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Directed acyclic graph (DAG) models have become widely studied and applied instatistics and machine learning -- indeed, their simplicity facilitatesefficient procedures for learning and inference. Unfortunately, these modelsare not closed under marginalization, making them poorly equipped to handlesystems with latent confounding. Acyclic directed mixed graph (ADMG) modelscharacterize margins of DAG models, making them far better suited to handlesuch systems. However, ADMG models have not seen wide-spread use due to theircomplexity and a shortage of statistical tools for their analysis. In thispaper, we introduce the m-connecting imset which provides an alternativerepresentation for the independence models induced by ADMGs. Furthermore, wedefine the m-connecting factorization criterion for ADMG models, characterizedby a single equation, and prove its equivalence to the global Markov property.The m-connecting imset and factorization criterion provide two new statisticaltools for learning and inference with ADMG models. We demonstrate theusefulness of these tools by formulating and evaluating a consistent scoringcriterion with a closed form solution.

The m-connecting imset and factorization for ADMG models