Posterior graph selection and estimation consistency for high-dimensional Bayesian DAG models
Posterior graph selection and estimation consistency for high-dimensional Bayesian DAG models
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DOI：
10.1214/18-aos1689
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发表时间：
2016-11
期刊：
影响因子：
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通讯作者：
Xuan Cao;K. Khare;M. Ghosh
中科院分区：
文献类型：
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作者：
Xuan Cao;K. Khare;M. Ghosh
高维多元数据集的协方差估计与选择是现代统计学中的一个基本问题。高斯有向无环图（DAG）模型是用于此目的的一类流行模型。高斯DAG模型在逆协方差矩阵的Cholesky因子中引入了稀疏性，而稀疏性模式反过来对应于对底层变量的特定条件独立性假设。近年来，人们对DAG模型中的贝叶斯推理进行了各种各样的先验研究，但这些模型的关键收敛性和稀疏性选择特性尚未得到深入的研究。这些先验中的大多数都是DAG上下文中Wishart分布的改编或推广。在本文中，我们考虑了一类具有多个形状参数的灵活且通用的“DAG-Wishart”先验。在温和的正则性假设下，我们建立了强图选择一致性，并建立了当变量数p随样本量n以适当的次指数速率增长时估计的后验收敛率。
Covariance estimation and selection for high-dimensional multivariate datasets is a fundamental problem in modern statistics. Gaussian directed acyclic graph (DAG) models are a popular class of models used for this purpose. 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 variables. A variety of priors have been developed in recent years for Bayesian inference in DAG models, yet crucial convergence and sparsity selection properties for these models have not been thoroughly investigated. Most of these priors are adaptations or generalizations of the Wishart distribution in the DAG context. In this paper, we consider a flexible and general class of these 'DAG-Wishart' priors with multiple shape parameters. Under mild regularity assumptions, we establish strong graph selection consistency and establish posterior convergence rates for estimation when the number of variables p is allowed to grow at an appropriate sub-exponential rate with the sample size n.