Decision theory principles and approaches
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 fund...
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| Otros Autores: | , |
| Formato: | Libro electrónico |
| Idioma: | Inglés |
| Publicado: |
Chichester, West Sussex :
John Wiley & Sons
c2009.
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| Edición: | 1st edition |
| Colección: | Wiley Series in Probability and Statistics
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| Materias: | |
| Ver en Biblioteca Universitat Ramon Llull: | https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009627609306719 |
Tabla de Contenidos:
- Decision Theory; Contents; Preface; Acknowledgments; 1 Introduction; 1.1 Controversies; 1.2 A guided tour of decision theory; Part One Foundations; 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; 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 means3.3 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; 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 health4.3.6 Difficulties in assessing utility; 4.4 Exercises; 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; 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
- 7 Decision functions7.1 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
- 8 Admissibility8.1 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; 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)
