A Data Driven Bayesian Graphical Ridge Estimator

TL;DR

This paper introduces a Bayesian graphical ridge estimator that emphasizes accurate association detection over sparsity, offering computational efficiency and improved performance for non-sparse precision matrices, demonstrated on biological data.

Contribution

It develops a novel Bayesian graphical ridge-type prior and a block Gibbs sampler, extending to an adaptive version, for efficient estimation of Gaussian graphical models.

Findings

Efficient Gibbs sampler for precision matrices

Superior performance on non-sparse matrices

Successful application to cell signaling data

Abstract

Bayesian methodologies prioritising accurate associations above sparsity in Gaussian graphical model (GGM) estimation remain relatively scarce in scientific literature. It is well accepted that the $\ell_2$ penalty enjoys a smaller computational footprint in GGM estimation, whilst the $\ell_1$ penalty encourages sparsity in the estimand. The Bayesian adaptive graphical lasso prior is used as a departure point in the formulation of a computationally efficient graphical ridge-type prior for events where accurate associations are prioritised over sparse representations. A novel block Gibbs sampler for simulating precision matrices is constructed using a ridge-type penalisation. The Bayesian graphical ridge-type prior is extended to a Bayesian adaptive graphical ridge-type prior. Synthetic experiments indicate that the graphical ridge-type estimators enjoy computational efficiency, in moderate dimensions, and numerical performance, for relatively non-sparse precision matrices, when compared to their lasso counterparts. The adaptive graphical ridge-type estimator is applied to cell signaling data to infer key associations between phosphorylated proteins in human T cell signalling. All computational workloads are carried out using the baygel R package.