Promote sign consistency in the joint estimation of precision matrices

Abstract

Abstract The Gaussian graphical model is a popular tool for inferring the relationships among random variables, where the precision matrix provides a natural interpretation of conditional independence. With high-dimensional data, sparsity of the precision matrix is often assumed, and various regularization methods have been applied for estimation. In several scenarios, it is desirable to conduct the joint estimation of multiple precision matrices. In joint estimation, entries corresponding to the same element of multiple precision matrices form a group, and group regularization methods have been applied for the estimation and identification of sparsity structures. In many practical examples, it can be difficult to interpret the results when parameters within the same group have conflicting signs. Unfortunately, existing methods lack an explicit mechanism in regards to sign consistency of group parameters. To tackle this problem, a novel regularization method is developed for the joint estimation of multiple precision matrices. It effectively enhances the sign consistency of group parameters and hence can lead to more interpretable results, while still allowing for conflicting signs to achieve full flexibility. The method’s consistency properties are rigorously established. Simulations show that the proposed method outperforms competing alternatives under a variety of settings. For the two data examples, the proposed approach leads to interpretable results that are different from the alternatives.