P.R de Waal. Marginals of DAG-isomorphic independence models. In:
C. Sossai, G. Chemello (editors). Proceedings
of
the Tenth European Conference on Symbolic and Quantitative
Approaches to Reasoning with Uncertainty, LNCS 5590, Springer,
New York,
pp. 192–203, 2009.
Probabilistic and graphical independence models both sat- isfy 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 inde- pendence model of a directed acyclic
graph. The original independence model can then be viewed as the
marginal of the model with latent vari- ables. We show that for some
independence models we need infinitely many latent variables to
accomplish this.
(full paper)