Decision Theory : Principles and Approaches

by Giovanni Parmigiani (John Hopkins School of Medicine, USA); Lurdes Inoue (University of Washington, USA)

• ISBN13: 9780470746677
• ISBN10: 047074667X
• Format: eBook
• Copyright: 2009-04-01
• Publisher: Wiley

Summary

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 fundamental concepts and outcomes of rational decision making under uncertainty, highlighting the implications for statistical practice. The authors have developed a series of self contained chapters focusing on bridging the gaps between the different fields that have contributed to rational decision making and presenting ideas in a unified framework and notation while respecting and highlighting the different and sometimes conflicting perspectives. This book: Provides a rich collection of techniques and procedures. Discusses the foundational aspects and modern day practice. Links foundations to practical applications in biostatistics, computer science, engineering and economics. Presents different perspectives and controversies to encourage readers to form their own opinion of decision making and statistics. Decision Theory is fundamental to all scientific disciplines, including biostatistics, computer science, economics and engineering. Anyone interested in the whys and wherefores of statistical science will find much to enjoy in this book.

Table of Contents

Preface

Introduction

Controversies

A guided tour of decision theory

Foundations

Coherence

The "Dutch Book" theorem

Temporal coherence

Scoring rules and the axioms of probabilities

Exercises

Utility

St. Petersburg paradox

Expected utility theory and the theory of means

The expected utility principle

The von Neumann-Morgenstern representation theorem

Allais' criticism

Extensions

Exercises

Utility in action

The "standard gamble"

Utility of money

Utility functions for medical decisions

Exercises

Ramsey and Savage

Ramsey's theory

Savage's theory

Allais revisited

Ellsberg paradox

Exercises

State independence

Horse lotteries

State-dependent utilities

State-independent utilities

Anscombe-Aumann representation theorem

Exercises

Statistical Decision Theory

Decision functions

Basic concepts

Data-based decisions

The travel insurance example

Randomized decision rules

Classification and hypothesis tests

Estimation

Minimax-Bayes connections

Exercises

Admissibility

Admissibility and completeness

Admissibility and minimax

Admissibility and Bayes

Complete classes

Using the same ? level across studies with different sample sizes is inadmissible

Exercises

Shrinkage

The Stein effect

Geometric and empirical Bayes heuristics

General shrinkage functions

Shrinkage with different likelihood and losses

Exercises

Scoring rules

Betting and forecasting

Scoring rules

Local scoring rules

Calibration and refinement

Exercises

Choosing models

The "true model" perspective

Model elaborations

Exercises

Optimal Design

Dynamic programming

History

The travel insurance example revisited

Dynamic programming

Trading off immediate gains and information

Sequential clinical trials

Variable selection in multiple regression

Computing

Exercises

Changes in utility as information

Measuring the value of information

Examples

Lindley information

Minimax and the value of information

Exercises

Sample size

Decision-theoretic approaches to sample size

Computing

Examples

Exercises

Stopping

Historical note

A motivating example

Bayesian optimal stopping

Examples

Sequential sampling to reduce uncertainty

The stopping rule principle

Exercises

Appendix

Notation

Relations

Probability (density) functions of some distributions

Conjugate updating

References

Index

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