Decision Theory

1Introduction

We statisticians, with our specific concern for uncertainty, are even more liable than other practical men to encounter philosophy, whether we like it or not.

(Savage 1981a)

1.1 Controversies

Statistics is a mature field but within it remain important controversies. These controversies stem from profoundly different perspectives on the meaning of learning and scientific inquiry, and often result in widely different ways of interpreting the same empirical observations.

For example, a controversy that is still very much alive involves how to evaluate the reliability of a prediction or guess. This is, of course, a fundamental issue for statistics, and has implications across a variety of practical activities. Many are captured by a case study on the evaluation of evidence from clinical trials (Ware 1989). We introduce the controversy with an example. You have to guess a secret number. You know it is an integer. You can perform an experiment that would yield either the number before it or the number after it, with equal probability. You know there is no ambiguity about the experimental result or about the experimental answer. You perform this type of experiment twice and get numbers 41 and 43. What is the secret number? Easy, it is 42. Now, how good an answer do you think this is? Are you tempted to say “It is a perfect answer, the secret number has to be 42”? It turns out that not all statisticians think this is so easy. There are at least two opposed perspectives ...