We present and develop a new class of models for random, sparse orthogonal matrices and utilize the framework to define new prior distributions over the eigenvectors of covariance matrices. Using a generalized Givens representation of square orthogonal matrices in terms of products of rotation matrices, the framework is open to incorporating varying degrees of sparsity in resulting eigenvector matrices. This is of interest in developing parsimonious representations for multivariate analysis, and in Bayesian factor analysis in particular. We discuss the model theory and connections with Gaussian graphical models, and then develop Bayesian model fitting via reversible jump Markov chain Monte Carlo methods. Some examples are given in analyses of gene expression data in a simple random sample and then a Gaussian mixture model context.

Research partially supported by the National Science Foundation under grant DMS 1106516 and the National Institutes of Health under grant 1RC1-AI086032. Any opinions, findings and conclusions or recommendations expressed in this work are those of the authors and do not necessarily reflect the views of the NIH.