A solution for the mean parametrization of the von Mises-Fisher distribution

The von Mises-Fisher distribution as an exponential family can be expressed in terms of either its natural or its mean parameters. Unfortunately, however, the normalization function for the distribution in terms of its mean parameters is not available in closed form, limiting the practicality of the mean parametrization and complicating maximum-likelihood estimation more generally. We derive a second-order ordinary differential equation, the solution to which yields the mean-parameter normalizer along with its first two derivatives, as well as the variance function of the family. We also provide closed-form approximations to the solution of the differential equation. This allows rapid evaluation of both densities and natural parameters in terms of mean parameters. We show applications to topic modeling with mixtures of von Mises-Fisher distributions using Bregman Clustering.