  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.

Methodology