论文标题
与网络结构协变量的高维回归模型的联合贝叶斯变量和DAG选择一致性
Joint Bayesian Variable and DAG Selection Consistency for High-dimensional Regression Models with Network-structured Covariates
论文作者
论文摘要
我们考虑了在高维回归模型中的协变量的回归系数和协方差矩阵的关节稀疏估计，其中预测因子都与感兴趣的响应变量相关，并且通过高斯定向的acyclic图（DAG）模型相互关联，并且在功能上相关。高斯DAG模型在反协方差矩阵的Cholesky因子中引入稀疏性，而稀疏模式又对应于基础预测因子上的特定条件独立性假设。近年来，贝叶斯推断在回归环境中识别这种网络结构的预测因子方面已经开发了多种方法，但是这些模型的关键稀疏选择属性尚未得到彻底研究。在本文中，我们在回归系数上考虑了带有尖峰和平板先验的分层模型，以及在逆协方差矩阵的Cholesky因子上具有多个形状参数的灵活而通用的DAG-WISHART分布。在轻度的规律性假设下，当允许预测因子的尺寸生长大得多时，我们建立了变量和基础dag的关节选择一致性。我们证明，我们的方法在选择几个模拟设置中选择网络结构化预测变量时优于现有方法。
We consider the joint sparse estimation of regression coefficients and the covariance matrix for covariates in a high-dimensional regression model, where the predictors are both relevant to a response variable of interest and functionally related to one another via a Gaussian directed acyclic graph (DAG) model. Gaussian DAG models introduce sparsity in the Cholesky factor of the inverse covariance matrix, and the sparsity pattern in turn corresponds to specific conditional independence assumptions on the underlying predictors. A variety of methods have been developed in recent years for Bayesian inference in identifying such network-structured predictors in regression setting, yet crucial sparsity selection properties for these models have not been thoroughly investigated. In this paper, we consider a hierarchical model with spike and slab priors on the regression coefficients and a flexible and general class of DAG-Wishart distributions with multiple shape parameters on the Cholesky factors of the inverse covariance matrix. Under mild regularity assumptions, we establish the joint selection consistency for both the variable and the underlying DAG of the covariates when the dimension of predictors is allowed to grow much larger than the sample size. We demonstrate that our method outperforms existing methods in selecting network-structured predictors in several simulation settings.
