Extended Bayesian Information Criteria for Gaussian Graphical Models

TLDR

Gaussian graphical models with sparse inverse covariance matrices are widely used, and information criteria guide structure recovery and tuning of methods such as the graphical lasso, especially when the number of non‑zero parameters can grow with the number of variables. The paper establishes the consistency of an extended Bayesian information criterion for Gaussian graphical models when both the number of variables and the sample size grow. The authors evaluate the criterion on simulated data with the graphical lasso, showing its superiority over cross‑validation and the ordinary BIC when both p and q scale with n. The criterion outperforms cross‑validation and ordinary BIC when both the number of variables and non‑zero parameters grow with the sample size.

Abstract

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.