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Sparse estimation of the precision matrix under high-dimensional scalingconstitutes a canonical problem in statistics and machine learning. Numerousregression and likelihood based approaches, many frequentist and some Bayesianin nature have been developed. Bayesian methods provide direct uncertaintyquantification of the model parameters through the posterior distribution andthus do not require a second round of computations for obtaining debiasedestimates of the model parameters and their confidence intervals. However, theyare computationally expensive for settings involving more than 500 variables.To that end, we develop B-CONCORD for the problem at hand, a Bayesian analogueof the CONvex CORrelation selection methoD (CONCORD) introduced by Khare et al.(2015). B-CONCORD leverages the CONCORD generalized likelihood functiontogether with a spike-and-slab prior distribution to induce sparsity in theprecision matrix parameters. We establish model selection and estimationconsistency under high-dimensional scaling; further, we develop a procedurethat refits only the non-zero parameters of the precision matrix, leading tosignificant improvements in the estimates in finite samples. Extensivenumerical work illustrates the computational scalability of the proposedapproach vis-a-vis competing Bayesian methods, as well as its accuracy.

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