Quasi-Bayesian estimation of large Gaussian graphical models

This paper deals with the Bayesian estimation of large precision matrices in Gaussian graphical models. We develop a quasi-Bayesian implementation of the neighborhood selection method of Meinshausen and Balmann (2016). The method produces a product form quasi-posterior distribution that can be efficiently explored by parallel computing. Under some restrictions on the true precision matrix, we show that the quasi-posterior distribution contracts in the spectral norm at the rate of O{s(*)root ln(p)/n}, where p is the number of nodes in the graph, n the sample size, and s(*) is the maximum degree of the undirected graph defined by the true precision matrix. We develop a Markov Chain Monte Carlo algorithm for approximate computations, following an approach from Atchade (2019). We illustrate the methodology using real and simulated data examples. (C) 2019 Elsevier Inc. All rights reserved.