Marginals of DAG-Isomorphic Independence Models

Abstract

Probabilistic and graphical independence models both satisfy the semi-graphoid axioms, but their respective modelling powers are not equal. For every graphical independence model that is represented by d-separation in a directed acyclic graph, there exists an isomorphic probabilistic independence model, i.e. it has exactly the same independence statements. The reverse does not hold, as there exist probability distributions for which there are no perfect maps. We investigate if a given probabilistic independence model can be augmented with latent variables to a new independence model that is isomorphic with a graphical independence model of a directed acyclic graph. The original independence model can then be viewed as the marginal of the model with latent variables. We show that for some independence models we need infinitely many latent variables to accomplish this