This study proposes the first Bayesian approach for learning high-dimensional
linear Bayesian networks. The proposed approach iteratively estimates each
element of the topological ordering from backward and its parent using the
inverse of a partial covariance matrix. The proposed method successfully
recovers the underlying structure when Bayesian regularization for the inverse
covariance matrix with unequal shrinkage is applied. Specifically, it shows
that the number of samples $n = \Omega( d_M^2 \log p)$ and $n = \Omega(d_M^2
p^{2/m})$ are sufficient for the proposed algorithm to learn linear Bayesian
networks with sub-Gaussian and 4m-th bounded-moment error distributions,
respectively, where $p$ is the number of nodes and $d_M$ is the maximum degree
of the moralized graph. The theoretical findings are supported by extensive
simulation studies including real data analysis. Furthermore the proposed
method is demonstrated to outperform state-of-the-art frequentist approaches,
such as the BHLSM, LISTEN, and TD algorithms in synthetic data.

该研究提出了第一个用于学习高维线性贝叶斯网络的贝叶斯方法，该方法通过反向逐步估计拓扑排序的每个元素及其父节点，使用了偏差协方差矩阵的逆。当应用于具有不等收缩的逆协方差矩阵的贝叶斯正则化时，该方法成功恢复了底层结构。具体来说，该方法表明样本数 n = Ω(d_M^2logp) 和 n = Ω(d_M^2p^{2/m}) 足以让该算法学习具有亚高斯和 4m 阶有界矩的线性贝叶斯网络，其中 p 是节点数，d_M 是道德化图的最大度数。理论发现得到了包括真实数据分析在内的大量模拟研究的支持。此外，该方法在合成数据中表现出优于 BHLSM、LISTEN 和 TD 等频频方法的性能。