The Generalized von Mises℃Fisher Distribution

In this chapter we introduce the broad class of generalized von Mises–Fisher (GvMF) distributions on the unit hypersphere S p − 1, which arises from a generalization of the von Mises–Fisher distribution. This class of distributions has some important information theoretic properties. It is shown that, under constraints on some moments along some fixed directions, and using Kullback–Leibler information as measure, the closest distribution to any predetermined one on S p − 1, has the GvMF form. Lower bounds for the Kullback–Leibler information in this context are also provided. Several connections between GvMF and other directional or linear distributions are given. GvMF distributions can be re-expressed in terms of generalized von Mises distributions when p = 2. GvMF distributions arise as offset multivariate normal distributions, as offset singular normal distributions and as offset distributions from an exponential spherical type of distributions. Various forms of GvMF densities which feature uni- and multimodality, with different shape of modes, girdle and with other particularities are graphically illustrated.