Bayesian inference on the Stiefel manifold

The Stiefel manifold $V_{p,d}$ is the space of all $d \times p$ orthonormal matrices, and includes the $d-1$ hypersphere and the space of all orthogonal matrices as special cases. It is often desirable to parametrically or nonparametrically estimate the distribution of variables taking values on this manifold. Unfortunately, the intractable normalizing constant in the likelihood makes Bayesian inference challenging. We develop a novel Markov chain Monte Carlo algorithm to sample from this doubly intractable distribution. By mixing the matrix Langevin with respect to a random probability measure, we define a flexible class of nonparametric models. Theory is provided justifying the flexibility of the model and its asymptotic properties, while we also extend the MCMC algorithm to this situation. We apply our ideas to simulated data and two real datasets, and also discuss how our sampling ideas are applicable to other doubly intractable problems.