Bayesian decision theory

Thomas Bayes is best known for his formula for calculating conditional probabilities. However, he also deserves recognition for his observation that the concepts of probability and utility can be defined in terms of preferences over a set of uncertain prospects:

The probability of any event is the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the chance of the thing expected upon it’s happening … If a person has an expectation depending on the happening of an event, the probability of the event is to the probability of its failure as his loss if it fails to his gain if it happens. (Bayes 1763:376-77)

The plan of this chapter is as follows. Section 2.1 gives a detailed but nontechnical introduction to the basic ideas of Bayesian decision theory. Sections 2.2 to 2.4 are devoted to the theories presented by Anscombe and Aumann, Savage, and Jeffrey, respectively. Section 2.5 presents an argument for considering alternative, non-Bayesian views; briefly put, the main complaint raised against the Bayesian approach is that it does not offer any substantial action-guidance to agents, not even to ideal ones. This is because in Bayesian theories, beliefs and desires are mere hypothetical entities ascribed to agents. Beliefs and desires do not figure as genuine reasons for selecting one alternative act over another.