Decision Theory

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.

• Preface xiiiAcknowledgments xvii1 Introduction 11.1 Controversies 11.2 A guided tour of decision theory 6Part One Foundations 112 Coherence 132.1 The "Dutch Book" theorem 152.2 Temporal coherence 242.3 Scoring rules and the axioms of probabilities 262.4 Exercises 273 Utility 333.1 St. Petersburg paradox 343.2 Expected utility theory and the theory of means 373.3 The expected utility principle 403.4 The von Neumann–Morgenstern representation theorem 423.5 Allais' criticism 483.6 Extensions 503.7 Exercises 504 Utility in action 554.1 The "standard gamble" 564.2 Utility of money 574.3 Utility functions for medical decisions 634.4 Exercises 705 Ramsey and Savage 755.1 Ramsey's theory 765.2 Savage's theory 815.3 Allais revisited 915.4 Ellsberg paradox 925.5 Exercises 936 State independence 976.1 Horse lotteries 986.2 State-dependent utilities 1006.3 State-independent utilities 1016.4 Anscombe–Aumann representation theorem 1036.5 Exercises 105Part Two Statistical Decision Theory 1097 Decision functions 1117.1 Basic concepts 1127.2 Data-based decisions 1207.3 The travel insurance example 1267.4 Randomized decision rules 1317.5 Classification and hypothesis tests 1337.6 Estimation 1407.7 Minimax–Bayes connections 1447.8 Exercises 1508 Admissibility 1558.1 Admissibility and completeness 1568.2 Admissibility and minimax 1588.3 Admissibility and Bayes 1598.4 Complete classes 1648.5 Using the same α level across studies with different sample sizes is inadmissible 1688.6 Exercises 1719 Shrinkage 1759.1 The Stein effect 1769.2 Geometric and empirical Bayes heuristics 1799.3 General shrinkage functions 1839.4 Shrinkage with different likelihood and losses 1889.5 Exercises 18810 Scoring rules 19110.1 Betting and forecasting 19210.2 Scoring rules 19310.3 Local scoring rules 19710.4 Calibration and refinement 20010.5 Exercises 20711 Choosing models 20911.1 The "true model" perspective 21011.2 Model elaborations 21611.3 Exercises 219Part Three Optimal Design 22112 Dynamic programming 22312.1 History 22412.2 The travel insurance example revisited 22612.3 Dynamic programming 23012.4 Trading off immediate gains and information 23512.5 Sequential clinical trials 24112.6 Variable selection in multiple regression 24512.7 Computing 24812.8 Exercises 25113 Changes in utility as information 25513.1 Measuring the value of information 25613.2 Examples 26513.3 Lindley information 27613.4 Minimax and the value of information 28313.5 Exercises 28514 Sample size 28914.1 Decision-theoretic approaches to sample size 29014.2 Computing 29814.3 Examples 30214.4 Exercises 31615 Stopping 32315.1 Historical note 32415.2 A motivating example 32615.3 Bayesian optimal stopping 32815.4 Examples 33215.5 Sequential sampling to reduce uncertainty 33715.6 The stopping rule principle 33915.7 Exercises 342Appendix 345A.1 Notation 345A.2 Relations 349A.3 Probability (density) functions of some distributions 350A.4 Conjugate updating 350References 353Index 367

“Also anyone interested in learning more about decision theoretic experimental design (a topic of growing interest for example in sequential clinical trials) will find a useful overview and a good starting point for further investigations.”  (Stat Papers, 2011)"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." (Mathematical Reviews, 2011)