Generation of random directions from the generalized von Mises-Fisher distribution

We introduce two algorithms for generating from the bimodal generalized von Mises-Fisher distribution on the sphere. This generalization of the von Mises-Fisher distribution is more flexible, in particular by allowing for multimodality, and it preserves many of the theoretical properties of the von Mises-Fisher. The first proposed generation algorithm is obtained from conditional simulation and acceptance-rejection. The second one is the Metropolis-Hastings with mixture of von Mises-Fisher as jumping or instrumental distribution. These two algorithms are compared through density estimations of generated polar angles.