von Mises-Fisher distributions and their statistical divergence ∗

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 quasiBayesian 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 R 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. ∗The authors extend their thanks to Yu-Lin Lin and Hugo Lopez for helpful discussion and suggestions. The authors also gratefully acknowledge financial support from the ESRC, via the ESRC Centre for Microdata Methods and Practice (CeMMAP) (grant number RES-589-28-0001) and an ESRC studentship, respectively. Kitagawa further acknowledges financial support from the European Research Council (Starting grant No. 715940). †Brown University, Department of Economics and University College London, Department of Economics. Email: toru kitagawa@brown.edu. ‡University College London, Department of Economics. 1 ar X iv :2 20 2. 05 19 2v 1 [ ec on . E M ] 1 0 Fe b 20 22 1 The von Mises-Fisher family of distributions The von Mises-Fisher family of distributions is well-known in the field of directional statistics but is foreign to economics and, as such, warrants some introduction. Also known as the Langevin family (Watson, 1984), the von Mises-Fisher family recognises those two titans of statistics, Sir Ronald Fisher and Richard von Mises, for their seminal contributions in considering Gaussianity on the circle (von Mises, 1918) and the sphere (Fisher, 1953). Subsequent work has generalised the von Mises-Fisher family to R, and has led to the definition of other related parametric distributions such as the Bingham family (Bingham, 1974) and the Fisher-Bingham or Kent family (Kent, 1982). The von Mises-Fisher distribution assigns probability mass to the surface of the unit ball – the hypersphere. As such, the von Mises-Fisher family is relevant to situations where the researcher is interested in either the sampling of directional vectors – i.e., vectors of unit length – or in the clustering of some phenomenon on a circular object, such as occurs if data is periodic. Applications range from the study of sea turtle navigation (Hillen et al., 2017), to the study of perihelia of long-tailed comets (Mardia, 1975) and near-earth objects (Sei et al., 2013), to the study of patient arrival data (Mardia, 1975). Sabelfeld (2018) even links the von Mises-Fisher family to the solving of high-dimensional diffusion-advection-reaction equations. The von MisesFisher family is a two-parameter family, summarised by a concentration parameter (or, simply, concentration), which we denote by κ > 0, and a mean direction, which we denote by μ ∈ R. The main contribution of this paper is to provide analytical expressions for the f -divergence of a von Mises-Fisher distribution from two relevant reference distributions given several common choices of function. We study the broad class of Rényi divergence of simple order as well as several other measures of (statistical) divergence that relate to the Rényi class – the χ-square 1See Mardia and Jupp (2009) for a summary of important results in the field of directional statistics.