Extended Bayesian Information Criteria for Gaussian Graphical Models

TL;DR

This paper introduces an extended Bayesian information criterion for Gaussian graphical models, demonstrating its consistency and superior performance over traditional criteria in high-dimensional settings with growing variables and parameters.

Contribution

It establishes the consistency of an extended BIC for Gaussian graphical models with increasing variables and non-zero parameters, improving model selection in high-dimensional contexts.

Findings

Extended BIC outperforms cross-validation and ordinary BIC in simulations.

The criterion remains consistent as both variables and non-zero parameters grow.

Performance verified with graphical lasso in high-dimensional scenarios.

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.