  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.
