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

arXiv:2404.07358v1 Announce Type: cross
Abstract: 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, …

abstract arxiv distribution family fisher form function however mean natural normalization parameters solution stat.co stat.ml terms type