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.

翻译：本研究提出了首个用于学习高维线性贝叶斯网络的贝叶斯方法。该方法通过逆偏协方差矩阵，从后向迭代估计拓扑排序中的每个元素及其父节点。当对逆协方差矩阵施加具有非均匀收缩的贝叶斯正则化时，所提方法能成功恢复底层结构。具体而言，在次高斯误差分布和四阶有界矩误差分布下，分别证明了样本量$n = \Omega( d_M^2 \log p)$和$n = \Omega(d_M^2 p^{2/m})$足以使该算法学习线性贝叶斯网络，其中$p$表示节点数，$d_M$表示道德化图的最大度。理论研究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，实验证明该方法在合成数据上的性能优于现有最先进的频率学派方法，如BHLSM、LISTEN和TD算法。