TUNING PARAMETER SELECTION FOR PENALIZED LIKELIHOOD ESTIMATION OF GAUSSIAN GRAPHICAL MODEL

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)) has been proposed in the literature. In this article, we establish the result that when p is fixed, using the Bayesian information criterion (BIG) to select the tuning parameter in penalized likelihood estimation with the SCAD penalty can lead to consistent graphical model selection. When p increases with the sample size, a modified BIC with an extra penalty term is proposed. It can consistently select the true graphical model under the condition that p tends to infinity and all the true edges are included in a bounded subset. We compare the empirical performance of BIC with the cross validation method and demonstrate the advantageous performance of BIC criterion for sparse graphical models through simulation studies.