Estimation of the covariance matrix for high-dimensional multivariate datasets is a challenging and important problem in modern statistics. In this paper, we focus on high-dimensional Gaussian DAG models where sparsity is induced on the Cholesky factor L of the inverse covariance matrix. In recent work, ([Cao, Khare, and Ghosh, 2019]), we established high-dimensional sparsity selection consistency for a hierarchical Bayesian DAG model, where an Erdos-Renyi prior is placed on the sparsity pattern in the Cholesky factor L, and a DAG-Wishart prior is placed on the resulting non-zero Cholesky entries. In this paper we significantly improve and extend this work, by (a) considering more diverse and effective priors on the sparsity pattern in L, namely the beta-mixture prior and the multiplicative prior, and (b) establishing sparsity selection consistency under significantly relaxed conditions on p, and the sparsity pattern of the true model. We demonstrate the validity of our theoretical results via numerical simulations, and also use further simulations to demonstrate that our sparsity selection approach is competitive with existing state-of-the-art methods including both frequentist and Bayesian approaches in various settings.

中文翻译：

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具有乘法和 Beta 混合先验的高维高斯 DAG 模型的一致贝叶斯稀疏选择

估计高维多元数据集的协方差矩阵是现代统计学中一个具有挑战性和重要的问题。在本文中，我们关注高维高斯 DAG 模型，其中在逆协方差矩阵的 Cholesky 因子 L 上引入了稀疏性。在最近的工作中，（[Cao、Khare 和 Ghosh，2019]），我们为分层贝叶斯 DAG 模型建立了高维稀疏选择一致性，其中 Erdos-Renyi 先验被放置在 Cholesky 因子 L 的稀疏模式上，并且 DAG-Wishart 先验被放置在所得的非零 Cholesky 条目上。在本文中，我们通过 (a) 在 L 中的稀疏模式上考虑更多样化和更有效的先验，即 beta 混合先验和乘法先验，显着改进和扩展了这项工作，(b) 在 p 上显着放松的条件下建立稀疏选择一致性，以及真实模型的稀疏模式。我们通过数值模拟证明了我们的理论结果的有效性，并且还使用进一步的模拟来证明我们的稀疏选择方法与现有的最先进方法（包括各种设置中的频率论和贝叶斯方法）具有竞争力。