Bayes Estimator - Generalized Bayes Estimators

Generalized Bayes Estimators

See also: Admissible decision rule#Bayes rules and generalized Bayes rules

The prior distribution has thus far been assumed to be a true probability distribution, in that

However, occasionally this can be a restrictive requirement. For example, there is no distribution (covering the set, R, of all real numbers) for which every real number is equally likely. Yet, in some sense, such a "distribution" seems like a natural choice for a non-informative prior, i.e., a prior distribution which does not imply a preference for any particular value of the unknown parameter. One can still define a function, but this would not be a proper probability distribution since it has infinite mass,

Such measures, which are not probability distributions, are referred to as improper priors.

The use of an improper prior means that the Bayes risk is undefined (since the prior is not a probability distribution and we cannot take an expectation under it). As a consequence, it is no longer meaningful to speak of a Bayes estimator that minimizes the Bayes risk. Nevertheless, in many cases, one can define the posterior distribution

This is a definition, and not an application of Bayes' theorem, since Bayes' theorem can only be applied when all distributions are proper. However, it is not uncommon for the resulting "posterior" to be a valid probability distribution. In this case, the posterior expected loss

is typically well-defined and finite. Recall that, for a proper prior, the Bayes estimator minimizes the posterior expected loss. When the prior is improper, an estimator which minimizes the posterior expected loss is referred to as a generalized Bayes estimator.