While a set of covariance matrices corresponding to different populations are unlikely to be exactly equal they can still exhibit a high degree of similarity. For example, some pairs of variables may be positively correlated across most groups, while the correlation between other pairs may be consistently negative. In such cases much of the similarity across covariance matrices can be described by similarities in their principal axes, the axes defined by the eigenvectors of the covariance matrices. Estimating the degree of across-population eigenvector heterogeneity can be helpful for a variety of estimation tasks. Eigenvector matrices can be pooled to form a central set of principal axes, and to the extent that the axes are similar, covariance estimates for populations having small sample sizes can be stabilized by shrinking their principal axes towards the across-population center. To this end, this article develops a hierarchical model and estimation procedure for pooling principal axes across several populations. The model for the across-group heterogeneity is based on a matrix-valued antipodally symmetric Bingham distribution that can flexibly describe notions of “center” and “spread” for a population of orthonormal matrices.
Some key words: Bayesian inference, copula, Markov chain Monte Carlo, principal components, random matrix, Stiefel manifold