Decision Theory: Principles and Approaches / Edition 1

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Overview

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.

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Editorial Reviews

From the Publisher

“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)
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Product Details

Table of Contents

Preface xiii

Acknowledgments xvii

1 Introduction 1

1.1 Controversies 1

1.2 A guided tour of decision theory 6

Part 1 Foundations 11

2 Coherence 13

2.1 The "Dutch Book" theorem 15

2.1.1 Betting odds 15

2.1.2 Coherence and the axioms of probability 17

2.1.3 Coherent conditional probabilities 20

2.1.4 The implications of Dutch Book theorems 21

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

3.2.1 Utility and means 37

3.2.2 Associative means 38

3.2.3 Functional means 39

3.3 The expected utility principle 40

3.4 The von Neumann-Morgenstern representation theorem 42

3.4.1 Axioms 42

3.4.2 Representation of preferences via expected utility 44

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

4.2.1 Certainty equivalents 57

4.2.2 Risk aversion 57

4.2.3 A measure of risk aversion 60

4.3 Utility functions for medical decisions 63

4.3.1 Length and quality of life 63

4.3.2 Standard gamble for health states 64

4.3.3 The time trade-off methods 64

4.3.4 Relation between QALYs and utilities 65

4.3.5 Utilities for time in ill health 66

4.3.6 Difficulties in assessing utility 69

4.4 Exercises 70

5 Ramsey and Savage 75

5.1 Ramsey's theory 76

5.2 Savage's theory 81

5.2.1 Notation and overview 81

5.2.2 The sure thing principle 82

5.2.3 Conditional and a posteriori preferences 85

5.2.4 Subjective probability 85

5.2.5 Utility and expected utility 90

5.3 Allais revisited 91

5.4 Ellsbergparadox 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 2 Statistical Decision Theory 109

7 Decision functions 111

7.1 Basic concepts 112

7.1.1 The loss function 112

7.1.2 Minimax 114

7.1.3 Expected utility principle 116

7.1.4 Illustrations 117

7.2 Data-based decisions 120

7.2.1 Risk 120

7.2.2 Optimality principles 121

7.2.3 Rationality principles and the Likelihood Principle 123

7.2.4 Nuisance parameters 125

7.3 The travel insurance example 126

7.4 Randomized decision rules 131

7.5 Classification and hypothesis tests 133

7.5.1 Hypothesis testing 133

7.5.2 Multiple hypothesis testing 136

7.5.3 Classification 139

7.6 Estimation 140

7.6.1 Point estimation 140

7.6.2 Interval inference 143

7.7 Minimax-Bayes connection 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

8.3.1 Proper Bayes rules 159

8.3.2 Generalized Bayes rules 160

8.4 Complete classes 164

8.4.1 Completeness and Bayes 164

8.4.2 Sufficiency and the Rao-Blackwell inequality 165

8.4.3 The Neyman-Pearson lemma 167

8.5 Using the same α 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

9.2.1 Is x too big for $$? 179

9.2.2 Empirical Bayes shrinkage 181

9.3 General shrinkage functions 183

9.3.1 Unbiased estimation of the risk of x+g(x) 183

9.3.2 Bayes and minimax shrinkage 185

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

10.2.1 Definition 193

10.2.2 Proper scoring rules 194

10.2.3 The quadratic scoring rules 195

10.2.4 Scoring rules that are not proper 196

10.3 Local scoring rules 197

10.4 Calibration and refinement 200

10.4.1 The well-calibrated forecaster 200

10.4.2 Are Bayesians well calibrated? 205

10.5 Exercises 207

11 Choosing models 209

11.1 The "true model" perspective 210

11.1.1 Model probabilities 210

11.1.2 Model selection and Bayes factors 212

11.1.3 Model averaging for prediction and selection 213

11.2 Model elaborations 216

11.3 Exercises 219

Part 3 Optimal Design 221

12 Dynamic programming 223

12.1 History 224

12.2 The travel insurance example revisited 226

12.3 Dynamic programming 230

12.3.1 Two-stage finite decision problems 230

12.3.2 More than two stages 233

12.4 Trading off immediate gains and information 235

12.4.1 The secretary problem 235

12.4.2 The prophet inequality 239

12.5 Sequential clinical trials 241

12.5.1 Two-armed bandit problems 241

12.5.2 Adaptive designs for binary outcomes 242

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

13.1.1 The value function 256

13.1.2 Information from a perfect experiment 258

13.1.3 Information from a statistical experiment 259

13.1.4 The distribution of information 264

13.2 Examples 265

13.2.1 Tasting grapes 265

13.2.2 Medical testing 266

13.2.3 Hypothesis testing 273

13.3 Lindley information 276

13.3.1 Definition 276

13.3.2 Properties 278

13.3.3 Computing 280

13.3.4 Optimal design 281

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

14.1.1 Sample size and power 290

14.1.2 Sample size as a decision problem 290

14.1.3 Bayes and minimax optimal sample size 292

14.1.4 A minimax paradox 293

14.1.5 Goal sampling 295

14.2 Computing 298

14.3 Examples 302

14.3.1 Point estimation with quadratic loss 302

14.3.2 Composite hypothesis testing 304

14.3.3 A two-action problem with linear utility 306

14.3.4 Lindley information for exponential data 309

14.3.5 Multicenter clinical trials 311

14.4 Exercises 316

15 Stopping 323

15.1 Historical note 324

15.2 A motivating example 326

15.3 Bayesian optimal stopping 328

15.3.1 Notation 328

15.3.2 Bayes sequential procedure 329

15.3.3 Bayes truncated procedure 330

15.4 Examples 332

15.4.1 Hypotheses testing 332

15.4.2 An example with equivalence between sequential and fixed sample size designs 336

15.5 Sequential sampling to reduce uncertainty 337

15.6 The stopping rule principle 339

15.6.1 Stopping rules and the Likelihood Principle 339

15.6.2 Sampling to a foregone conclusion 340

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

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