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Abstract:
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We consider a Bayesian framework for estimating the precision matrix of a high dimensional vector, in which adaptive shrinkage and sparsity are induced by using spike and slab Laplace priors. Besides discussing our formulation from the Bayesian standpoint, we investigate the resultant posterior from a penalized likelihood perspective that gives rise to a new non-convex penalty approximating the L_0 penalty. Optimal error rates for estimation consistency in terms of \ell_\infty, $\ell_F$ and \ell_2 norms along with selection consistency for sparsity structure recovery are shown under mild conditions. For fast and efficient computation, an EM algorithm tis proposed. Through extensive simulation studies and a real application to a call center data, we have demonstrated the fine performance of our method compared with the existing alternatives.
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