A short note on parameter approximation for von Mises-Fisher distributions: and a fast implementation of Is(x)

In high-dimensional directional statistics one of the most basic probability distributions is the von Mises-Fisher (vMF) distribution. Maximum likelihood estimation for the vMF distribution turns out to be surprisingly hard because of a difficult transcendental equation that needs to be solved for computing the concentration parameter κ. This paper is a followup to the recent paper of Tanabe et al. (Comput Stat 22(1):145–157, 2007), who exploited inequalities about Bessel function ratios to obtain an interval in which the parameter estimate for κ should lie; their observation lends theoretical validity to the heuristic approximation of Banerjee et al. (JMLR 6:1345–1382, 2005). Tanabe et al. (Comput Stat 22(1):145–157, 2007) also presented a fixed-point algorithm for computing improved approximations for κ. However, their approximations require (potentially significant) additional computation, and in this short paper we show that given the same amount of computation as their method, one can achieve more accurate approximations using a truncated Newton method. A more interesting contribution of this paper is a simple algorithm for computing I s (x): the modified Bessel function of the first kind. Surprisingly, our naïve implementation turns out to be several orders of magnitude faster for large arguments common to high-dimensional data, than the standard implementations in well-established software such as Mathematica ©, Maple ©, and Gp/Pari