Chain graphs (CG) (= adicyclic graphs) use undirected and directed edges to represent both structural and associative dependences. Like acyclic directed graphs (ADGs), the CG associated with a statistical Markov model may not be unique, so CGs fall into Markov equivalence classes, which may be superexponentially large, leading to unidentifiability and computational inefficiency in model search and selection. It is shown here that, under the Andersson–Madigan–Perlman (AMP) interpretation of a CG, each Markov-equivalence class can be uniquely represented by a single distinguished CG, the AMP essential graph, that is itself simultaneously Markov equivalent to all CGs in the AMP Markov equivalence class. A complete characterization of AMP essential graphs is obtained. Like the essential graph previously introduced for ADGs, the AMP essential graph will play a fundamental role for inference and model search and selection for AMP CG models.
"Characterizing Markov equivalence classes for AMP chain graph models." Ann. Statist. 34 (2) 939 - 972, April 2006. https://doi.org/10.1214/009053606000000173