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  "body": "\n## Definition\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a [[statistical model|JZ6R7J36K5]] on $(S, \\mathcal{S})$ and let $X$ be the data.\nA statistic $T = T(X)$ is *minimal sufficient* if:\n\n1. $T$ is [[sufficient|QNMR5F525C]] for $\\theta$, and\n2. for every sufficient statistic $U = U(X)$ there is a measurable function $\\varphi$ with $T = \\varphi(U)$ $P_\\theta$-almost surely for every $\\theta \\in \\Theta$.\n\n## Remarks\n\nExample: for $X_1, \\dots, X_n$ i.i.d. Bernoulli($p$), $p \\in (0, 1)$, the sum $T = \\sum_i X_i$ is minimal sufficient.\nIndeed, the likelihood ratio $P_p(X = x)/P_p(X = y) = (p/(1 - p))^{\\sum_i x_i - \\sum_i y_i}$ is free of $p$ only when $\\sum_i x_i = \\sum_i y_i$, and by the criterion below this makes $T$ minimal.\nNon-example: for $n \\ge 2$, the whole sample $X$ is sufficient but not minimal.\nClause 2 fails with $U = T$: $X$ is not a function of $T$, since $(1, 0, \\dots, 0)$ and $(0, 1, 0, \\dots, 0)$ have the same sum.\nA common criterion, for a model with densities $f_\\theta$: if $T(x) = T(y)$ holds exactly when $f_\\theta(x)/f_\\theta(y)$ does not depend on $\\theta$, then $T$ is minimal sufficient.\nA minimal sufficient statistic is unique up to one-to-one measurable functions.\n",
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