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 non-convex 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.

本文针对高维稀疏精度矩阵的估计问题，构建了一种贝叶斯框架，该框架通过拉普拉斯先验混合模型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方法进行理论分析外，本文还从惩罚似然视角对最大后验估计器（maximum a posteriori, MAP）展开研究，进而导出一种近似ℓ₀惩罚的新型非凸惩罚函数。在温和条件下，针对该唯一MAP估计器，本文推导了各类矩阵范数下估计一致性的最优误差率，以及稀疏结构恢复的选择一致性保证。为实现快速高效的计算，本文提出一种期望最大化算法（Expectation-Maximization, EM），用于求解精度矩阵的MAP估计值，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一项呼叫中心数据集的真实应用案例，本文验证了所提方法相较于现有同类方法的优异性能。