📂Probability Distribution

Bingham-Mardia Distribution

Definition ¹

The Bingham-Mardia Distribution is a multivariate distribution $\text{BM}_{p} \left( \mu , \kappa, \nu \right)$ that has a probability density function for Unique Mode $\mu \in S^{p-1}$, Concentration $\kappa > 0$, and radius $\nu > 0$ as follows. $$ f \left( \mathbf{x} \right) = {{ 1 } \over { \alpha \left( \kappa , \nu \right) }} \exp \left( - \kappa \left( \mu^{T} \mathbf{x} - \nu \right)^{2} \right) \qquad , \mathbf{x} \in S^{p-1} $$ Here, $\alpha \left( \kappa , \nu \right) > 0$ is the normalizing constant that makes $\int_{S^{p-1}} f(\mathbf{x}) d \mathbf{x} = 1$.

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• $S^{p-1} \subset \mathbb{R}^{p}$ is the unit sphere.
• $\mu ^{T}$ is the transpose of the vector $\mu$.

Explanation

The Bingham-Mardia distribution is a probability distribution that forms clusters in the shape of Small Circles on the sphere.

von Mises-Fisher distribution’s probability density function: $$ f \left( \mathbf{x} \right) = \left( {{ \kappa } \over { 2 }} \right)^{p/2-1} {{ 1 } \over { \gamma \left( p/2 \right) I_{p/2-1} \left( \kappa \right) }} \exp \left( \kappa \mu^{T} \mathbf{x} \right) \qquad , \mathbf{x} \in S^{p-1} $$

When compared to the von Mises-Fisher distribution, which felt like a normal distribution on the sphere, it will be easily understood that in the probability density function of the Bingham-Mardia distribution, $\kappa \left( \mu^{T} \mathbf{x} - \nu \right)^{2}$ plays a role in forming the circle shape.