The chain graph model admits both undirected and directed edges in one graph, where symmetric conditional dependencies are encoded via undirected edges and asymmetric causal relations are encoded via directed edges. Though frequently encountered in practice, the chain graph model has been largely under investigated in literature, possibly due to the lack of identifiability conditions between undirected and directed edges. In this paper, we first establish a set of novel identifiability conditions for the Gaussian chain graph model, exploiting a low rank plus sparse decomposition of the precision matrix. Further, an efficient learning algorithm is built upon the identifiability conditions to fully recover the chain graph structure. Theoretical analysis on the proposed method is conducted, assuring its asymptotic consistency in recovering the exact chain graph structure. The advantage of the proposed method is also supported by numerical experiments on both simulated examples and a real application on the Standard & Poor 500 index data.

翻译：链图模型允许在一个图中同时包含无向边和有向边，其中对称条件依赖关系通过无向边编码，非对称因果关系通过有向边编码。尽管在实践应用中频繁出现，但链图模型在文献中尚未得到充分研究，这可能是由于无向边与有向边之间缺乏可识别性条件所致。本文首先利用精度矩阵的低秩加稀疏分解，为高斯链图模型建立了一组全新的可识别性条件。进一步地，基于这些可识别性条件构建了一种高效的学习算法，以完整恢复链图结构。对提出的方法进行了理论分析，确保其在实际恢复链图结构时具有渐近一致性。通过模拟算例和标准普尔500指数数据的实际应用进行的数值实验，也验证了所提方法的优势。