在比例高维情形下，基于尺度混合误差分布的同时收缩协方差估计的贝叶斯有向无环图结构学习
Bayesian DAG Structure Learning with Simultaneous Shrinkage Covariance Estimation under Scale-Mixture Error Distributions in the Proportional High-Dimensional Regime
AI总结：
该研究针对高维比例渐近情形，提出鲁棒DAG - 乔列斯基马蹄铁框架，通过特定先验和逆伽马尺度混合实现DAG结构学习与精度矩阵估计，设计吉布斯采样器，模拟和实际数据应用均显示其优于其他方法。
摘要：
我们提出了一个统一的贝叶斯框架，即鲁棒DAG - 乔列斯基马蹄铁（R - DACH），用于在高维比例渐近情形$p/n \to c \in (0,\infty)$下，在正态误差的尺度混合情况下进行联合有向无环图（DAG）结构学习和精度矩阵估计。该构造直接在DAG - 马尔可夫精度矩阵的修正乔列斯基因子的严格下三角元素上放置全局 - 局部马蹄铁型先验，使得乔列斯基参数中的稀疏性诱导与变量拓扑排序一致的连贯父集选择。每个观测值的逆伽马尺度混合对重尾和受污染观测值具有自动鲁棒性，并将学生 - t、拉普拉斯和斜线分布作为特殊情况包含在内。我们设计了一个部分折叠的块吉布斯采样器，它遍历排序、稀疏模式和连续参数的联合空间。对$(n,p)$配置（$p$高达数百）的模拟证实了理论速率，并表明在污染情况下比图形马蹄铁、DAG - 威沙特和基于PC的竞争方法有显著优势。对来自《癌症基因组图谱》的RNA - seq基因表达数据的应用揭示了竞争方法无法恢复的具有生物学可解释性的调控结构。
英文摘要：
We propose a unified Bayesian framework namely robust DAG-Cholesky horseshoe (R-DACH) for joint directed acyclic graph (DAG) structure learning and precision matrix estimation in the high-dimensional proportional asymptotic regime $p/n \to c \in (0,\infty)$, under the scale mixture of normal errors. The construction places a global-local horseshoe-type prior directly on the strictly lower-triangular entries of the modified Cholesky factor of the DAG-Markov precision matrix, so that sparsity in the Cholesky parameters induces a coherent parent-set selection consistent with a topological ordering of the variables. A per-observation inverse-gamma scale mixture yields automatic robustness to heavy-tailed and contaminated observations and admits Student-$t$, Laplace, and slash distributions as special cases. We design a partially-collapsed blocked Gibbs sampler that traverses the joint space of orderings, sparsity patterns and continuous parameters. Simulations across $(n,p)$ configurations with $p$ up to several hundreds confirm the theoretical rates and demonstrate substantial gains over graphical-horseshoe, DAG-Wishart, and PC-based competitors under contamination. An application to RNA-seq gene-expression data from \emph{The Cancer Genome Atlas} reveals biologically interpretable regulatory structure that competing methods fail to recover.
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