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  "body": "\n## Definition\n\nLet $x_1, \\dots, x_n \\in \\R^p$ be known regressor vectors.\nThe *Poisson regression model* for responses $y_1, \\dots, y_n \\in \\set{0, 1, 2, \\dots}$ states three things.\nThe $y_i$ are independent, $y_i$ has the [[Poisson distribution|0XB9GQTN9G]] with mean $\\mu_i \u003e 0$, and\n$$\\log \\mu_i = x_i^T \\beta$$\nfor an unknown $\\beta \\in \\R^p$.\n\n## Remarks\n\nIt is the [[generalized linear model|K6C48KMRKC]] with Poisson responses and the log link, which is the canonical link.\nThe log-likelihood is $\\ell(\\beta) = \\sum_i \\big( y_i x_i^T \\beta - e^{x_i^T \\beta} - \\log y_i! \\big)$, a concave function.\nThe [[maximum likelihood|0FREPFKNS7]] estimate solves $\\sum_i (y_i - \\mu_i) x_i = 0$.\n\nExample: with only an intercept, $x_i = 1$, the likelihood equation gives $\\sum_i (y_i - e^{\\beta}) = 0$, so $e^{\\hat\\beta} = \\bar y$, the mean count.\n\nWhen $y_i$ counts events over an exposure $t_i \u003e 0$, one takes $\\log \\mu_i = \\log t_i + x_i^T \\beta$; the known term $\\log t_i$ is an *offset*, and $e^{x_i^T \\beta}$ is the rate per unit of exposure.\n\nNon-example: if the counts have variance larger than their mean, the hypothesis that $y_i$ is Poisson fails (overdispersion).\nThe point estimates remain consistent when the mean is correctly specified, but the usual standard errors are too small.\n",
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