A Bayesian approach for partial Gaussian graphical models with sparsity

TL;DR

This paper develops Bayesian hierarchical models with spike-and-slab priors for estimating sparse partial Gaussian graphical models, applicable to small or high-dimensional data, with theoretical guarantees and demonstrated effectiveness.

Contribution

Introduces novel Bayesian hierarchical approaches with sparsity and group sparsity for partial Gaussian graphical models, including theoretical model selection guarantees and efficient Gibbs sampling algorithms.

Findings

Models achieve high support recovery accuracy.

Bayesian methods outperform existing approaches in simulations.

Effective in small and high-dimensional settings.

Abstract

We explore various Bayesian approaches to estimate partial Gaussian graphical models. Our hierarchical structures enable to deal with single-output as well as multiple-output linear regressions, in small or high dimension, enforcing either no sparsity, sparsity, group sparsity or even sparse-group sparsity for a bi-level selection through partial correlations (direct links) between predictors and responses, thanks to spike-and-slab priors corresponding to each setting. Adaptative and global shrinkages are also incorporated in the Bayesian modeling of the direct links. An existing result for model selection consistency is reformulated to stick to our sparse and group-sparse settings, providing a theoretical guarantee under some technical assumptions. Gibbs samplers are developed and a simulation study shows the efficiency of our models which give very competitive results, especially in terms of support recovery. To conclude, a real dataset is investigated.