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  "body": "\n## Definition\n\nLet $(\\mathcal{X}, \\mathcal{F})$ be a measurable space with a [[σ-finite measure|WEBBT18D4X]] $\\mu$, and let $\\set{P_\\theta : \\theta \\in \\Theta}$ be a [[parametric family|MSC0MR8VMH]] of probability measures on it.\nThe family is a *$k$-parameter exponential family* if there are\n\n1. a measurable function $h \\colon \\mathcal{X} \\to [0, \\infty)$,\n2. measurable functions $T_1, \\dots, T_k \\colon \\mathcal{X} \\to \\R$,\n3. functions $\\eta_1, \\dots, \\eta_k, A \\colon \\Theta \\to \\R$,\n\nsuch that each $P_\\theta$ has a [[density|1NTKG64F7N]] with respect to $\\mu$ of the form\n$$p_\\theta(x) = h(x) \\exp\\Big( \\sum_{i=1}^{k} \\eta_i(\\theta) T_i(x) - A(\\theta) \\Big), \\qquad x \\in \\mathcal{X}.$$\n\n## Remarks\n\nThe function $A$ is fixed by normalization: $A(\\theta) = \\log \\int h(x) \\exp\\big(\\sum_i \\eta_i(\\theta) T_i(x)\\big) \\, \\mu(dx)$.\nThe numbers $\\eta_i = \\eta_i(\\theta)$ are the *natural parameters*, and writing the density in terms of $\\eta$ gives the *canonical form*.\n\nExample: the [[Bernoulli distribution|0386Y0Z7Y7]] with parameter $p \\in (0, 1)$, with $\\mu$ the counting measure on $\\set{0, 1}$.\nIts density is $p^x (1-p)^{1-x} = \\exp\\big(x \\log \\tfrac{p}{1-p} + \\log(1-p)\\big)$.\nSo it is a $1$-parameter exponential family with $h = 1$, $T(x) = x$, $\\eta(p) = \\log \\frac{p}{1-p}$ and $A(p) = -\\log(1-p)$.\n\nNon-example: the uniform distributions on $[0, \\theta]$ for $\\theta \u003e 0$.\nIn an exponential family the set $\\set{x : p_\\theta(x) \u003e 0} = \\set{x : h(x) \u003e 0}$ does not depend on $\\theta$, up to a $\\mu$-null set, because the exponential factor is positive.\nFor the uniform family this set is $[0, \\theta]$, which moves with $\\theta$, so no choice of $h$, $T_i$, $\\eta_i$, $A$ works.\n",
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