Selecting the tuning parameter in penalized Gaussian graphical models

Penalized inference of Gaussian graphical models is a way to assess the conditional independence structure in multivariate problems. In this setting, the conditional independence structure, corresponding to a graph, is related to the choice of the tuning parameter, which determines the model complexity or degrees of freedom. There has been little research on the degrees of freedom for penalized Gaussian graphical models. In this paper, we propose an estimator of the degrees of freedom in l1-penalized Gaussian graphical models. Specically, we derive an estimator inspired by the generalized information criterion and propose to use this estimator as the bias term for two information criteria. We called these tuning parameter selectors GAIC and GBIC. These selectors can be used to choose the tuning parameter, i.e. the optimal tuning parameter is the minimizer of GAIC or GBIC. A simulation study shows that GAIC tends to improve the performance of both AIC-type and CV-type model selectors, in terms of estimation quality (entropy loss function) in the high-dimensional setting. Moreover, GBIC model selector improves the performance of both BIC-type and CV-type model selectors, in terms of support recovery (Fscore). A data analysis shows that GBIC selects a tuning parameter that produces a sparser graph with respect to BIC and a CV-type model selector (KLCV).