Large-scale Bayesian Structure Learning for Gaussian Graphical Models using Marginal Pseudo-likelihood

Bayesian methods for learning Gaussian graphical models offer a comprehensive framework that addresses model uncertainty and incorporates prior knowledge. Despite their theoretical strengths, the applicability of Bayesian methods is often constrained by computational demands, especially in modern contexts involving thousands of variables. To overcome this issue, we introduce two novel Markov chain Monte Carlo (MCMC) search algorithms with a significantly lower computational cost than leading Bayesian approaches. Our proposed MCMC-based search algorithms use the marginal pseudo-likelihood approach to bypass the complexities of computing intractable normalizing constants and iterative precision matrix sampling. These algorithms can deliver reliable results in mere minutes on standard computers, even for large-scale problems with one thousand variables. Furthermore, our proposed method efficiently addresses model uncertainty by exploring the full posterior graph space. We establish the consistency of graph recovery, and our extensive simulation study indicates that the proposed algorithms, particularly for large-scale sparse graphs, outperform leading Bayesian approaches in terms of computational efficiency and accuracy. We also illustrate the practical utility of our methods on medium and large-scale applications from human and mice gene expression studies. The implementation supporting the new approach is available through the R package BDgraph.