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Type of "good" decision rule in Bayesian statistics

In statistical decision theory, an admissible decision rule is a rule for making a decision such that there is no other rule that is always "better" than it (or at least sometimes better and never worse), in the precise sense of "better" defined below. This concept is analogous to Pareto efficiency.

Let be a probability distribution on the states of nature. From a Bayesian point of view, we would regard it as a prior distribution. That is, it is our believed probability distribution on the states of nature, prior to observing data. For a frequentist, 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.

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 case, the Bayes risk is not even well defined, nor is there any well defined distribution over However, the posterior —and hence the expected loss—may be well defined for each , so that it is still possible to define a generalized Bayes rule.

According to the complete class theorems, under mild conditions every admissible rule is a (generalized) Bayes rule (with respect to some prior —possibly an improper one—that favors distributions where that rule achieves low risk). Thus, in frequentist decision theory it is sufficient to consider only (generalized) Bayes rules. Conversely, while Bayes rules with respect to proper priors are virtually always admissible, generalized Bayes rules corresponding to improper priors need not yield admissible procedures. Stein's example is one such famous situation.

The James–Stein estimator is a nonlinear estimator of the mean of Gaussian random vectors and can be shown to dominate the ordinary least squares technique with respect to a mean squared error loss function. Thus least squares estimation is not an admissible estimation procedure in this context. Some others of the standard estimates associated with the normal distribution are also inadmissible: for example, the sample estimate of the variance when the population mean and variance are unknown.