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  "body": "\n## Statement\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a family of probability measures for the data $X$.\nLet $T = T(X)$ be a [[sufficient|QNMR5F525C]] statistic that is boundedly [[complete|PF4RHTZWXH]], and let $V = V(X)$ be an [[ancillary|NR6MGAB5V0]] statistic.\nThen, for every $\\theta \\in \\Theta$, the statistics $T$ and $V$ are [[independent|0XCE070V8B]] under $P_\\theta$.\n\n## Remarks\n\nProof idea: fix a measurable set $B$ and let $q = P_\\theta(V \\in B)$, which does not depend on $\\theta$ since $V$ is ancillary.\nBy sufficiency, the [[conditional probability|0YAMJRK04E]] $k(T) = P_\\theta(V \\in B \\mid T)$ can be chosen free of $\\theta$.\nThen $k(T) - q$ is bounded and $E_\\theta[k(T) - q] = 0$ for all $\\theta$, so bounded completeness gives $k(T) = q$ almost surely; that is, $V \\in B$ is independent of $T$.\nExample: for $X_1, \\dots, X_n$ i.i.d. $\\mathcal{N}(\\mu, \\sigma^2)$ with $\\sigma^2$ known and $\\mu \\in \\R$, $\\bar{X}$ is complete and sufficient for $\\mu$, and the sample variance $S^2$ is ancillary; so $\\bar{X}$ and $S^2$ are independent.\nSince $\\sigma^2$ was arbitrary, this proves the independence of $\\bar{X}$ and $S^2$ for every normal sample.\nCompleteness cannot be dropped: for $n \\ge 2$ the whole sample $T = (X_1, \\dots, X_n)$ is sufficient, and $V = X_1 - X_2$ is ancillary, but $V$ is a non-constant function of $T$, so they are not independent.\n\n## Explanations\n\n### L1 human\n\nA complete sufficient statistic holds all the information about the parameter, and an ancillary statistic holds none.\nBasu's theorem says that two such statistics do not even influence each other: they are independent.\nThis gives quick proofs of independence results that are otherwise hard to compute.\n\n### L2 human\n\nThe best-known use is the independence of the sample mean and the sample variance of a normal sample, which underlies the $t$-distribution of $\\sqrt{n}(\\bar{X} - \\mu)/S$.\nMore generally, in an exponential family of full rank the natural sufficient statistic is complete, so it is independent of every ancillary statistic.\nThe theorem also allows one to compute $E_\\theta[V \\mid T]$ as $E_\\theta[V]$ for an integrable ancillary $V$, which simplifies Rao–Blackwell computations.\n",
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