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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 $\\delta$.\nA rule $\\delta'$ *dominates* a rule $\\delta$ if both conditions hold:\n\n1. $R(\\theta, \\delta') \\le R(\\theta, \\delta)$ for every $\\theta \\in \\Theta$;\n2. $R(\\theta_0, \\delta') \u003c R(\\theta_0, \\delta)$ for some $\\theta_0 \\in \\Theta$.\n\nA rule $\\delta$ is *admissible* if no rule dominates it, and *inadmissible* otherwise.\n\n## Remarks\n\nAdmissibility depends on the [[decision problem|CPFSBBHXT5]]: on the model, on the loss and on the class of rules allowed.\n\nExample: let $X$ be [[normal|05AQE0QR0D]] with mean $\\theta \\in \\R$ and variance $1$, with squared error loss, and let $\\delta \\equiv \\theta_0$ be a constant.\nIts risk is $(\\theta - \\theta_0)^2$, which is $0$ at $\\theta_0$.\nA rule $\\delta'$ with $R(\\theta_0, \\delta') \\le 0$ must equal $\\theta_0$ almost surely under $P_{\\theta_0}$.\nAll $P_\\theta$ have the same null sets, so $\\delta' = \\theta_0$ almost surely for every $\\theta$, and $\\delta'$ has the same risk as $\\delta$.\nSo nothing dominates $\\delta$, and it is admissible.\n\nNon-example: in the same problem, $\\delta(X) = 2X$ has risk $E_\\theta[(2X - \\theta)^2] = 4 + \\theta^2$.\nThe rule $X$ has risk $1 \u003c 4 + \\theta^2$ for every $\\theta$, so it dominates $2X$, which is inadmissible.\n",
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