Towards standard imsets for maximal ancestral graphs

Abstract

The imsets of Studený (Probabilistic Conditional Independence Structures (2005) Springer) are an algebraic method for representing conditional independence models. They have many attractive properties when applied to such models, and they are particularly nice for working with directed acyclic graph (DAG) models. In particular, the 'standard' imset for a DAG is in one-to-one correspondence with the independences it induces, and hence is a label for its Markov equivalence class. We first present a proposed extension to standard imsets for maximal ancestral graph (MAG) models, using the parameterizing set representation of Hu and Evans (In Proc. 36th Conf. Uncertainty in Artificial Intelligence (2020) PMLR). We show that for many such graphs our proposed imset is perfectly Markovian with respect to the graph, including a class of graphs we refer to as simple MAGs, which includes DAGs as a special case. In these cases the imset provides a scoring criteria by measuring the discrepancy for a list of independences that define the model; this gives an alternative to the usual BIC score that is also consistent, and much easier to compute. We also show that, of independence models that do represent the MAG, the imset we give is minimal. Unfortunately, for some graphs the representation does not represent all the independences in the model, and in certain cases does not represent any at all. For these general MAGs, we refine the reduced ordered local Markov property (Richardson in (Scand. J. Stat. 30 (2003) 145–157)) by a novel graphical tool called power DAGs, and this results in an imset that induces the correct model and which, under a mild condition, can be constructed in polynomial time.