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  "body": "\n## Statement\n\nLet $\\set{f_\\theta : \\theta \\in \\Theta}$ be a family of [[probability densities|0ZY9DX8FBF]] on $\\R$ with the following properties.\n\n1. There is an open interval $I$, the same for all $\\theta$, such that $f_\\theta(x) \u003e 0$ for $x \\in I$ and $f_\\theta(x) = 0$ for $x \\notin I$.\n2. Each $f_\\theta$ is continuously differentiable on $I$.\n\nLet $n \\ge 2$ and let $X_1, \\dots, X_n$ be [[independent and identically distributed|00MRN0FARV]] with density $f_\\theta$.\nAssume that there is a [[sufficient statistic|WP4TGXJQJG]] $T = (T_1, \\dots, T_k) \\colon I^n \\to \\R^k$ for $\\theta$ that is continuously differentiable, with $k \u003c n$.\nThen the family is an [[exponential family|EP9VPZ0S6G]] with at most $k$ terms: there are functions $h$, $u_1, \\dots, u_k$, $\\eta_1, \\dots, \\eta_k$ and $A$ such that\n$$f_\\theta(x) = h(x) \\exp\\Big( \\sum_{i=1}^{k} \\eta_i(\\theta) u_i(x) - A(\\theta) \\Big), \\qquad x \\in I.$$\n\n## Remarks\n\nThe converse is easy.\nFor an exponential family, the sums $\\sum_{j=1}^{n} u_i(X_j)$, $i = 1, \\dots, k$, form a sufficient statistic of dimension $k$ for every sample size $n$, by the factorization criterion.\nSo, among families with a fixed support, exponential families are exactly those with a sufficient statistic whose dimension does not grow with $n$.\n\nThe idea of the proof is to write the factorization of the joint density and take logarithms.\nDifferentiating in the $x_j$ shows that $\\partial_x \\log f_\\theta(x)$ lies, as a function of $\\theta$, in a space of dimension at most $k$.\nIntegrating in $x$ gives the exponential form.\n\nThe fixed support is necessary.\nFor the uniform densities on $(0, \\theta)$, the maximum $\\max_j X_j$ is sufficient and one-dimensional for every $n$.\nYet the support moves with $\\theta$, and this family is not an exponential family.\n\n## Explanations\n\n### L1 human\n\nSometimes a whole sample can be squeezed into a few numbers without losing information about the parameter, for any sample size.\nThe theorem says that, when the range of possible values does not depend on the parameter, this only happens for exponential families.\nSo exponential families are the natural home of data compression in statistics.\n\n### L2 human\n\nThe theorem explains why exponential families appear everywhere in classical parametric statistics.\nSufficient statistics of fixed dimension give conjugate priors, complete statistics and explicit minimum-variance unbiased estimators.\nThe result is named after Pitman, Koopman and Darmois.\nThe precise regularity conditions differ from one textbook to another, but some smoothness of the sufficient statistic is always needed to exclude pathological maps from $\\R^n$ to $\\R^k$.\n",
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