A random \(d\)-dimensional unit length vector \(x\) has a von
Mises-Fisher (or Langevin, short: vMF) distribution with parameter
\(\theta\) if its density with respect to the uniform distribution
on the unit hypersphere is given by
$$f(x|\theta) = \exp(\theta'x) / {}_0F_1(; d/2; \|\theta\|^2/4),$$
where \({}_0F_1\) is a generalized hypergeometric function
(e.g.,
https://en.wikipedia.org/wiki/Generalized_hypergeometric_function)
and related to the modified Bessel function \(I_\nu\) of the first
kind via
$${}_0F_1(; \nu+1; z^2/4) =
I_\nu(z)\Gamma(\nu+1) / (z/2)^\nu.$$
With this parametrization, the von Mises-Fisher family is the natural
exponential family through the uniform distribution on the unit
sphere, with cumulant transform
$$M(\theta) = \log({}_0F_1(; d/2; \|\theta\|^2/4)).$$
We note that the vMF distribution is commonly parametrized by the
mean direction parameter \(\mu = \theta / \|\theta\|\) (which however is not well-defined if \(\theta =
0\)) and the concentration parameter \(\kappa = \|\theta\|\), e.g.,
https://en.wikipedia.org/wiki/Von_Mises%E2%80%93Fisher_distribution
(which also uses the un-normalized Haar measure on the unit sphere as
the reference distribution, and hence includes the “area” of
the unit sphere as an additional normalizing constant).
dmovMF computes the (log) density of mixtures of vMF
distributions.
rmovMF generates samples from finite mixtures of vMF
distributions, using Algorithm VM* in Wood (1994) for sampling from
the vMF distribution.
Arguments theta and alpha are recycled to a common
number of mixture components.