  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.
