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  "body": "\n## Definition\n\nLet $p \\ge 2$ and let $S^{p-1} \\subseteq \\R^p$ be the unit [[sphere|T62369RFH6]] with its surface measure $\\sigma$.\nLet $\\mu \\in S^{p-1}$ and $\\kappa \u003e 0$.\nLet $I_\\nu$ be the modified Bessel function of the first kind of order $\\nu$.\nThe *von Mises–Fisher distribution* $\\operatorname{vMF}(\\mu, \\kappa)$ is the [[distribution|0DVDCMA2DK]] on $S^{p-1}$ with this [[density|0ZY9DX8FBF]] with respect to $\\sigma$:\n$$f(x) = C_p(\\kappa) \\exp(\\kappa \\mu^T x), \\qquad C_p(\\kappa) = \\frac{\\kappa^{p/2 - 1}}{(2\\pi)^{p/2} I_{p/2 - 1}(\\kappa)}.$$\nFor $\\kappa = 0$ it is defined as the uniform distribution on $S^{p-1}$.\nThe vector $\\mu$ is the *mean direction* and $\\kappa$ is the *concentration parameter*.\n\n## Remarks\n\nExample: for $p = 3$, $I_{1/2}(\\kappa) = \\sqrt{2/(\\pi \\kappa)} \\sinh \\kappa$, so $C_3(\\kappa) = \\kappa / (4 \\pi \\sinh \\kappa)$.\nCheck: in spherical coordinates with $\\mu$ as north pole, $\\int_{S^2} e^{\\kappa \\cos \\theta} \\, d\\sigma = 2\\pi \\int_0^\\pi e^{\\kappa \\cos \\theta} \\sin \\theta \\, d\\theta = 4 \\pi \\sinh(\\kappa)/\\kappa$.\nNon-example: a density proportional to $\\exp(\\kappa (\\mu^T x)^2)$ with $\\kappa \u003e 0$ is not von Mises–Fisher.\nIt takes the same value at $x$ and $-x$, so its maximum is attained at both $\\mu$ and $-\\mu$, while a von Mises–Fisher density has a single maximum.\nThe density is largest at $x = \\mu$ and smallest at $x = -\\mu$, and it depends on $x$ only through the angle to $\\mu$.\nFor $p = 2$, writing $x = (\\cos \\theta, \\sin \\theta)$ gives the von Mises density $e^{\\kappa \\cos(\\theta - \\theta_0)} / (2 \\pi I_0(\\kappa))$.\nThe maximum likelihood estimate of $\\mu$ from a sample is its mean direction.\n",
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