Von Mises-Like Probability Density Functions on Surfaces

Abstract : Directional densities were introduced in the pioneering work of von Mises, with the definition of a rotationally invariant probability distribution on the circle. It was further generalized to more complex objects like the torus or the hyperbolic space. The purpose of the present work is to give a construction of equivalent objects on surfaces with genus larger than or equal to 2, for which an hyperbolic structure exists. Although the directional densities on the torus were introduced by several authors and are closely related to the original von Mises distribution, allowing more than one hole is challenging as one cannot simply add more angular coordinates. The approach taken here is to use a wrapping as in the case of the circular wrapped Gaussian density, but with a summation taken over all the elements of the group that realizes the surface as a quotient of the hyperbolic plane.