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：链图模型允许在同一图中同时包含无向边和有向边，其中对称条件依赖性通过无向边编码，非对称因果关系通过有向边编码。尽管在实践中频繁出现，但链图模型在文献中研究不足，这可能是由于缺乏无向边与有向边之间的可识别性条件。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新颖的可识别性条件。进一步，基于这些可识别性条件构建了一种高效的学习算法，以完全恢复链图结构。对提出的方法进行了理论分析，确保其在恢复精确链图结构方面具有渐近一致性。数值实验（包括模拟示例和标准普尔500指数数据的实际应用）也支持了该方法的优势。