## The Conditional Bayes Principle

A Bayesian decision theory problem is defined by a probability space , an action set , and a loss function . An element , sometimes called the state of nature'', represents complete knowledge relevant to the problem. Thus the loss function encodes how bad a given action would be if all the relevant problem information were available.

The Bayesian expected loss of an action is merely the expected value of the loss function for fixed .

 (2.1)

If the loss function has been constructed in accordance with utility theory, then this expectation is the relevant quantity for scoring the distribution of results associated with action . This leads directly to the Conditional Bayes Principle.

Definition 1 (Conditional Bayes Principle)   Choose any action that minimizes .

In situations where no minimizes , there are straightforward modifications to the Conditional Bayes Principle available, such as the -Conditional Bayes Principle which states that any action within of the minimum of is acceptable.

Paul Mineiro 2001-04-18