Index
Note: The entries in italics denote figures
a posteriori probabilities 23–4
Actions 40–1
multistage problems 230, 231, 233
NM theory 42
Savage’s theory 81
Adaptive design 245
Admissibility, decision rules 155–73, 284
Allais, M. 48
Allais paradox
NM theory 48–50
Anscombe–Aumann representation theorem 103–5
Anscombe–Aumann theory 98–108
Archimedean axiom
alternative version 50
Ramsey’s theory (R8) 79
Savage’s theory (S6) 89–90
Arrow–Pratt measure 61
see also Local risk aversion (risk-aversion functions)
Associative means 38–9
Axioms
Anscombe–Aumann theory 100–1, 102, 103–8
Ramsey’s theory 78–9
Savage’s theory 82–4, 85, 86–7, 89–90
Backwards induction 223, 224, 225, 230, 238, 249, 330
Bather, J. 240
Bayes estimators 149, 160, 164, 181–2, 183, 187
Bayesian decision theory (Chernoff quote) 6
Bayesian information criteria (BICs) 213
Bayes risks 121, 126, 132, 145–6
Bayes rules 21, 23, 121–2, 125, 126, 129–31
admissibility 155–6, 159–60, 164–5
relationship to expected utility principle 122
relationship to minimax rules 144, 146, 150
uses
binary classification problems 139
coherent forecasters 206
information analysis 277
interval estimation 144
multistage problems 227, 228, 232
obtaining estimators 181, 185, 187
sample size determination 292–3, 294, 296, 298, 304–5
sequential sampling 329, 330, 332
see also Formal Bayes rules; Generalized Bayes rules
Bayes sequential procedures 329
Bayes’ theorem 31
Bayes truncated procedures 331, 332, 342–4
Before/after axiom, Savage’s theory 85, 92, 130, 226
Bellman, R. E. 224–6
Bernoulli, Daniel 34–6
Bernoulli, Nicholas 36
Bernoulli trials 5
Berry, D. 339
Binary classification problems 139
Blake, William 3
Bonferroni, C. 38–9
Borrowing strength 181
Bounded sequential decision procedures 330
Brier, G. 191–2
Brier scores, see Quadratic scoring rules (Brier scores)
Calibration, probability forecasters 200, 203, 204, 205–7
Certainty equivalents 57, 59, 61, 68, 70
Ramsey’s theory 79
Chance nodes/points 127, 128
Chisini, O. 39
Clinical trials 241–5, 249, 311–16
Coherent probability distributions 20
Complete classes, decision rules 157, 158, 164–5
Completeness, preference relations 43, 51
Conditional expectations 21
Conditionality principle 24
Conditional preferences 84, 85
Conditional probabilities 20, 23–4, 30
Ramsey’s theory 80
Savage’s theory 85
Conditional random variables 21
Conditional values
of perfect information 259
of sample/imperfect information 261
Conjugate updating 350–1
Constantly risk-averse utility functions 62, 272
Constrained backwards induction 251
Continuous spaces 50
Countable additivity 20
Cromwell’s Rule 20
Dawid, A. 207
Decision nodes/points 127
Decision rules/functions 120–2, 123, 126, 129–30
dynamic programming 224
hypothesis testing 133, 169, 172
sequential sampling 329
see also Bayes rules; Minimax rules
Decision trees 127–8, 223, 226, 228–9, 230–1, 233–4, 236, 247, 269, 317
Decreasingly risk-averse utility functions 60
de Finetti, B. 14, 15, 16, 22, 39–40
DeGroot, M. H. 132–3, 200, 207
Discoveries (hypothesis testing) 136–7
Dominance axiom, generalized NM theory 50
Dutch Book theorems 18–19, 20, 22–3, 191
Dynamic programming 223–53, 259, 331, 336
computational complexity 248–51
Edgeworth, F. Y. 111
Ellsberg, D. 92–3
Ellsberg paradox 92–3
Empirical Bayes approach 181–3
Essentially complete classes, decision rules 157, 166, 168
Estimators 140, 141, 147–9, 175–89, 283
admissibility 160, 161–4, 172, 175–6, 179, 188
Ethically neutral propositions 78
Events, de Finetti’s definition 22
Expected utility principle 40–1, 115–17, 121
nuisance parameters 125
relationship to Bayes rules 122
relationship to likelihood principle 123
Expected utility scores 37, 39
Expected values
of perfect information 259, 268
of (sample) information 263, 270, 271, 276
Experiments 258–64
Exponential survival data 309
Extensive form of analysis, multistage problems 230
Fienberg, S. E. 200
Finite multistage problems 223
Finite partition condition 89
Forecasters, see Probability forecasters
Foregone conclusions 341–2
Formal Bayes rules 122, 123, 171, 175, 179, 232, 330
Forward sampling algorithms 249–50
Framing effects 69
Frequentist approach 13, 14, 117, 121, 125, 156, 160, 172, 175
point estimation 141
randomized decision rules 131
sample size determination 293, 295, 296–7
sequential sampling 241, 338, 339, 340
Friedman, Milton 324–5
Game theory 111
Generalized Bayes estimators 187
Generalized Bayes rules 160–1
Goal sampling 295–8
Grape tasting example 265–6
Grundy, P. 291
Horse lotteries 98
Hypothesis testing 133–9, 151, 167–71, 172, 273–6
sample size determination 293–5, 296, 304
sequential sampling 332–6
see also Neyman–Pearson hypothesis testing
Independence axiom, NM theory (NM2) 43, 44, 48–50, 51, 84
Indifference relations 78
Information analysis 255–87
Interval estimation 143–4
James–Stein
estimators 179, 180, 182, 185, 186
theorem 176
Kadane, J. B. 341
Kullback–Leibler (KL) divergence 218, 277–8
Laplace, P.-S. 37–8
Least favorable distributions 146–7, 150
Least refined forecasters 202
Lee, S. 297–8
Likelihood
functions 4, 120, 123, 126, 166, 246, 281, 292, 339
multistage decisions 232
stopping rules 339
ratio statistics 216
Lindley, D. V. 150–1, 197, 225, 226, 228–9, 239, 295
Lindley information 276–83
Linearized diagnostics (model evaluation) 218, 219
Local risk aversion (risk-aversion functions) 61, 62
Local scoring rules 197–9
Logarithmic scoring rules 199, 215, 218
Long-run calibration, probability forecasters 206–7
Loss functions 113, 114, 117, 118, 125, 127, 132, 158, 164, 165, 167
binary classification problems 139
hypothesis testing 133, 137–8, 296, 304
interval estimation 143–4
model choice 211, 212–13, 214, 215
multiple regression 246
point estimation 140, 141, 142, 302
sample size determination 290–1, 293, 294, 295, 296, 297, 298
sequential sampling 338
see also Quadratic loss functions; Regret loss functions
Markov-chain Monte Carlo methods 280
Maximum likelihood
estimators 147–8, 160, 175, 188
MCMC methods (integration) 126
McNeil, B. J. 63
see also Clinical trials; Travel insurance examples
Medical insurance 126–31
Minimal complete classes, decision rules 157, 158, 164, 168
Minimax actions 114, 117, 118, 119, 120
Minimax estimators 147–8, 149, 186
Minimax loss actions 114
Minimax principle 112, 114–16, 130, 150–1
information analysis 283–5
sample sizes 292–5
sequential sampling 341
shrinkage 185–7
Minimax regret actions 114
Minimax regret principle 285
admissibility 158–9
relationship to Bayes rules 144, 146, 150
uses
obtaining estimators 185, 186, 187
point estimation 143
Minimax theorem 150
Models
choice 209–16
comparison 213
elaboration 216–19
Model-specific predictions 214
Model-specific priors 211–12
Monotone likelihood ratio property 165
Monotonicity axiom, Anscombe–Aumann theory (AA5) 102, 103, 104, 107–8
Monte Carlo methods
dynamic programming 249–50
information analysis 280
sample size determination 298–302
Most refined forecasters 202
Multicenter clinical trials 311–16
Multicriteria decisions 137–8
Multiple regression 245–8
Multiplication rule 20
Multistage problems 223, 224, 230–5
Nagumo–Kolmogorov characterization 44
Nested decision problems 248
Neyman factorization theorem 166
Neyman–Pearson
hypothesis testing 134, 138, 168
sample size determination 290
lemma 167–8
NM representation theorem 44–8, 56
generalizations 50
NM theory 33, 41, 42, 43–8, 55, 56
generalizations 50
relationship to Anscombe–Aumann theory 98
relationship to Savage’s theory 82
Normal form of analysis, multistage problems 230
Notation 345–9
Nuisance parameters 125–6, 137, 211
Observed information 260–2, 264
Outcomes 40
Patient horizons (clinical trials) 242
Perfect sharpness, probability forecasters 202
Point estimation 140–3
sample size determination 302–4
Point and stab priors 151
Posterior expected losses 122, 123, 126, 130, 160, 214, 215, 217
binary classification problems 139
hypothesis testing 138, 304, 305
multistage problems 228, 232, 233
point estimation 140, 141, 143, 303
sample size determination 294, 297, 301, 303, 304, 305
Poultry paradox (Lindley) 150–1
Precision parameter, sequential sampling 336
Predictive distributions 214
Preference relations 17, 43, 51
Savage’s theory 82
Preventative surgery decisions 272
Principle of optimality 226
Prior expected losses 116, 117, 118
Probabilistic sensitivity analysis 272–3
Probability
density functions 350
premiums 72
Proper local scoring rules 198–9, 257
Proper scoring rules 194–5, 197, 199, 203, 215
Proper stopping rules 329
Prophet inequality 239–40
Quadratic loss functions 140, 145, 147, 175, 182, 302, 338
Quadratic scoring rules (Brier scores) 195–6, 197–8, 203
Qualitative probabilities 88, 89
Quality-adjusted life years (QALYs) 64–6
Ramsey, F. P. 3, 14, 76–7, 80, 263
Ramsey’s theory 76–81
relationship to Savage’s theory 82
Randomized rules 119–20, 131–3
admissibility 157
Rao–Blackwell theorem 166–7
R-better decision rules 156–7
Receiver operating characteristics curves (ROC curves) 204–5
Refinement, probability forecasters 201–3
Regret loss functions 113, 114–15, 123
Relations 349–50
Relative frequencies 13–14
Risk-aversion functions 62
see also Local risk aversion (risk-aversion functions)
Risk functions 121, 122, 124, 145, 148, 150, 156–7, 158, 159, 160, 169, 170, 171, 177, 189
uses
hypothesis testing 134
sample size determination 292, 303
Risk neutrality 58
Robert, C. P. 175
ROC (receiver operating characteristics) curves 204–5
St. Petersburg game 35
Sample size determination 281, 289–321
Savage density ratio 218
Savage, L. J. 1, 15, 36, 48, 76, 97–8, 114, 115–16, 283
Schervish, M. J. 103
Schuyler, G. L. 324
Scoring rules 26–7, 28, 30, 192–9, 215
Secretary problem 235–9
Seidenfeld, T. 206–7
Sensitivity analyses 69
Sensitivity, medical tests 128, 269
Sequential probability ratio test 325, 338
Sequential problems, see Multistage problems; Sequential sampling
Sequential sampling 323–44
Shannon information 337
Shrinkage, estimators 176, 179–88
“Small world” models (Savage) 209–10, 216
Specificity, medical tests 128, 269
Spherically symmetric distributions 188
Standard gamble approach 56–7
medical decisions 65–6, 67–8, 72
State-dependent utilities 101
State-independent utilities 101–3, 113
Statistical models 3–4
Statistical Research Group, Columbia University 324–5
Stein effect 175–9, 180, 183, 188
Stochastic transformations 201–2
Stopping rules/principle 328, 329, 330, 332, 336, 339
Stopping times 328
Strict risk aversion 58
Strict risk seeking 58
Student’s t distribution 210
Subjective probability (de Finetti) 15
Sufficient statistics 165–7, 171
Sure thing principle 82–3, 93–4
Temporal coherence 24–6
Ramsey’s theory 80
Terminal
decision rules 293, 301, 305, 328
Time trade-off method 64–6
Transition tables 72–3
Transitivity, preference relations 43, 51
Travel insurance examples 126–31, 226–30, 266–73
Truncated James–Stein estimators 179
Truncated procedures 331
Two-armed bandit problem 241–2
Two-stage problems 226, 230–3, 259, 266–7, 292
Type I error probabilities 168–70
Unbiased estimators 183–5
Unknown variances 188
Utilitarianism 111
Utility functions 39–40
Anscombe–Aumann theory 103
continuous spaces 50
relationship to loss functions 113
relationship to value functions 256
risk averse utility functions 58, 59, 60, 61, 62, 272
Savage’s theory 90
uses
information analysis 261, 264, 271, 272, 274, 277
multistage problems 237, 246, 248, 249
sample size determination 307, 308, 312
sequential sampling 326, 328, 330, 333, 334
Utility theory 33–54
von Neumann–Morgenstern representation theorem, see NM representation theorem
von Neumann–Morgenstern theory, see NM theory
Wald, A. 111–13, 117, 121, 325
Wallis, W. A. 324–5
Weight functions, see Loss functions
Well-calibrated forecasters 200–5
Winkler, R. L. 195
Wolpert, R. L. 340
Zelen, M. 297–8
Zero sharpness, forecasters 202