Decision Theory Principles and Approaches

by Parmigiani, Giovanni; Inoue, Lurdes

• ISBN13: 9780471496571
• ISBN10: 047149657X
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-05-26
• Publisher: Wiley

Summary

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 Biography

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.

Table of Contents

Preface	p. xiii
Acknowledgments	p. xvii
Introduction	p. 1
Controversies	p. 1
A guided tour of decision theory	p. 6
Foundations	p. 11
Coherence	p. 13
The "Dutch Book" theorem	p. 15
Betting odds	p. 15
Coherence and the axioms of probability	p. 17
Coherent conditional probabilities	p. 20
The implications of Dutch Book theorems	p. 21
Temporal coherence	p. 24
Scoring rules and the axioms of probabilities	p. 26
Exercises	p. 27
Utility	p. 33
St. Petersburg paradox	p. 34
Expected utility theory and the theory of means	p. 37
Utility and means	p. 37
Associative means	p. 38
Functional means	p. 39
The expected utility principle	p. 40
The von Neumann-Morgenstern representation theorem	p. 42
Axioms	p. 42
Representation of preferences via expected utility	p. 44
Allais' criticism	p. 48
Extensions	p. 50
Exercises	p. 50
Utility in action	p. 55
The "standard gamble"	p. 56
Utility of money	p. 57
Certainty equivalents	p. 57
Risk aversion	p. 57
A measure of risk aversion	p. 60
Utility functions for medical decisions	p. 63
Length and quality of life	p. 63
Standard gamble for health states	p. 64
The time trade-off methods	p. 64
Relation between QALYs and utilities	p. 65
Utilities for time in ill health	p. 66
Difficulties in assessing utility	p. 69
Exercises	p. 70
Ramsey and Savage	p. 75
Ramsey's theory	p. 76
Savage's theory	p. 81
Notation and overview	p. 81
The sure thing principle	p. 82
Conditional and a posteriori preferences	p. 85
Subjective probability	p. 85
Utility and expected utility	p. 90
Allais revisited	p. 91
Ellsberg paradox	p. 92
Exercises	p. 93
State independence	p. 97
Horse lotteries	p. 98
State-dependent utilities	p. 100
State-independent utilities	p. 101
Anscombe-Aumann representation theorem	p. 103
Exercises	p. 105
Statistical Decision Theory	p. 109
Decision functions	p. 111
Basic concepts	p. 112
The loss function	p. 112
Minimax	p. 114
Expected utility principle	p. 116
Illustrations	p. 117
Data-based decisions	p. 120
Risk	p. 120
Optimality principles	p. 121
Rationality principles and the Likelihood Principle	p. 123
Nuisance parameters	p. 125
The travel insurance example	p. 126
Randomized decision rules	p. 131
Classification and hypothesis tests	p. 133
Hypothesis testing	p. 133
Multiple hypothesis testing	p. 136
Classification	p. 139
Estimation	p. 140
Point estimation	p. 140
Interval inference	p. 143
Minimax-Bayes connection	p. 144
Exercises	p. 150
Admissibility	p. 155
Admissibility and completeness	p. 156
Admissibility and minimax	p. 158
Admissibility and Bayes	p. 159
Proper Bayes rules	p. 159
Generalized Bayes rules	p. 160
Complete classes	p. 164
Completeness and Bayes	p. 164
Sufficiency and the Rao-Blackwell inequality	p. 165
The Neyman-Pearson lemma	p. 167
Using the same ¿ level across studies with different sample sizes is inadmissible	p. 168
Exercises	p. 171
Shrinkage	p. 175
The Stein effect	p. 176
Geometric and empirical Bayes heuristics	p. 179
Is x too big for $$?	p. 179
Empirical Bayes shrinkage	p. 181
General shrinkage functions	p. 183
Unbiased estimation of the risk of x+g(x)	p. 183
Bayes and minimax shrinkage	p. 185
Shrinkage with different likelihood and losses	p. 188
Exercises	p. 188
Scoring rules	p. 191
Betting and forecasting	p. 192
Scoring rules	p. 193
Definition	p. 193
Proper scoring rules	p. 194
The quadratic scoring rules	p. 195
Scoring rules that are not proper	p. 196
Local scoring rules	p. 197
Calibration and refinement	p. 200
The well-calibrated forecaster	p. 200
Are Bayesians well calibrated?	p. 205
Exercises	p. 207
Choosing models	p. 209
The "true model" perspective	p. 210
Model probabilities	p. 210
Model selection and Bayes factors	p. 212
Model averaging for prediction and selection	p. 213
Model elaborations	p. 216
Exercises	p. 219
Optimal Design	p. 221
Dynamic programming	p. 223
History	p. 224
The travel insurance example revisited	p. 226
Dynamic programming	p. 230
Two-stage finite decision problems	p. 230
More than two stages	p. 233
Trading off immediate gains and information	p. 235
The secretary problem	p. 235
The prophet inequality	p. 239
Sequential clinical trials	p. 241
Two-armed bandit problems	p. 241
Adaptive designs for binary outcomes	p. 242
Variable selection in multiple regression	p. 245
Computing	p. 248
Exercises	p. 251
Changes in utility as information	p. 255
Measuring the value of information	p. 256
The value function	p. 256
Information from a perfect experiment	p. 258
Information from a statistical experiment	p. 259
The distribution of information	p. 264
Examples	p. 265
Tasting grapes	p. 265
Medical testing	p. 266
Hypothesis testing	p. 273
Lindley information	p. 276
Definition	p. 276
Properties	p. 278
Computing	p. 280
Optimal design	p. 281
Minimax and the value of information	p. 283
Exercises	p. 285
Sample size	p. 289
Decision-theoretic approaches to sample size	p. 290
Sample size and power	p. 290
Sample size as a decision problem	p. 290
Bayes and minimax optimal sample size	p. 292
A minimax paradox	p. 293
Goal sampling	p. 295
Computing	p. 298
Examples	p. 302
Point estimation with quadratic loss	p. 302
Composite hypothesis testing	p. 304
A two-action problem with linear utility	p. 306
Lindley information for exponential data	p. 309
Multicenter clinical trials	p. 311
Exercises	p. 316
Stopping	p. 323
Historical note	p. 324
A motivating example	p. 326
Bayesian optimal stopping	p. 328
Notation	p. 328
Bayes sequential procedure	p. 329
Bayes truncated procedure	p. 330
Examples	p. 332
Hypotheses testing	p. 332
An example with equivalence between sequential and fixed sample size designs	p. 336
Sequential sampling to reduce uncertainty	p. 337
The stopping rule principle	p. 339
Stopping rules and the Likelihood Principle	p. 339
Sampling to a foregone conclusion	p. 340
Exercises	p. 342
	p. 345
Notation	p. 345
Relations	p. 349
Probability (density) functions of some distributions	p. 350
Conjugate updating	p. 350
References	p. 353
Index	p. 367
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