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  "body": "\n## Definition\n\nLet $\\set{f(\\cdot \\mid \\theta) : \\theta \\in \\Theta}$ be a [[parametric model|Q0DXKAQXE1]] with $\\Theta \\subseteq \\R^k$ open, regular enough that its [[Fisher information|G1Z3WSTE8Q]] matrix $I(\\theta)$ exists for every $\\theta \\in \\Theta$.\nThe *Jeffreys prior* is the [[prior|00NBBSXRQS]] on $\\Theta$ with density with respect to Lebesgue measure\n$$\\pi(\\theta) \\propto \\sqrt{\\det I(\\theta)},$$\nwhere $\\det$ is the [[determinant|0C3CFHDN6R]].\nIf $\\int_\\Theta \\sqrt{\\det I(\\theta)} \\, d\\theta = \\infty$, it is an improper prior, used only when the resulting posterior is proper.\n\n## Remarks\n\nExample: for one Bernoulli observation with chance $p \\in (0, 1)$, $I(p) = 1/(p(1 - p))$.\nSo the Jeffreys prior is proportional to $p^{-1/2}(1 - p)^{-1/2}$, which is the proper distribution $\\mathrm{Beta}(1/2, 1/2)$.\nFor the mean $\\mu$ of $\\mathcal{N}(\\mu, \\sigma^2)$ with $\\sigma$ known, $I(\\mu) = 1/\\sigma^2$ is constant, so the Jeffreys prior is the improper flat prior on $\\R$.\nFor the scale $\\sigma \u003e 0$ of $\\mathcal{N}(0, \\sigma^2)$, $I(\\sigma) = 2/\\sigma^2$, so the Jeffreys prior is proportional to $1/\\sigma$.\nInvariance: let $\\varphi = h(\\theta)$ for a smooth bijection $h$ with smooth inverse.\nThen the Jeffreys prior computed in the parameter $\\varphi$ is the image of the Jeffreys prior in $\\theta$.\nThe reason is that $\\det I$ is multiplied by the square of the Jacobian determinant of the change of variables.\nNon-example: the flat prior on $p \\in (0, 1)$ is not the Jeffreys prior of the Bernoulli model, since its density $1$ is not proportional to $p^{-1/2}(1 - p)^{-1/2}$.\n",
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