Bayesian Scalable Precision Factor Analysis for Massive Sparse Gaussian Graphical Models

• Activity Number: 501 - Bayesian Penalized Likelihood Methods for Gaussian Graphical Models
• Type: Invited
• Date/Time: Thursday, August 11, 2022 : 8:30 AM to 10:20 AM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Noirrit Kiran Chandra* and Abhra Sarkar and Peter Mueller
• Companies: The University of Texas at Austin and The University of Texas at Austin and The University of Texas at Austin
• Keywords: Factor models ; False discovery rate control; Gaussian graphical models; Markov chain Monte Carlo; Precision matrix estimation; Scalable computation

Abstract

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 however not true for precision matrices due to the lack of computationally convenient representations which restricts inference to low-to-moderate dimensional problems. We address this remarkable gap in the literature by building on a latent variable representation for such decomposition for precision matrices. 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 also allows the model uncertainty to be easily assessed. The decomposition crucially additionally 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.