Scalable Bayesian Structure Learning for Gaussian Graphical Models Using Marginal Pseudo-likelihood

TL;DR

This paper introduces scalable Bayesian algorithms for learning Gaussian graphical models by integrating out precision matrices using a marginal pseudo-likelihood, enabling efficient exploration of large graph spaces with uncertainty quantification.

Contribution

It develops novel MCMC algorithms based on marginal pseudo-likelihood for scalable Bayesian structure learning in Gaussian graphical models, with theoretical guarantees and practical implementation.

Findings

Algorithms scale to graphs with over 1,000 nodes.

Significant computational gains over existing methods.

Effective recovery of meaningful biological network structures.

Abstract

Bayesian methods for learning Gaussian graphical models offer a principled framework for quantifying model uncertainty and incorporating prior knowledge. However, their scalability is constrained by the computational cost of jointly exploring graph structures and precision matrices. To address this challenge, we perform inference directly on the graph by integrating out the precision matrix. We adopt a marginal pseudo-likelihood approach, eliminating the need to compute intractable normalizing constants and perform computationally intensive precision matrix sampling. Building on this framework, we develop continuous-time (birth-death) and discrete-time (reversible jump) Markov chain Monte Carlo (MCMC) algorithms that efficiently explore the posterior over graph space. We establish theoretical guarantees for posterior contraction, convergence, and graph selection consistency. The algorithms scale to large graph spaces, enabling parallel exploration for graphs with over 1,000 nodes, while providing uncertainty quantification and supporting flexible prior specification over the graph space. Extensive simulations show substantial computational gains over state-of-the-art Bayesian approaches without sacrificing graph recovery accuracy. Applications to human and mouse gene expression datasets demonstrate the ability of our approach to recover biologically meaningful structures and quantify uncertainty in complex networks. An implementation is available in the R package BDgraph.