  The von Mises-Fisher (vMF) is a well-known density model for directional
random variables. The recent surge of the deep embedding methodologies for
high-dimensional structured data such as images or texts, aimed at extracting
salient directional information, can make the vMF model even more popular. In
this article, we will review the vMF model and its mixture, provide detailed
recipes of how to train the models, focusing on the maximum likelihood
estimators, in Python/PyTorch. In particular, implementation of vMF typically
suffers from the notorious numerical issue of the Bessel function evaluation in
the density normalizer, especially when the dimensionality is high, and we
address the issue using the MPMath library that supports arbitrary precision.
For the mixture learning, we provide both minibatch-based large-scale SGD
learning, as well as the EM algorithm which is a full batch estimator. For each
estimator/methodology, we test our implementation on some synthetic data, while
we also demonstrate the use case in a more realistic scenario of image
clustering. Our code is publicly available in
https://github.com/minyoungkim21/vmf-lib.
