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Part Two: Statistical Decision Theory
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Part Two: Statistical Decision Theory
by Lurdes Inoue, Giovanni Parmigiani
Decision Theory
Cover Page
Title Page
Copyright Page
Contents
Preface
Acknowledgments
Chapter 1: Introduction
1.1 Controversies
1.2 A guided tour of decision theory
Part One: Foundations
Chapter 2: Coherence
2.1 The “Dutch Book” theorem
2.1.1 Betting odds
2.1.2 Coherence and the axioms of probability
2.1.3 Coherent conditional probabilities
2.1.4 The implications of Dutch Book theorems
2.2 Temporal coherence
2.3 Scoring rules and the axioms of probabilities
2.4 Exercises
Chapter 3: Utility
3.1 St. Petersburg paradox
3.2 Expected utility theory and the theory of means
3.2.1 Utility and means
3.2.2 Associative means
3.2.3 Functional means
3.3 The expected utility principle
3.4 The von Neumann–Morgenstern representation theorem
3.4.1 Axioms
3.4.2 Representation of preferences via expected utility
3.5 Allais’ criticism
3.6 Extensions
3.7 Exercises
Chapter 4: Utility in action
4.1 The “standard gamble”
4.2 Utility of money
4.2.1 Certainty equivalents
4.2.2 Risk aversion
4.2.3 A measure of risk aversion
4.3 Utility functions for medical decisions
4.3.1 Length and quality of life
4.3.2 Standard gamble for health states
4.3.3 The time trade-off methods
4.3.4 Relation between QALYs and utilities
4.3.5 Utilities for time in ill health
4.3.6 Difficulties in assessing utility
4.4 Exercises
Chapter 5: Ramsey and Savage
5.1 Ramsey’s theory
5.2 Savage’s theory
5.2.1 Notation and overview
5.2.2 The sure thing principle
5.2.3 Conditional and a posteriori preferences
5.2.4 Subjective probability
5.2.5 Utility and expected utility
5.3 Allais revisited
5.4 Ellsberg paradox
5.5 Exercises
Chapter 6: State independence
6.1 Horse lotteries
6.2 State-dependent utilities
6.3 State-independent utilities
6.4 Anscombe–Aumann representation theorem
6.5 Exercises
Part Two: Statistical Decision Theory
Chapter 7: Decision functions
7.1 Basic concepts
7.1.1 The loss function
7.1.2 Minimax
7.1.3 Expected utility principle
7.1.4 Illustrations
7.2 Data-based decisions
7.2.1 Risk
7.2.2 Optimality principles
7.2.3 Rationality principles and the Likelihood Principle
7.2.4 Nuisance parameters
7.3 The travel insurance example
7.4 Randomized decision rules
7.5 Classification and hypothesis tests
7.5.1 Hypothesis testing
7.5.2 Multiple hypothesis testing
7.5.3 Classification
7.6 Estimation
7.6.1 Point estimation
7.6.2 Interval inference
7.7 Minimax–Bayes connections
7.8 Exercises
Chapter 8: Admissibility
8.1 Admissibility and completeness
8.2 Admissibility and minimax
8.3 Admissibility and Bayes
8.3.1 Proper Bayes rules
8.3.2 Generalized Bayes rules
8.4 Complete classes
8.4.1 Completeness and Bayes
8.4.2 Sufficiency and the Rao–Blackwell inequality
8.4.3 The Neyman–Pearson lemma
8.5 Using the same α level across studies with different sample sizes is inadmissible
8.6 Exercises
Chapter 9: Shrinkage
9.1 The Stein effect
9.2 Geometric and empirical Bayes heuristics
9.2.1 Is x too big for θ?
9.2.2 Empirical Bayes shrinkage
9.3 General shrinkage functions
9.3.1 Unbiased estimation of the risk of x + g(x)
9.3.2 Bayes and minimax shrinkage
9.4 Shrinkage with different likelihood and losses
9.5 Exercises
Chapter 10: Scoring rules
10.1 Betting and forecasting
10.2 Scoring rules
10.2.1 Definition
10.2.2 Proper scoring rules
10.2.3 The quadratic scoring rules
10.2.4 Scoring rules that are not proper
10.3 Local scoring rules
10.4 Calibration and refinement
10.4.1 The well-calibrated forecaster
10.4.2 Are Bayesians well calibrated?
10.5 Exercises
Chapter 11: Choosing models
11.1 The “true model” perspective
11.1.1 Model probabilities
11.1.2 Model selection and Bayes factors
11.1.3 Model averaging for prediction and selection
11.2 Model elaborations
11.3 Exercises
Part Three: Optimal Design
Chapter 12: Dynamic programming
12.1 History
12.2 The travel insurance example revisited
12.3 Dynamic programming
12.3.1 Two-stage finite decision problems
12.3.2 More than two stages
12.4 Trading off immediate gains and information
12.4.1 The secretary problem
12.4.2 The prophet inequality
12.5 Sequential clinical trials
12.5.1 Two-armed bandit problems
12.5.2 Adaptive designs for binary outcomes
12.6 Variable selection in multiple regression
12.7 Computing
12.8 Exercises
Chapter 13: Changes in utility as information
13.1 Measuring the value of information
13.1.1 The value function
13.1.2 Information from a perfect experiment
13.1.3 Information from a statistical experiment
13.1.4 The distribution of information
13.2 Examples
13.2.1 Tasting grapes
13.2.2 Medical testing
13.2.3 Hypothesis testing
13.3 Lindley information
13.3.1 Definition
13.3.2 Properties
13.3.3 Computing
13.3.4 Optimal design
13.4 Minimax and the value of information
13.5 Exercises
Chapter 14: Sample size
14.1 Decision-theoretic approaches to sample size
14.1.1 Sample size and power
14.1.2 Sample size as a decision problem
14.1.3 Bayes and minimax optimal sample size
14.1.4 A minimax paradox
14.1.5 Goal sampling
14.2 Computing
14.3 Examples
14.3.1 Point estimation with quadratic loss
14.3.2 Composite hypothesis testing
14.3.3 A two-action problem with linear utility
14.3.4 Lindley information for exponential data
14.3.5 Multicenter clinical trials
14.4 Exercises
Chapter 15: Stopping
15.1 Historical note
15.2 A motivating example
15.3 Bayesian optimal stopping
15.3.1 Notation
15.3.2 Bayes sequential procedure
15.3.3 Bayes truncated procedure
15.4 Examples
15.4.1 Hypotheses testing
15.4.2 An example with equivalence between sequential and fixed sample size designs
15.5 Sequential sampling to reduce uncertainty
15.6 The stopping rule principle
15.6.1 Stopping rules and the Likelihood Principle
15.6.2 Sampling to a foregone conclusion
15.7 Exercises
Appendix
A.1 Notation
A.2 Relations
A.3 Probability (density) functions of some distributions
A.4 Conjugate updating
References
Index
Wiley Series in Probability and Statistics
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Chapter 6: State independence
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Chapter 7: Decision functions
Part Two
Statistical Decision Theory
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