Bayesian Regularization for Graphical Models With Unequal Shrinkage

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 nonconvex penalty approximating the ℓ0 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. Supplementary materials for this article are available online.

本文针对高维稀疏精度矩阵（precision matrix）的估计问题，构建贝叶斯推断框架，该框架通过拉普拉斯先验（Laplace prior）的混合分布实现自适应收缩与稀疏性诱导。除从贝叶斯视角对所提建模框架开展理论分析外，本文还从惩罚似然视角考察最大后验（maximum a posteriori, MAP）估计器，推导出一种可近似ℓ0范数惩罚的新型非凸惩罚函数。在温和正则条件下，针对该唯一最大后验估计器，本文证明了其在多种矩阵范数下的估计一致性最优误差率，以及稀疏结构恢复的选择一致性。为实现快速高效的计算，本文提出期望最大化（expectation-maximization, EM）算法，用于求解精度矩阵的最大后验估计，以及潜在稀疏结构边的（近似）后验概率。通过大量模拟实验与呼叫中心（call center）真实数据集的应用案例，本文验证了所提方法相较于现有同类方法的优异性能。本文补充材料可在线获取。