Discrete mixture representations of spherical distributions

TL;DR

This paper introduces a unified method for representing various spherical probability distributions as discrete mixtures, using surface harmonic expansions and connections to stochastic processes.

Contribution

It provides a general approach to isotropic distributions on spheres via density expansions and links to stochastic processes like random walks and diffusions.

Findings

Discrete mixture representations for von Mises--Fisher, Watson, and angular Gaussian distributions.

Connections established between mixture representations and spherical stochastic processes.

Framework enables new insights into the structure of spherical distributions.

Abstract

We obtain discrete mixture representations for parametric families of probability distributions on Euclidean spheres, such as the von Mises--Fisher, the Watson and the angular Gaussian families. In addition to several special results we present a general approach to isotropic distribution families that is based on density expansions in terms of special surface harmonics. We discuss the connections to stochastic processes on spheres, in particular random walks, discrete mixture representations derived from spherical diffusions, and the use of Markov representations for the mixing base to obtain representations for families of spherical distributions.