  Directed acyclic graph (DAG) models have become widely studied and applied in
statistics and machine learning -- indeed, their simplicity facilitates
efficient procedures for learning and inference. Unfortunately, these models
are not closed under marginalization, making them poorly equipped to handle
systems with latent confounding. Acyclic directed mixed graph (ADMG) models
characterize margins of DAG models, making them far better suited to handle
such systems. However, ADMG models have not seen wide-spread use due to their
complexity and a shortage of statistical tools for their analysis. In this
paper, we introduce the m-connecting imset which provides an alternative
representation for the independence models induced by ADMGs. Furthermore, we
define the m-connecting factorization criterion for ADMG models, characterized
by a single equation, and prove its equivalence to the global Markov property.
The m-connecting imset and factorization criterion provide two new statistical
tools for learning and inference with ADMG models. We demonstrate the
usefulness of these tools by formulating and evaluating a consistent scoring
criterion with a closed form solution.
