# von Mises–Fisher distribution

In directional statistics, the von Mises–Fisher distribution (named after Ronald Fisher and Richard von Mises), is a probability distribution on the $(p-1)$ -sphere in $\mathbb {R} ^{p}$ . If $p=2$ the distribution reduces to the von Mises distribution on the circle.

The probability density function of the von Mises–Fisher distribution for the random p-dimensional unit vector $\mathbf {x} \,$ is given by:

$f_{p}(\mathbf {x} ;{\boldsymbol {\mu }},\kappa )=C_{p}(\kappa )\exp \left({\kappa {\boldsymbol {\mu }}^{T}\mathbf {x} }\right),$ where $\kappa \geq 0,\left\Vert {\boldsymbol {\mu }}\right\Vert =1\,$ and the normalization constant $C_{p}(\kappa )\,$ is equal to

$C_{p}(\kappa )={\frac {\kappa ^{p/2-1}}{(2\pi )^{p/2}I_{p/2-1}(\kappa )}},$ where $I_{v}$ denotes the modified Bessel function of the first kind at order $v$ . If $p=3$ , the normalization constant reduces to

$C_{3}(\kappa )={\frac {\kappa }{4\pi \sinh \kappa }}={\frac {\kappa }{2\pi (e^{\kappa }-e^{-\kappa })}}.$ The parameters $\mu \,$ and $\kappa \,$ are called the mean direction and concentration parameter, respectively. The greater the value of $\kappa \,$ , the higher the concentration of the distribution around the mean direction $\mu \,$ . The distribution is unimodal for $\kappa >0\,$ , and is uniform on the sphere for $\kappa =0\,$ .

The von Mises–Fisher distribution for $p=3$ , also called the Fisher distribution, was first used to model the interaction of electric dipoles in an electric field (Mardia&Jupp, 1999). Other applications are found in geology, bioinformatics, and text mining.

## Relation to normal distribution

Starting from a normal distribution

$G_{p}(\mathbf {x} ;{\boldsymbol {\mu }},\kappa )=\left({\sqrt {\frac {\kappa }{2\pi }}}\right)^{p}\exp \left(-\kappa {\frac {(\mathbf {x} -{\boldsymbol {\mu }})^{2}}{2}}\right),$ the von Mises-Fisher distribution is obtained by expanding

$(\mathbf {x} -{\boldsymbol {\mu }})^{2}=\mathbf {x} ^{2}+{\boldsymbol {\mu }}^{2}-2{\boldsymbol {\mu }}^{T}\mathbf {x} ,$ using the fact that $\mathbf {x}$ and ${\boldsymbol {\mu }}$ are unit vectors, and recomputing the normalization constant by integrating $\mathbf {x}$ over the unit sphere.

## Estimation of parameters

A series of N independent measurements $x_{i}$ are drawn from a von Mises–Fisher distribution. Define

$A_{p}(\kappa )={\frac {I_{p/2}(\kappa )}{I_{p/2-1}(\kappa )}}.\,$ Then (Mardia&Jupp, 1999) the maximum likelihood estimates of $\mu \,$ and $\kappa \,$ are given by the sufficient statistic

${\bar {x}}={\frac {1}{N}}\sum _{i}^{N}x_{i},$ as

$\mu ={\bar {x}}/{\bar {R}},{\text{where }}{\bar {R}}=\|{\bar {x}}\|,$ and

$\kappa =A_{p}^{-1}({\bar {R}}).$ Thus $\kappa \,$ is the solution to

$A_{p}(\kappa )={\frac {\|\sum _{i}^{N}x_{i}\|}{N}}={\bar {R}}.$ A simple approximation to $\kappa$ is (Sra, 2011)

${\hat {\kappa }}={\frac {{\bar {R}}(p-{\bar {R}}^{2})}{1-{\bar {R}}^{2}}},$ but a more accurate measure can be obtained by iterating the Newton method a few times

${\hat {\kappa }}_{1}={\hat {\kappa }}-{\frac {A_{p}({\hat {\kappa }})-{\bar {R}}}{1-A_{p}({\hat {\kappa }})^{2}-{\frac {p-1}{\hat {\kappa }}}A_{p}({\hat {\kappa }})}},$ ${\hat {\kappa }}_{2}={\hat {\kappa }}_{1}-{\frac {A_{p}({\hat {\kappa }}_{1})-{\bar {R}}}{1-A_{p}({\hat {\kappa }}_{1})^{2}-{\frac {p-1}{{\hat {\kappa }}_{1}}}A_{p}({\hat {\kappa }}_{1})}}.$ For N ≥ 25, the estimated spherical standard error of the sample mean direction can be computed as

${\hat {\sigma }}=\left({\frac {d}{N{\bar {R}}^{2}}}\right)^{1/2}$ where

$d=1-{\frac {1}{N}}\sum _{i}^{N}(\mu ^{T}x_{i})^{2}$ It's then possible to approximate a $100(1-\alpha )\%$ confidence cone about $\mu$ with semi-vertical angle

$q=\arcsin(e_{\alpha }^{1/2}{\hat {\sigma }}),$ where $e_{\alpha }=-\ln(\alpha ).$ For example, for a 95% confidence cone, $\alpha =0.05,e_{\alpha }=-\ln(0.05)=2.996,$ and thus $q=\arcsin(1.731{\hat {\sigma }}).$ ## Generalizations

The matrix von Mises-Fisher distribution has the density

$f_{n,p}(\mathbf {X} ;\mathbf {F} )\propto \exp(\operatorname {tr} (\mathbf {F} ^{T}\mathbf {X} ))$ supported on the Stiefel manifold of $n\times p$ orthonormal p-frames $\mathbf {X}$ , where $\mathbf {F}$ is an arbitrary $n\times p$ real matrix.