Contents
1.2 A guided tour of decision theory
2.1.2 Coherence and the axioms of probability
2.1.3 Coherent conditional probabilities
2.1.4 The implications of Dutch Book theorems
2.3 Scoring rules and the axioms of probabilities
3.2 Expected utility theory and the theory of means
3.3 The expected utility principle
3.4 The von Neumann–Morgenstern representation theorem
3.4.2 Representation of preferences via expected utility
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
5.2.2 The sure thing principle
5.2.3 Conditional and a posteriori preferences
5.2.5 Utility and expected utility
6.3 State-independent utilities
6.4 Anscombe–Aumann representation theorem
Part Two Statistical Decision Theory
7.1.3 Expected utility principle
7.2.3 Rationality principles and the Likelihood Principle
7.3 The travel insurance example
7.5 Classification and hypothesis tests
7.5.2 Multiple hypothesis testing
8.1 Admissibility and completeness
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
9.2 Geometric and empirical Bayes heuristics
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
10.2.3 The quadratic scoring rules
10.2.4 Scoring rules that are not proper
10.4 Calibration and refinement
10.4.1 The well-calibrated forecaster
10.4.2 Are Bayesians well calibrated?
11.1 The “true model” perspective
11.1.2 Model selection and Bayes factors
11.1.3 Model averaging for prediction and selection
12.2 The travel insurance example revisited
12.3.1 Two-stage finite decision problems
12.4 Trading off immediate gains and information
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
13 Changes in utility as information
13.1 Measuring the value of information
13.1.2 Information from a perfect experiment
13.1.3 Information from a statistical experiment
13.1.4 The distribution of information
13.4 Minimax and the value of information
14.1 Decision-theoretic approaches to sample size
14.1.2 Sample size as a decision problem
14.1.3 Bayes and minimax optimal sample size
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
15.3 Bayesian optimal stopping
15.3.2 Bayes sequential procedure
15.3.3 Bayes truncated procedure
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