  Graphical models describe associations between variables through the notion
of conditional independence. Gaussian graphical models are a widely used class
of such models where the relationships are formalized by non-null entries of
the precision matrix. However, in high dimensional cases, standard covariance
estimates are typically unstable. Moreover, it is natural to expect only a few
significant associations to be present in many realistic applications. This
necessitates the injection of sparsity techniques into the estimation.
Classical frequentist methods use penalization for this purpose; in contrast,
fully Bayesian methods are computationally slow, typically requiring iterative
sampling over a quadratic number of parameters in a space constrained by
positive definiteness. We propose a Bayesian graph estimation method based on
an ensemble of Bayesian neighborhood regressions. An attractive feature of our
methods is the ability for easy parallelization across separate graphical
neighborhoods, invoking computational efficiency greater than most existing
methods. Our strategy induces sparsity with a Horseshoe shrinkage prior and
includes a novel variable selection step based on the marginal likelihood from
the predictors ranks. Our method appropriately combines the estimated
regression coefficients to produce a graph estimate and a matrix of partial
correlation estimates for inference. Performance of various methods are
assessed using measures like FDR and TPR. Competitive performance across a
variety of cases is demonstrated through extensive simulations. Lastly, we
apply these methods to investigate the dependence structure across genetic
expressions for women with triple negative breast cancer.
