Bayesian estimation

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 ${\displaystyle \theta }$ is known to have a prior distribution ${\displaystyle \pi }$. Let ${\displaystyle {\widehat {\theta }}={\widehat {\theta }}(x)}$ be an estimator of ${\displaystyle \theta }$ (based on some measurements x), and let ${\displaystyle L(\theta ,{\widehat {\theta }})}$ be a loss function, such as squared error. The Bayes risk of ${\displaystyle {\widehat {\theta }}}$ is defined as ${\displaystyle E_{\pi }(L(\theta ,{\widehat {\theta }}))}$, where the expectation is taken over the probability distribution of ${\displaystyle \theta }$: this defines the risk function as a function of ${\displaystyle {\widehat {\theta }}}$. An estimator ${\displaystyle {\widehat {\theta }}}$ 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 ${\displaystyle E(L(\theta ,{\widehat {\theta }})|x)}$ for each ${\displaystyle x}$ also minimizes the Bayes risk and therefore is a Bayes estimator.

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