On block ordering of variables in graphical modelling

In graphical modelling, the existence of substantive background knowledge on block ordering of variables is used to perform structural learning within the family of chain graphs (CGs) in which every block corresponds to an undirected graph and edges joining vertices in different blocks are directed in accordance with the ordering. We show that this practice may lead to an inappropriate restriction of the search space and introduce the concept of labelled block ordering B corresponding to a family of B-consistent CGs in which every block may be either an undirected graph or a directed acyclic graph or, more generally, a CG. In this way we provide a flexible tool for specifying subsets of chain graphs, and we observe that the most relevant subsets of CGs considered in the literature are families of B-consistent CGs for the appropriate choice of B. Structural learning within a family of B-consistent CGs requires to deal with Markov equivalence. We provide a graphical characterization of equivalence classes of B-consistent CGs, namely the B-essential graphs, as well as a procedure to construct the S-essential graph for any given equivalence class of B-consistent chain graphs. Both largest CGs and essential graphs turn out to be special cases of B-essential graphs.