Precision Matrix Estimation under the Horseshoe-like Prior-Penalty Dual

The problem of precision matrix estimation in a multivariate Gaussian model is fundamental to network estimation. Although there exist both Bayesian and frequentist approaches to this, it is difficult to obtain good Bayesian and frequentist properties under the same prior-penalty dual, complicating justification. It is well known, for example, that the Bayesian version of the popular lasso estimator has poor posterior concentration properties. To bridge this gap for the precision matrix estimation problem, our contribution is a novel prior-penalty dual that closely approximates the popular graphical horseshoe prior and penalty, and performs well in both Bayesian and frequentist senses. A chief difficulty with the horseshoe prior is a lack of closed form expression of the density function, which we overcome in this article, allowing us to directly study the penalty function. In terms of theory, we establish posterior convergence rate of the precision matrix that matches the oracle rate, in addition to the frequentist consistency of the maximum a posteriori estimator. In addition, our results also provide theoretical justifications for previously developed approaches that have been unexplored so far, e.g. for the graphical horseshoe prior. Computationally efficient Expectation Conditional Maximization and Markov chain Monte Carlo algorithms are developed respectively for the penalized likelihood and fully Bayesian estimation problems, using the same latent variable framework. In numerical experiments, the horseshoe-based approaches echo their superior theoretical properties by comprehensively outperforming the competing methods. A protein-protein interaction network estimation in B-cell lymphoma is considered to validate the proposed methodology.