📂Probability Distribution

Von Mises-Fisher Distribution

Definition ¹

The Von Mises-Fisher Distribution is defined as the multivariate distribution $\text{vMF}_{p} \left( \mu , \kappa \right)$ with the following probability density function for Unique Mode $\mu \in S^{p-1}$ and Concentration $\kappa > 0$. $$ f \left( \mathbf{x} \right) = \left( {{ \kappa } \over { 2 }} \right)^{p/2-1} {{ 1 } \over { \Gamma \left( p/2 \right) I_{p/2-1} \left( \kappa \right) }} \exp \left( \kappa \mu^{T} \mathbf{x} \right) \qquad , \mathbf{x} \in S^{p-1} $$

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• $S^{p-1} \subset \mathbb{R}^{p}$ is the unit sphere.
• $\mu ^{T}$ is the transpose of the vector $\mu$.
• $\Gamma$ is the gamma function.
• $I_{\nu}$ is the modified Bessel function of the first kind of order $\nu$. The reason for the usage of such a complex function is explained in the post Why the Modified Bessel Function of the First Kind Appears in Directional Statistics.

Explanation

The Von Mises-Fisher Distribution is called the Von Mises Distribution when $p=2$ and Fisher Distribution when $p=3$. Similar to how the Von Mises Distribution is the normal distribution on the circle, the Fisher Distribution becomes the multivariate normal distribution on the sphere, and although generalizing to $p > 3$ is geometrically hard to imagine, it still holds a similar meaning.

When talking about the normal distribution in Directional Statistics, the Von Mises-Fisher Distribution naturally comes to mind first. The Fisher Distribution, meaning $p=3$, is the normal distribution on the sphere, making it easy to conceive its utility in planetary-scale research, especially that involving Earth.