Quasi-Bayesian estimation of large Gaussian graphical models

Abstract

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 Bühlmann (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⋆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 Atchadé (2019). We illustrate the methodology using real and simulated data examples.