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  "body": "\n## Definition\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a [[statistical model|JZ6R7J36K5]] and let $X$ be the data, with law $P_\\theta$ for a fixed unknown $\\theta$.\n*Frequentist inference* is the approach to [[statistical inference|FXCQWFY5RV]] in which:\n\n1. the parameter $\\theta$ is treated as fixed, and no probability distribution is placed on $\\Theta$;\n2. a procedure $\\delta(X)$ is assessed through its [[sampling distribution|FC9NRV5GSP]] under $P_\\theta$;\n3. a guarantee about $\\delta$ must hold for every $\\theta \\in \\Theta$, or be stated as a function of $\\theta$.\n\n## Remarks\n\nExample: a [[confidence set|0K4NMMJBG2]] of level $1 - \\alpha$ is a set-valued statistic $C(X)$ with $P_\\theta(\\theta \\in C(X)) \\ge 1 - \\alpha$ for every $\\theta \\in \\Theta$.\nFor $n$ i.i.d. $\\mathcal{N}(\\theta, 1)$ observations, $C(X) = [\\bar{X}_n - z/\\sqrt{n}, \\bar{X}_n + z/\\sqrt{n}]$, with $z$ the $1 - \\alpha/2$ quantile of $\\mathcal{N}(0, 1)$, has coverage exactly $1 - \\alpha$ for every $\\theta$, since $\\sqrt{n}(\\bar{X}_n - \\theta) \\sim \\mathcal{N}(0, 1)$.\nNon-example: a Bayesian credible interval is computed from a posterior distribution on $\\Theta$; it violates clause 1, and in general it has no coverage guarantee that holds for every $\\theta$.\nA test of level $\\alpha$ is likewise a procedure whose probability of rejecting a true hypothesis is at most $\\alpha$ under every $\\theta$ of the hypothesis.\n",
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