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  "body": "\n## Definition\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a statistical model with a loss function, and let $R(\\theta, \\delta)$ be the [[risk function|N26TSVRH2S]] of a decision rule $\\delta$.\nLet $\\Theta$ carry a σ-algebra, and let $\\pi$ be a probability measure on $\\Theta$, the [[prior|00NBBSXRQS]], such that $\\theta \\mapsto R(\\theta, \\delta)$ is measurable for the rules considered.\nThe *Bayes risk* of $\\delta$ under $\\pi$ is\n$$r(\\pi, \\delta) = \\int_\\Theta R(\\theta, \\delta) \\, \\pi(d\\theta) \\in [0, \\infty].$$\nThe *Bayes risk of the prior* $\\pi$ is $r(\\pi) = \\inf_\\delta r(\\pi, \\delta)$, the infimum over all decision rules.\n\n## Remarks\n\nThe integral is a [[Lebesgue integral|DWJF7ABPR1]] of a nonnegative function, so it always exists, possibly infinite.\nBy Fubini's theorem, $r(\\pi, \\delta)$ is also the expected loss when $\\theta$ is drawn from $\\pi$ and then $X$ from $P_\\theta$.\n\nExample: let $X$ be [[binomial|0WCWA1KT3M]] with parameters $n$ and $\\theta \\in [0, 1]$, with squared error loss and $\\delta(X) = X/n$.\nThen $R(\\theta, \\delta) = \\theta(1 - \\theta)/n$.\nFor the uniform prior, $r(\\pi, \\delta) = \\frac{1}{n} \\int_0^1 \\theta(1 - \\theta) \\, d\\theta = \\frac{1}{6n}$.\n\nNon-example: the maximum $\\sup_\\theta R(\\theta, \\delta) = \\frac{1}{4n}$ is not a Bayes risk of $\\delta$ for the uniform prior.\nIt is the worst-case risk used in the minimax principle; it equals $r(\\pi, \\delta)$ only for priors that put all their mass where the risk is maximal.\n",
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