book

Decision Theory

Name: Decision Theory
ISBN: 9780471496571

by Giovanni Parmigiani, Lurdes Inoue

May 2009

Intermediate to advanced

404 pages

11h 10m

English

Wiley

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Cover Page
Title Page
Copyright Page
Contents
Preface
Acknowledgments
Chapter 1: Introduction
1.1 Controversies1.2 A guided tour of decision theory
Part One: Foundations
Chapter 2: Coherence
2.1 The “Dutch Book” theorem2.1.1 Betting odds2.1.2 Coherence and the axioms of probability2.1.3 Coherent conditional probabilities2.1.4 The implications of Dutch Book theorems2.2 Temporal coherence2.3 Scoring rules and the axioms of probabilities2.4 Exercises
Chapter 3: Utility
3.1 St. Petersburg paradox3.2 Expected utility theory and the theory of means3.2.1 Utility and means3.2.2 Associative means3.2.3 Functional means3.3 The expected utility principle3.4 The von Neumann–Morgenstern representation theorem3.4.1 Axioms3.4.2 Representation of preferences via expected utility3.5 Allais’ criticism3.6 Extensions3.7 Exercises

Chapter 4: Utility in action
4.1 The “standard gamble”4.2 Utility of money4.2.1 Certainty equivalents4.2.2 Risk aversion4.2.3 A measure of risk aversion4.3 Utility functions for medical decisions4.3.1 Length and quality of life4.3.2 Standard gamble for health states4.3.3 The time trade-off methods4.3.4 Relation between QALYs and utilities4.3.5 Utilities for time in ill health4.3.6 Difficulties in assessing utility4.4 Exercises
Chapter 5: Ramsey and Savage
5.1 Ramsey’s theory5.2 Savage’s theory5.2.1 Notation and overview5.2.2 The sure thing principle5.2.3 Conditional and a posteriori preferences5.2.4 Subjective probability5.2.5 Utility and expected utility5.3 Allais revisited5.4 Ellsberg paradox5.5 Exercises
Chapter 6: State independence
6.1 Horse lotteries6.2 State-dependent utilities6.3 State-independent utilities6.4 Anscombe–Aumann representation theorem6.5 Exercises
Part Two: Statistical Decision Theory
Chapter 7: Decision functions
7.1 Basic concepts7.1.1 The loss function7.1.2 Minimax7.1.3 Expected utility principle7.1.4 Illustrations7.2 Data-based decisions7.2.1 Risk7.2.2 Optimality principles7.2.3 Rationality principles and the Likelihood Principle7.2.4 Nuisance parameters7.3 The travel insurance example7.4 Randomized decision rules7.5 Classification and hypothesis tests7.5.1 Hypothesis testing7.5.2 Multiple hypothesis testing7.5.3 Classification7.6 Estimation7.6.1 Point estimation7.6.2 Interval inference7.7 Minimax–Bayes connections7.8 Exercises
Chapter 8: Admissibility
8.1 Admissibility and completeness8.2 Admissibility and minimax8.3 Admissibility and Bayes8.3.1 Proper Bayes rules8.3.2 Generalized Bayes rules8.4 Complete classes8.4.1 Completeness and Bayes8.4.2 Sufficiency and the Rao–Blackwell inequality8.4.3 The Neyman–Pearson lemma8.5 Using the same α level across studies with different sample sizes is inadmissible8.6 Exercises
Chapter 9: Shrinkage
9.1 The Stein effect9.2 Geometric and empirical Bayes heuristics9.2.1 Is x too big for θ?9.2.2 Empirical Bayes shrinkage9.3 General shrinkage functions9.3.1 Unbiased estimation of the risk of x + g(x)9.3.2 Bayes and minimax shrinkage9.4 Shrinkage with different likelihood and losses9.5 Exercises
Chapter 10: Scoring rules
10.1 Betting and forecasting10.2 Scoring rules10.2.1 Definition10.2.2 Proper scoring rules10.2.3 The quadratic scoring rules10.2.4 Scoring rules that are not proper10.3 Local scoring rules10.4 Calibration and refinement10.4.1 The well-calibrated forecaster10.4.2 Are Bayesians well calibrated?10.5 Exercises
Chapter 11: Choosing models
11.1 The “true model” perspective11.1.1 Model probabilities11.1.2 Model selection and Bayes factors11.1.3 Model averaging for prediction and selection11.2 Model elaborations11.3 Exercises
Part Three: Optimal Design
Chapter 12: Dynamic programming
12.1 History12.2 The travel insurance example revisited12.3 Dynamic programming12.3.1 Two-stage finite decision problems12.3.2 More than two stages12.4 Trading off immediate gains and information12.4.1 The secretary problem12.4.2 The prophet inequality12.5 Sequential clinical trials12.5.1 Two-armed bandit problems12.5.2 Adaptive designs for binary outcomes12.6 Variable selection in multiple regression12.7 Computing12.8 Exercises
Chapter 13: Changes in utility as information
13.1 Measuring the value of information13.1.1 The value function13.1.2 Information from a perfect experiment13.1.3 Information from a statistical experiment13.1.4 The distribution of information13.2 Examples13.2.1 Tasting grapes13.2.2 Medical testing13.2.3 Hypothesis testing13.3 Lindley information13.3.1 Definition13.3.2 Properties13.3.3 Computing13.3.4 Optimal design13.4 Minimax and the value of information13.5 Exercises
Chapter 14: Sample size
14.1 Decision-theoretic approaches to sample size14.1.1 Sample size and power14.1.2 Sample size as a decision problem14.1.3 Bayes and minimax optimal sample size14.1.4 A minimax paradox14.1.5 Goal sampling14.2 Computing14.3 Examples14.3.1 Point estimation with quadratic loss14.3.2 Composite hypothesis testing14.3.3 A two-action problem with linear utility14.3.4 Lindley information for exponential data14.3.5 Multicenter clinical trials14.4 Exercises
Chapter 15: Stopping
15.1 Historical note15.2 A motivating example15.3 Bayesian optimal stopping15.3.1 Notation15.3.2 Bayes sequential procedure15.3.3 Bayes truncated procedure15.4 Examples15.4.1 Hypotheses testing15.4.2 An example with equivalence between sequential and fixed sample size designs15.5 Sequential sampling to reduce uncertainty15.6 The stopping rule principle15.6.1 Stopping rules and the Likelihood Principle15.6.2 Sampling to a foregone conclusion15.7 Exercises
Appendix
A.1 NotationA.2 RelationsA.3 Probability (density) functions of some distributionsA.4 Conjugate updating
References
Index
Wiley Series in Probability and Statistics

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Decision Theory

by Giovanni Parmigiani, Lurdes Inoue

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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