  In a Gaussian graphical model, the conditional independence between two
variables are characterized by the corresponding zero entries in the inverse
covariance matrix. Maximum likelihood method using the smoothly clipped
absolute deviation (SCAD) penalty (Fan and Li, 2001) and the adaptive LASSO
penalty (Zou, 2006) have been proposed in literature. In this article, we
establish the result that using Bayesian information criterion (BIC) to select
the tuning parameter in penalized likelihood estimation with both types of
penalties can lead to consistent graphical model selection. We compare the
empirical performance of BIC with cross validation method and demonstrate the
advantageous performance of BIC criterion for tuning parameter selection
through simulation studies.
