Towards standard imsets for maximal ancestral graphs

TL;DR

This paper extends the algebraic imset framework to maximal ancestral graphs (MAGs), providing a new scoring criterion for MAG models, and introduces power DAGs for better representation of independences in complex MAGs.

Contribution

It proposes a novel extension of standard imsets to MAGs using the set representation, and introduces power DAGs to improve independence modeling in complex MAGs.

Findings

The extended imset provides a consistent scoring criterion for MAGs.

The imset is minimal among models representing the same independences.

Power DAGs enable polynomial-time construction of correct models under mild conditions.

Abstract

The imsets of Studen\'y (2005) 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 (2020). 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 (2003) 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.