On Block Ordering of Variables in Graphical Modelling

ALBERTO ROVERATO, LUCA LA ROCCA
2006 Scandinavian Journal of Statistics  
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 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
more » ... -consistent chain graphs in which every block may be either an undirected graph or a directed acyclic graph or, more generally, a chain graph. In this way we provide a flexible tool for specifying subsets of chain graphs, and we observe that the most relevant subsets of chain graphs considered in the literature are families of B-consistent chain graphs for the appropriate choice of B. Structural learning within a family of B-consistent chain graphs requires to deal with Markov equivalence. We provide a graphical characterisation of equivalence classes of B-consistent chain graphs, namely the B-essential graphs, as well as a procedure to construct the B-essential graph for any given equivalence class of B-consistent chain graphs. Both largest chain graphs and essential graphs turn out to be special cases of B-essential graphs.
doi:10.1111/j.1467-9469.2006.00478.x fatcat:dzzgovre6vc5zmfo2wjheezxji