Von Mises-Fisher distribution



The von Mises-Fisher distribution is a probability distribution on the $$(p-1)$$-dimensional sphere in $$\mathbb{R}^{p}$$. If $$p=2$$ the distribution reduces to the von Mises distribution on the circle. The distribution belongs to the field of directional statistics.

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}; \mu, \kappa)=C_{p}(\kappa)\exp \left( {\kappa \mu^T \mathbf{x} } \right)

$$

where $$ \kappa \ge 0, \left \Vert \mu \right \Vert =1 \,$$ 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 and order $$v$$.

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 $$\kappa=0\,$$. If $$p=3$$, the distribution is also called the Fisher distribution.

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