  The von Mises-Fisher family is a parametric family of distributions on the
surface of the unit ball, summarised by a concentration parameter and a mean
direction. As a quasi-Bayesian prior, the von Mises-Fisher distribution is a
convenient and parsimonious choice when parameter spaces are isomorphic to the
hypersphere (e.g., maximum score estimation in semi-parametric discrete choice,
estimation of single-index treatment assignment rules via empirical welfare
maximisation, under-identifying linear simultaneous equation models). Despite a
long history of application, measures of statistical divergence have not been
analytically characterised for von Mises-Fisher distributions. This paper
provides analytical expressions for the $f$-divergence of a von Mises-Fisher
distribution from another, distinct, von Mises-Fisher distribution in
$\mathbb{R}^p$ and the uniform distribution over the hypersphere. This paper
also collect several other results pertaining to the von Mises-Fisher family of
distributions, and characterises the limiting behaviour of the measures of
divergence that we consider.
