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.
Read more about this topic: Bayes Estimator
Famous quotes containing the word definition:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)