it is merely a function on with no such special interpretation. The Bayes risk of the decision rule with respect to is the expectation A decision rule that minimizes is called a Bayes rule with respect to . There may be more than one such Bayes rule. If the Bayes risk is infinite for all , then no Bayes rule is defined. Generalized Bayes rules In the Bayesian approach to decision theory, the observed is considered fixed. Whereas the frequentist approach (i.e., risk) averages over possible samples , the Bayesian would fix the observed sample and average over hypotheses . Thus, the Bayesian approach is to consider for our observed the expected loss where the expectation is over the posterior of given (obtained from and using Bayes' theorem). Having made explicit the expected loss for each given separately, we can define a decision rule by specifying for each an action that minimizes the expected loss. This is known as a generalized Bayes rule with respect to . There may be more than one generalized Bayes rule, since there may be multiple choices of that achieve the same expected loss. At first, this may appear rather different from the Bayes rule approach of the previous section, not a generalization. However, notice that the Bayes risk already averages over in Bayesian fashion, and the Bayes risk may be recovered as the expectation over of the expected loss (where and ). Roughly speaking, minimizes this expectation of expected loss (i.e., is a Bayes rule) if and only if it minimizes the expected loss for each separately (i.e., is a generalized Bayes rule). Then why is the notion of generalized Bayes rule an improvement? It is indeed equivalent to the notion of Bayes rule when a Bayes rule exists and all have positive probability. However, no Bayes rule exists if the Bayes risk is infinite (for all ). In this case it is still useful to define a generalized Bayes rule , which at least chooses a minimum-expected-loss action for those for which a finite-expected-loss action does exist. In addition, a generalized Bayes rule may be desirable because it must choose a minimum-expected-loss action for every , whereas a Bayes rule would be allowed to deviate from this policy on a set of measure 0 without affecting the Bayes risk. More important, it is sometimes convenient to use an improper prior . In this