  We consider the problem of estimating high-dimensional covariance matrices of
a particular structure, which is a summation of low rank and sparse matrices.
This covariance structure has a wide range of applications including factor
analysis and random effects models. We propose a Bayesian method of estimating
the covariance matrices by representing the covariance model in the form of a
factor model with 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 sparse component estimation.
Simulation studies show that our method can recover the rank as well as the
sparsity of the two components respectively. We further extend our method to a
graphical factor model where the graphical model of the residuals as well as
selecting the number of factors is of interest. We employ a hyper-inverse
Wishart prior for modeling decomposable graphs of the residuals, and a Bayesian
graphical lasso selection method for unrestricted graphs. We show through
simulations that the extended models can recover both the number of latent
factors and the graphical model of the residuals successfully when the sample
size is sufficient relative to the dimension.
