An Introduction to Decision Theory

There are almost as many definitions of Bayesianism as there are decision theorists. The following broad definition proposed by Richard Bradley is, however, a fairly uncontroversial characterization of the main idea.

According to this definition, Bayesianism has two distinct components. The first tells you what your state of mind ought to be like, while the second tells you how you ought to act given that state of mind. Let us call the two components the epistemic and the deliberative component, respectively. The epistemic component of Bayesianism is a claim about what rational agents ought to believe, and which combinations of beliefs and desires are rationally permissible. In essence, the theory holds that one is free to believe whatever one wishes as long as one's beliefs can be represented by a subjective probability function, and those beliefs are updated in accordance with Bayes’ theorem (see Chapter 6). This means that Bayesian epistemology offers virtually no substantial advice on how one ought to go about when exploring the world. The theory merely provides a set of structural restrictions on what it is permissible to believe, and how one is permitted to revise those beliefs in light of new information. Bayesians thus maintain that all beliefs come in degrees, and that the theory of subjective probability provides an accurate account of how degrees of belief ought to be revised. (Some Bayesians advocate objective versions of the theory, according to which probabilities are objective features of the world such as relative frequencies or physical propensities. Naturally, the mathematical framework for how probabilities ought to be updated is the same.)