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  "body": "\n## Statement\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a [[statistical model|JZ6R7J36K5]] on a measurable space $(S, \\mathcal{S})$ that is dominated by a $\\sigma$-finite measure $\\mu$: each $P_\\theta$ has a [[density|ECDJ8DD9RV]] $f_\\theta = dP_\\theta/d\\mu$.\nLet $T \\colon S \\to \\mathcal{T}$ be a measurable map into a measurable space $\\mathcal{T}$.\nThen $T$ is a [[sufficient statistic|QNMR5F525C]] for $\\theta$ if and only if the following holds.\nThere are measurable functions $g_\\theta \\colon \\mathcal{T} \\to [0, \\infty)$, for $\\theta \\in \\Theta$, and $h \\colon S \\to [0, \\infty)$ such that, for every $\\theta \\in \\Theta$,\n$$f_\\theta(x) = g_\\theta\\left(T(x)\\right) h(x) \\quad \\text{for } \\mu\\text{-almost every } x \\in S.$$\n\n## Remarks\n\nIn the discrete case the proof is a direct computation: if the factorization holds, then $P_\\theta(X = x \\mid T = t) = h(x)/\\sum_{y : T(y) = t} h(y)$, which does not depend on $\\theta$; conversely one takes $g_\\theta(t) = P_\\theta(T = t)$ and $h(x) = P(X = x \\mid T = T(x))$.\nThe general case is due to Halmos and Savage.\nIt first replaces $\\mu$ by a probability measure $\\lambda = \\sum_j c_j P_{\\theta_j}$, a countable convex combination equivalent to the whole family.\nIt then uses Radon–Nikodym derivatives with respect to $\\lambda$.\nExample: for i.i.d. $\\mathcal{N}(\\mu, \\sigma^2)$ data,\n$$f_{(\\mu, \\sigma^2)}(x) = (2\\pi\\sigma^2)^{-n/2} \\exp\\left(-\\frac{\\sum_i x_i^2 - 2\\mu \\sum_i x_i + n\\mu^2}{2\\sigma^2}\\right),$$\nso $T(x) = \\left(\\sum_i x_i, \\sum_i x_i^2\\right)$ is sufficient for $(\\mu, \\sigma^2)$, with $h = 1$.\nExample: for i.i.d. uniform data on $[0, \\theta]$, $f_\\theta(x) = \\theta^{-n}\\mathbf{1}\\set{\\max_i x_i \\le \\theta}\\mathbf{1}\\set{\\min_i x_i \\ge 0}$, so $\\max_i x_i$ is sufficient.\nThe domination hypothesis is needed for the densities $f_\\theta$ to exist, so without it the criterion cannot even be stated.\n\n## Explanations\n\n### L1 human\n\nTo check whether a summary of the data keeps all the information about a parameter, write down the likelihood.\nIf the parameter enters the likelihood only through the summary, the summary is sufficient, and conversely.\nThis turns a question about conditional distributions into simple algebra.\n\n### L2 human\n\nThe factorization criterion is the standard way to find sufficient statistics in practice.\nFor an [[exponential family|EP9VPZ0S6G]] with density $h(x)\\exp(\\eta(\\theta) \\cdot T(x) - A(\\theta))$, it shows at once that $T$ is sufficient.\nIt also shows that the likelihood function, up to a factor free of $\\theta$, depends on the data only through any sufficient statistic, which links sufficiency with the [[likelihood principle|YR5RAT9HWZ]].\nIt is the first step towards the [[Rao–Blackwell theorem|Y3TX4J41TP]] and the [[Lehmann–Scheffé theorem|QQZPWDWS29]].\n",
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