B-CONCORD -- A scalable Bayesian high-dimensional precision matrix estimation procedure

• Main Authors: Jalali, Peyman, Khare, Kshitij, Michailidis, George
• Format: Journal Article
• Language: English
• Published: 18-05-2020
• Subjects: Statistics - Methodology
• DOI: 10.48550/arxiv.2005.09017

Sparse estimation of the precision matrix under high-dimensional scaling
constitutes a canonical problem in statistics and machine learning. Numerous
regression and likelihood based approaches, many frequentist and some Bayesian
in nature have been developed. Bayesian methods provide direct uncertainty
quantification of the model parameters through the posterior distribution and
thus do not require a second round of computations for obtaining debiased
estimates of the model parameters and their confidence intervals. However, they
are computationally expensive for settings involving more than 500 variables.
To that end, we develop B-CONCORD for the problem at hand, a Bayesian analogue
of the CONvex CORrelation selection methoD (CONCORD) introduced by Khare et al.
(2015). B-CONCORD leverages the CONCORD generalized likelihood function
together with a spike-and-slab prior distribution to induce sparsity in the
precision matrix parameters. We establish model selection and estimation
consistency under high-dimensional scaling; further, we develop a procedure
that refits only the non-zero parameters of the precision matrix, leading to
significant improvements in the estimates in finite samples. Extensive
numerical work illustrates the computational scalability of the proposed
approach vis-a-vis competing Bayesian methods, as well as its accuracy.