2024/04/10 by Marcel Nonnenmacher, Maneesh Sahani, Nonnenmacher, Marcel +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Census and Population Estimation #Computation (stat.CO) #FOS: Computer and information sciences #Machine Learning (stat.ML) #Statistical Methods and Bayesian Inference
paper · pdf · doi:10.48550/arxiv.2404.07358
openalex publication_date 2024/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.