Novel sampling method for the von Mises–Fisher distribution

The von Mises–Fisher distribution is a widely used probability model in directional statistics. An algorithm for generating pseudo-random vectors from this distribution was suggested by Wood (Commun Stat Simul Comput 23(1):157–164, 1994), which is based on a rejection sampling scheme. This paper proposes an alternative to this rejection sampling approach for drawing pseudo-random vectors from arbitrary von Mises–Fisher distributions. A useful mixture representation is derived, which is a mixture of beta distributions with mixing weights that follow a confluent hypergeometric distribution. A condensed table-lookup method is adopted for sampling from the confluent hypergeometric distribution. A theoretical analysis investigates the amount of computation required to construct the condensed lookup table. Through numerical experiments, we demonstrate that the proposed algorithm outperforms the rejection-based method when generating a large number of pseudo-random vectors from the same distribution.