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.

中文翻译：

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具有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

我们在高维回归模型中考虑回归系数的联合稀疏估计和协变量的协方差矩阵，其中预测变量既与感兴趣的响应变量相关，又通过高斯有向无环图 (DAG) 在功能上相互关联模型。高斯 DAG 模型在逆协方差矩阵的 Cholesky 因子中引入了稀疏性，而稀疏模式又对应于对基础预测变量的特定条件独立假设。近年来，已经开发了多种方法用于贝叶斯推理，以识别回归设置中的此类网络结构预测变量，但尚未彻底研究这些模型的关键稀疏选择特性。在本文中，我们考虑在回归系数上具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通用的 DAG-Wishart 分布类。在温和的规律性假设下，当允许预测变量的维度增长远大于样本大小时，我们为变量和协变量的潜在 DAG 建立联合选择一致性。我们证明了我们的方法在几种模拟设置中选择网络结构预测器方面优于现有方法。当允许预测变量的维度增长远大于样本大小时，我们为变量和协变量的基础 DAG 建立联合选择一致性。我们证明了我们的方法在几种模拟设置中选择网络结构预测器方面优于现有方法。当允许预测变量的维度增长远大于样本大小时，我们为变量和协变量的基础 DAG 建立联合选择一致性。我们证明了我们的方法在几种模拟设置中选择网络结构预测器方面优于现有方法。