methods	p. 64
4.3.4 Relation between QALYs and utilities	p. 65
4.3.5 Utilities for time in ill health	p. 66
4.3.6 Difficulties in assessing utility	p. 69
4.4 Exercises	p. 70
5 Ramsey and Savage	p. 75
5.1 Ramsey's theory	p. 76
5.2 Savage's theory	p. 81
5.2.1 Notation and overview	p. 81
5.2.2 The sure thing principle	p. 82
5.2.3 Conditional and a posteriori preferences	p. 85
5.2.4 Subjective probability	p. 85
5.2.5 Utility and expected utility	p. 90
5.3 Allais revisited	p. 91
5.4 Ellsberg paradox	p. 92
5.5 Exercises	p. 93
6 State independence	p. 97
6.1 Horse lotteries	p. 98
6.2 State-dependent utilities	p. 100
6.3 State-independent utilities	p. 101
6.4 Anscombe-Aumann representation theorem	p. 103
6.5 Exercises	p. 105
Part 2 Statistical Decision Theory	p. 109
7 Decision functions	p. 111
7.1 Basic concepts	p. 112
7.1.1 The loss function	p. 112
7.1.2 Minimax	p. 114
7.1.3 Expected utility principle	p. 116
7.1.4 Illustrations	p. 117
7.2 Data-based decisions	p. 120
7.2.1 Risk	p. 120
7.2.2 Optimality principles	p. 121
7.2.3 Rationality principles and the Likelihood Principle	p. 123
7.2.4 Nuisance parameters	p. 125
7.3 The travel insurance example	p. 126
7.4 Randomized decision rules	p. 131
7.5 Classification and hypothesis tests	p. 133
7.5.1 Hypothesis testing	p. 133
7.5.2 Multiple hypothesis testing	p. 136
7.5.3 Classification	p. 139
7.6 Estimation	p. 140
7.6.1 Point estimation	p. 140
7.6.2 Interval inference	p. 143
7.7 Minimax-Bayes connection	p. 144
7.8 Exercises	p. 150
8 Admissibility	p. 155
8.1 Admissibility and completeness	p. 156
8.2 Admissibility and minimax	p. 158
8.3 Admissibility and Bayes	p. 159
8.3.1 Proper Bayes rules	p. 159
8.3.2 Generalized Bayes rules	p. 160
8.4 Complete classes	p. 164
8.4.1 Completeness and Bayes	p. 164
8.4.2 Sufficiency and the Rao-Blackwell inequality	p. 165
8.4.3 The Neyman-Pearson lemma	p. 167
8.5 Using the same ¿ level across studies with different sample sizes is inadmissible	p. 168
8.6 Exercises	p. 171
9 Shrinkage	p. 175
9.1 The Stein effect	p. 176
9.2 Geometric and empirical Bayes heuristics	p. 179
9.2.1 Is x too big for $$?	p. 179
9.2.2 Empirical Bayes shrinkage	p. 181
9.3 General shrinkage functions	p. 183
9.3.1 Unbiased estimation of the risk of x+g(x)	p. 183
9.3.2 Bayes and minimax shrinkage	p. 185
9.4 Shrinkage with different likelihood and losses	p. 188
9.5 Exercises	p. 188
10 Scoring rules	p. 191
10.1 Betting and forecasting	p. 192
10.2 Scoring rules	p. 193
10.2.1 Definition	p. 193
10.2.2 Proper scoring rules	p. 194
10.2.3 The quadratic scoring rules	p. 195
10.2.4 Scoring rules that are not proper	p. 196
10.3 Local scoring rules	p. 197
10.4 Calibration and refinement	p. 200
10.4.1 The well-calibrated forecaster	p. 200
10.4.2 Are Bayesians well calibrated?	p. 205
10.5 Exercises	p. 207
11 Choosing models	p. 209
11.1 The "true model" perspective	p. 210
11.1.1 Model probabilities	p. 210
11.1.2 Model selection and Bayes factors	p. 212
11.1.3 Model averaging for prediction and selection	p. 213
11.2 Model elaborations	p. 216
11.3 Exercises	p. 219
Part 3 Optimal Design	p. 221
12 Dynamic programming	p. 223
12.1 History	p. 224
12.2 The travel insurance example revisited	p. 226
12.3 Dynamic programming	p. 230
12.3.1 Two-stage finite decision problems	p. 230
12.3.2 More than two stages	p. 233
12.4 Trading off immediate gains and information	p. 235
12.4.1 The secretary problem	p. 235
12.4.2 The prophet inequality	p. 239
12.5 Sequential clinical trials	p. 241
12.5.1 Two-armed bandit problems	p. 241
12.5.2 Adaptive designs for binary outcomes	p. 242
12.6 Variable selection in multiple regression	p. 245
12.7 Computing	p. 248
12.8 Exercises	p. 251
13 Changes in utility as information	p. 255
13.1 Measuring the value of information	p. 256
13.1.1 The value function	p. 256
13.1.2 Information from a perfect experiment	p. 258
13.1.3 Information from a statistical experiment	p. 259
13.1.4 The distribution of information	p. 264
13.2 Examples	p. 265
13.2.1 Tasting grapes	p. 265
13.2.2 Medical testing	p. 266
13.2.3 Hypothesis testing	p. 273
13.3 Lindley information	p. 276
13.3.1 Definition	p. 276
13.3.2 Properties	p. 278
13.3.3 Computing	p. 280
13.3.4 Optimal design	p. 281
13.4 Minimax and the value of information	p. 283
13.5 Exercises	p. 285
14 Sample size	p. 289
14.1 Decision-theoretic approaches to sample size	p. 290
14.1.1 Sample size and power	p. 290
14.1.2 Sample size as a decision problem	p. 290
14.1.3 Bayes and minimax optimal sample size	p. 292
14.1.4 A minimax paradox	p. 293
14.1.5 Goal sampling	p. 295
14.2 Computing	p. 298
14.3 Examples	p. 302
14.3.1 Point estimation with quadratic loss	p. 302
14.3.2 Composite hypothesis testing	p. 304
14.3.3 A two-action problem with linear utility	p. 306
14.3.4 Lindley information for exponential data	p. 309
14.3.5 Multicenter clinical trials	p. 311
14.4 Exercises	p. 316
15 Stopping	p. 323
15.1 Historical note	p. 324
15.2 A motivating example	p. 326
15.3 Bayesian optimal stopping	p. 328
15.3.1 Notation	p. 328
15.3.2 Bayes sequential procedure	p. 329
15.3.3 Bayes truncated procedure	p. 330
15.4 Examples	p. 332
15.4.1 Hypotheses testing	p. 332
15.4.2 An example with equivalence between sequential and fixed sample size designs	p. 336
15.5 Sequential sampling to reduce uncertainty	p. 337
15.6 The stopping rule principle	p. 339
15.6.1 Stopping rules and the Likelihood Principle	p. 339
15.6.2 Sampling to a foregone conclusion	p. 340
15.7 Exercises	p. 342
Appendix p. 345
A.1 Notation	p. 345
A.2 Relations	p. 349
A.3 Probability (density) functions of some distributions	p. 350
A.4 Conjugate updating	p. 350
References	p. 353
Index	p. 367

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