  Despite major methodological developments, Bayesian inference for Gaussian
graphical models remains challenging in high dimension due to the tremendous
size of the model space. This article proposes a method to infer the marginal
and conditional independence structures between variables by multiple testing
of hypotheses. Specifically, we introduce closed-form Bayes factors under the
Gaussian conjugate model to evaluate the null hypotheses of marginal and
conditional independence between variables. Their computation for all pairs of
variables is shown to be extremely efficient, thereby allowing us to address
large problems with thousands of nodes. Moreover, we derive exact tail
probabilities from the null distributions of the Bayes factors. These allow the
use of any multiplicity correction procedure to control error rates for
incorrect edge inclusion. We demonstrate the proposed approach to graphical
model selection on various simulated examples as well as on a large gene
expression data set from The Cancer Genome Atlas.
