Contraction of a quasi-Bayesian model with shrinkage priors in precision matrix estimation

Abstract

Currently several Bayesian approaches are available to estimate large sparse precision matrices, including Bayesian graphical Lasso (Wang, 2012), Bayesian structure learning (Banerjee and Ghosal, 2015), and graphical horseshoe (Li et al., 2019). Although these methods have exhibited nice empirical performances, in general they are computationally expensive. Moreover, only a few theoretical results are available for Bayesian graphical models with continuous shrinkage priors. A very recent work (Sagar et al., 2021) studies the posterior concentration properties of the graphical horseshoe prior and graphical horseshoe-like priors under a full likelihood model. In this paper, we propose a new method that integrates some commonly used continuous shrinkage priors into a quasi-Bayesian framework featured by a pseudo-likelihood. Under mild conditions, we establish an optimal posterior contraction rate for the proposed method. Compared to existing approaches, our method has two main advantages. First, our method is computationally efficient while achieving similar error rate; second, our framework is amenable to theoretical analysis. Extensive simulation experiments and the analysis on a real data set are supportive of our theoretical results.