Models of Random Sparse Eigenmatrices with Application to Bayesian Factor Analysis

We discuss a new class of models for random covariance structures defined by probability distributions over sparse eigenmatrices. The decomposition of orthogonal square matrices in terms of Givens rotations defines a natural, interpretable framework for defining prior distributions over the sparsity structure of random eigenmatrices. We explore some theoretical aspects and implications for conditional independence graphical features arising in Gaussian models, and develop classes of priors over these matrices to underlie Bayesian modeling of resulting variance matrices. We discuss exploratory data analysis in this context, and define and implement Bayesian analysis based on reversible jump Markov chain Monte Carlo. This analysis is extended to the context of multivariate normal mixture models using the novel sparsity priors for variance matrices of the normal components, and this is applied in a study of a 20−dimensional gene expression data set in breast cancer classification.