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  "body": "\n## Definition\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a statistical model with a [[loss function|3BTG6VVCGZ]] $L$, and let $\\pi$ be a [[prior|00NBBSXRQS]] on $\\Theta$.\nFor an [[estimator|0SSS6ERQ0S]] $\\delta$, let $r(\\pi, \\delta)$ be its [[Bayes risk|VEBDNCJJTN]], the [[risk|N26TSVRH2S]] $R(\\theta, \\delta)$ averaged over $\\pi$.\nAn estimator $\\delta_\\pi$ is a *Bayes estimator* with respect to $\\pi$ if $r(\\pi, \\delta_\\pi) \u003c \\infty$ and\n$$r(\\pi, \\delta_\\pi) = \\inf_{\\delta} r(\\pi, \\delta),$$\nwhere the infimum is over all estimators $\\delta$.\n\n## Remarks\n\nIf $r(\\pi) \u003c \\infty$, an estimator that minimizes the [[posterior|0KJ7ZS8AJA]] expected loss $E[L(\\theta, \\delta(x)) \\mid X = x]$ for almost every $x$ is a Bayes estimator.\nUnder squared error loss $L(\\theta, a) = (a - \\theta)^2$, the Bayes estimator is the posterior mean $E[\\theta \\mid X]$.\nUnder absolute error loss, it is a posterior median.\n\nExample: let $X$ be [[binomial|0WCWA1KT3M]] with parameters $n$ and $\\theta$, and let $\\pi$ be the [[beta distribution|0EW824Z7G2]] with parameters $a, b \u003e 0$.\nThe posterior is the beta distribution with parameters $a + X$ and $b + n - X$.\nSo under squared error loss the Bayes estimator is its mean $(a + X)/(a + b + n)$.\n\nNon-example: for the uniform prior, $a = b = 1$, the Bayes estimator is $(X + 1)/(n + 2)$.\nThe proportion $X/n$ differs from it whenever $X \\ne n/2$, and for $n \\ge 1$ its Bayes risk $1/(6n)$ is larger, so $X/n$ is not the Bayes estimator for this prior.\n",
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