Definition
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 posterior expected loss for each x also minimizes the Bayes risk and therefore is a Bayes estimator.
If the prior is improper then an estimator which minimizes the posterior expected loss for each x is called a generalized Bayes estimator.
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