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  "body": "\n## Definition\n\nLet $\\set{f(\\cdot \\mid \\theta) : \\theta \\in \\Theta}$ be a [[parametric model|Q0DXKAQXE1]] given by densities or mass functions, with [[likelihood|05FSTXM92W]] $L(\\theta \\mid x) = f(x \\mid \\theta)$.\nLet $\\mathcal{C}$ be a family of probability distributions on $\\Theta$.\nThe family $\\mathcal{C}$ is *conjugate* for the model if the following holds for every [[prior|00NBBSXRQS]] $\\pi \\in \\mathcal{C}$.\nFor every observation $x$ with $0 \u003c \\int_\\Theta f(x \\mid \\theta) \\, \\pi(d\\theta) \u003c \\infty$, the [[posterior|0KJ7ZS8AJA]] $\\pi(\\cdot \\mid x)$ belongs to $\\mathcal{C}$.\nA *conjugate prior* is a member of such a family, usually a family indexed by finitely many hyperparameters.\n\n## Remarks\n\nExample: for $k$ successes in $n$ Bernoulli trials with chance $p \\in (0, 1)$, the [[beta distributions|0PP5ZAJPP7]] form a conjugate family.\nIndeed, with prior density proportional to $p^{a - 1}(1 - p)^{b - 1}$, the posterior density is proportional to $p^{a + k - 1}(1 - p)^{b + n - k - 1}$, so the posterior is $\\mathrm{Beta}(a + k, b + n - k)$.\nNon-example: for a Poisson observation $x$ with rate $\\lambda \u003e 0$, the log-normal family is not conjugate.\nThe posterior density is proportional to $\\lambda^{x} e^{-\\lambda}$ times a log-normal density, and the factor $e^{-\\lambda}$ cannot be written in log-normal form.\nThe gamma family is conjugate for the Poisson rate: a $\\mathrm{Gamma}(\\alpha, \\beta)$ prior, with rate $\\beta$, gives the posterior $\\mathrm{Gamma}(\\alpha + x, \\beta + 1)$.\nThe family of all distributions on $\\Theta$ is conjugate, because every posterior is a distribution on $\\Theta$; so the notion is useful only for small families.\n",
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