Dually flat structure of von Mises-Fisher distributions

The von Mises--Fisher (vMF) distribution is the natural exponential family analogue of the normal distribution on the unit hypersphere. We bring together previous results of its dually flat structure and establish new results on the information geometry of the family of vMF distributions. In particular, we characterize the exponential and mixture autoparallel submanifolds, and give the classification of doubly autoparallel submanifolds. Furthermore, we derive explicit formulas for the unique minimizers of the Kullback-Leibler divergence on these submanifolds and establish the corresponding projection theorems.