Bayesian Regularization for Graphical Models with Unequal Shrinkage
This work addresses the challenge of sparse structure recovery in graphical models for statisticians and data analysts, representing an incremental improvement with a novel penalty formulation.
The authors tackled the problem of estimating high-dimensional sparse precision matrices by introducing a Bayesian framework with adaptive shrinkage via Laplace mixture priors, which yields a non-convex penalty approximating the ℓ0 norm, and demonstrated its performance through simulations and a call center application.
We consider a Bayesian framework for estimating a high-dimensional sparse precision matrix, in which adaptive shrinkage and sparsity are induced by a mixture of Laplace priors. Besides discussing our formulation from the Bayesian standpoint, we investigate the MAP (maximum a posteriori) estimator from a penalized likelihood perspective that gives rise to a new non-convex penalty approximating the $\ell_0$ penalty. Optimal error rates for estimation consistency in terms of various matrix norms along with selection consistency for sparse structure recovery are shown for the unique MAP estimator under mild conditions. For fast and efficient computation, an EM algorithm is proposed to compute the MAP estimator of the precision matrix and (approximate) posterior probabilities on the edges of the underlying sparse structure. Through extensive simulation studies and a real application to a call center data, we have demonstrated the fine performance of our method compared with existing alternatives.