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  "body": "\n## Summary\n\nAn exponential family is a parametric family of probability distributions whose densities have the form $h(x) \\exp\\big(\\sum_i \\eta_i(\\theta) T_i(x) - A(\\theta)\\big)$.\nMost standard distributions belong to one: normal, binomial, Poisson, gamma, beta and many more.\nThe statistics $T_i$ are sufficient, and the structure makes estimation, testing and Bayesian updating explicit.\n\n## Explanations\n\n### L1 human\n\nIn an exponential family the parameter and the data meet only through a few products $\\eta_i(\\theta) T_i(x)$ in an exponent.\nSo the data can be compressed to the values of $T_1, \\dots, T_k$ without losing any information about the parameter.\nFor example, the Poisson probabilities $e^{-\\lambda} \\lambda^x / x!$ can be written as $\\frac{1}{x!} \\exp(x \\log \\lambda - \\lambda)$, with $T(x) = x$ and $\\eta = \\log \\lambda$.\nFor a sample of Poisson counts, only the total count matters for $\\lambda$.\n",
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