Bayesian sparse covariance decomposition with a graphical structure

We consider the problem of estimating covariance matrices of a particular structure that is a summation of a low-rank component and a sparse component. This is a general covariance structure encountered in multiple statistical models including factor analysis and random effects models, where the low-rank component relates to the correlations among variables coming from the latent factors or random effects and the sparse component displays the correlations of the remaining residuals. We propose a Bayesian method for estimating the covariance matrices of such structures by representing the covariance model in the form of a factor model with an unknown number of latent factors. We introduce binary indicators for factor selection and rank estimation for the low-rank component, combined with a Bayesian lasso method for the estimation of the sparse component. Simulation studies show that our method can recover the rank as well as the sparsity of the two respective components. We further extend our method to a latent-factor Markov graphical model, with a focus on the sparse conditional graphical model of the residuals as well as selecting the number of factors. We show through simulations that our Bayesian model can successfully recover both the number of latent factors and the Markov graphical model of the residuals.