  Directional data emerges in a wide array of applications, ranging from
atmospheric sciences to medical imaging. Modeling such data, however, poses
unique challenges by virtue of their being constrained to non-Euclidean spaces
like manifolds. Here, we present a unified Bayesian framework for inference on
the Stiefel manifold using the Matrix Langevin distribution. Specifically, we
propose a novel family of conjugate priors and establish a number of
theoretical properties relevant to statistical inference. %Importantly, these
include the propriety of these priors and concentration characterization.
Conjugacy enables the translation of these properties to their corresponding
posteriors, which we exploit to develop the posterior inference scheme. For the
implementation of the posterior computation, including the posterior sampling,
we adopt a novel computational procedure for evaluating the hypergeometric
function of matrix arguments that appears as normalization constants in the
relevant densities.
