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  "body": "\n## Definition\n\nLet $\\mathcal{P} = \\set{P_\\theta : \\theta \\in \\Theta}$ be a [[statistical model|JZ6R7J36K5]] on a sample space $S$.\nLet $x \\in S$ be an observed value of a random element $X$ whose law is $P_\\theta$ for an unknown $\\theta$.\n*Statistical inference* is the derivation, from $x$, of statements about $\\theta$ or about a function $\\tau(\\theta)$.\nA rule of inference is a map defined on $S$ that does not depend on $\\theta$; its main kinds are these.\n\n1. A point estimate: a value $\\hat{\\tau}(x)$ of an [[estimator|0TTNRP5PFQ]] of $\\tau(\\theta)$.\n2. A set estimate: a set $C(x)$ of plausible values of $\\tau(\\theta)$.\n3. A test: a decision $\\varphi(x) \\in \\set{0, 1}$ between a hypothesis $\\theta \\in \\Theta_0$ and its alternative $\\theta \\in \\Theta \\setminus \\Theta_0$.\n\n## Remarks\n\nExample: for $n$ independent $\\mathcal{N}(\\theta, 1)$ observations, the sample mean $\\bar{x}$ is a point estimate of $\\theta$, and $[\\bar{x} - 1.96/\\sqrt{n}, \\bar{x} + 1.96/\\sqrt{n}]$ is a set estimate of $\\theta$.\nBoth are computed from the data alone, as the definition requires.\nNon-example: the rule \"report $\\theta$\" is not a rule of inference, since it uses the unknown parameter.\nInference rules are built from [[statistics|036756NFD6]] and judged either by their behaviour over repeated samples or through a posterior distribution.\n",
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