Simulation of the Matrix Bingham–von Mises–Fisher Distribution, With Applications to Multivariate and Relational Data

Orthonormal matrices play an important role in reduced-rank matrix approximations and the analysis of matrix-valued data. A matrix Bingham–von Mises–Fisher distribution is a probability distribution on the set of orthonormal matrices that includes linear and quadratic terms in the log-density, and arises as a posterior distribution in latent factor models for multivariate and relational data. This article describes rejection and Gibbs sampling algorithms for sampling from this family of distributions, and illustrates their use in the analysis of a protein–protein interaction network. Supplemental materials, including code and data to generate all of the numerical results in this article, are available online.