Bayesian sparse graphical models and their mixtures using lasso selection priors

TL;DR

This paper introduces Bayesian methods with lasso priors for sparse Gaussian graphical models, enabling simultaneous model selection and estimation of the precision matrix, extended to mixture models for clustered data.

Contribution

It develops a novel selection prior that induces sparsity and positive definiteness in the precision matrix, and extends Bayesian graphical models to mixture models with Dirichlet process priors.

Findings

Effective in inducing sparsity in precision matrices

Performs well in simulations and real data applications

Allows for model selection and estimation simultaneously

Abstract

We propose Bayesian methods for Gaussian graphical models that lead to sparse and adaptively shrunk estimators of the precision (inverse covariance) matrix. Our methods are based on lasso-type regularization priors leading to parsimonious parameterization of the precision matrix, which is essential in several applications involving learning relationships among the variables. In this context, we introduce a novel type of selection prior that develops a sparse structure on the precision matrix by making most of the elements exactly zero, in addition to ensuring positive definiteness -- thus conducting model selection and estimation simultaneously. We extend these methods to finite and infinite mixtures of Gaussian graphical models for clustered data using Dirichlet process priors. We discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models, including the evaluation of normalizing constants that are functions of parameters of interest which result from the restrictions on the correlation matrix. We evaluate the operating characteristics of our method via several simulations and in application to real data sets.