Gaussian graphical models with sparsity in the inverse covariance matrix are
of significant interest in many modern applications. For the problem of
recovering the graphical structure, information criteria provide useful
optimization objectives for algorithms searching through sets of graphs or for
selection of tuning parameters of other methods such as the graphical lasso,
which is a likelihood penalization technique. In this paper we establish the
consistency of an extended Bayesian information criterion for Gaussian
graphical models in a scenario where both the number of variables p and the
sample size n grow. Compared to earlier work on the regression case, our
treatment allows for growth in the number of non-zero parameters in the true
model, which is necessary in order to cover connected graphs. We demonstrate
the performance of this criterion on simulated data when used in conjunction
with the graphical lasso, and verify that the criterion indeed performs better
than either cross-validation or the ordinary Bayesian information criterion
when p and the number of non-zero parameters q both scale with n