The Conditional Bayes Principle

A Bayesian decision theory problem is defined by a probability space $(\Omega, F, P)$, an action set $A$, and a loss function $L:A \times \Omega \to R^+$. An element $\omega \in \Omega$, 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 $a \in A$ is merely the expected value of the loss function for fixed $a$.

$$\rho (a) = E^P [L (a, \omega)]$$ (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 $a$. This leads directly to the Conditional Bayes Principle.

Definition 1 (Conditional Bayes Principle)   Choose any action $a \in A$ that minimizes $\rho (a)$.

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