Efficient von Mises-Fisher concentration parameter estimation using Taylor series

A simple method is outlined for constructing a Taylor series for the Maximum Likelihood Estimate of the von Mises–Fisher concentration parameter based around an initial heuristic estimate. While existing treatments require multiple computationally intensive calculations of a Bessel ratio, this method provides accurate results using only one such calculation. The accuracy of the method is tested extensively, and the reuse of the Taylor series for multiple calculations is explored.