Bayesian Statistics: An Introduction, 4th Edition

7.5 Bayesian decision theory

7.5.1 The elements of game theory

Only a very brief account of this important topic is included here; readers who want to know more should begin by consulting Berger (1985) and Ferguson (1967).

The elements of decision theory are very similar to those of the mathematical theory of games as developed by von Neumann and Morgenstern (1953), although for statistical purposes one of the players is nature (in some sense) rather than another player. Only those aspects of the theory of games which are strictly necessary are given here; an entertaining popular account is given by Williams (1966). A two-person zero-sum game has the following three basic elements:

1. A non-empty set θ of possible states of nature θ, sometimes called the parameter space;

2. A non-empty set A of actions available to the statistician;

3. A loss function L, which defines the loss which a statistician suffers if he takes action a when the true state of nature is θ (this loss being expressed as a real number).

A statistical decision problem or a statistical game is a game coupled with an experiment whose result x lies in a sample space and is randomly distributed with a density ...