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(November 2009)| In estimation theory and decision theory, a
Bayes estimator or a Bayes action is an estimator or decision
rule that minimizes the posterior expected value of a loss
function (i.e., the posterior expected loss). Equivalently, it
maximizes the posterior expectation of a utility function. An
alternative way of formulating an estimator within Bayesian
statistics is Maximum a posteriori estimation. Suppose an
unknown parameter θ is known to have a prior distribution . Let
be an estimator of θ (based on some measurements x), and let be
a loss function, such as squared error. The Bayes risk of is
defined as , where the expectation is taken over the probability
distribution of : this defines the risk function as a function
of . An estimator is said to be a Bayes estimator if it
minimizes the Bayes risk among all estimators. Equivalently, the
estimator which minimizes the posterio