Bayesian Scalable Precision Factor Analysis for Massive Sparse Gaussian Graphical Models

We propose a novel approach to estimating the precision matrix of multivariate Gaussian data that relies on decomposing them into a low-rank and a diagonal component. Such decompositions are very popular for modeling large covariance matrices as they admit a latent factor based representation that allows easy inference. The same is not true for precision matrices, due to the lack of computationally convenient representation, which restricts the use to low to moderate dimensional problems. We address this remarkable gap in the literature by introducing a novel latent variable representation for such decomposition for precision matrices as well. The construction leads to an efficient Gibbs sampler that scales very well to high-dimensional problems far beyond the limits of the current state-of-the-art. The ability to efficiently explore the full posterior space allows the model uncertainty to be easily assessed. The decomposition also crucially allows us to adapt sparsity inducing priors to shrink the insignificant entries of the precision matrix toward zero, making the approach adaptable to high-dimensional small-sample-size sparse settings. Exact zeros in the matrix encoding the underlying conditional independence graph are then determined via a novel posterior false discovery rate control procedure. We evaluate the method's empirical performance through synthetic experiments and illustrate its practical utility in data sets from two different application domains.