Towards standard imsets for maximal ancestral graphs

The imsets of \citet{studeny2006probabilistic} 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 present a proposed extension to standard imsets for maximal ancestral graph (MAG) models, using the parameterizing set representation of \citet{hu2020faster}. By construction, our imset also represents the Markov equivalence class of the MAG. We show that for many such graphs our proposed imset is \emph{perfectly Markovian} with respect to the graph thus providing 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. Unfortunately, for some models the representation does not work, and in certain cases does not represent any independences at all. We prove that it does work for \emph{simple} MAGs where there are only heads of size less than three, as well as for a large class of purely bidirected models. We also show that of independence models that do represent the MAG, the one we give is the simplest possible, in a manner we make precise. Further we refine the ordered local Markov property, which relates to finding the best imsets representing general MAGs.