Causal models with hidden variables

Directed acyclic graph models (DAG models, also called Bayesian networks) are widely used in the context of causal inference, since their structure admits a calculus of intervention.  They are also used to describe 'classical' models of reality in quantum physics.  However these models are not closed under marginalization, in the sense that they cannot faithfully represent the constraints imposed on margins of distributions which jointly follow a DAG.  Other classes of graphs have been introduced to deal withthis, including ancestral graphs and ADMGs, but we show that these are not sufficiently rich to capture the relevant models.

Since the conditional independence constraints which define DAGs are polynomials in probabilities, the model defined by the margin of a DAG is itself a semi-algebraic object: that is, it is defined by a mixture of polynomial equalities and inequalities.  We provide a complete characterization of the equality constraints, as well as some incomplete results giving inequality constraints.  We use these to provide partial results on the equivalence of models, and give a class of hypergraphs (mDAGs) that is sufficient to describe the equivalence classes, includingunder causal interventions on the observed variables.  Understanding such equivalences is critical for the use of causal structure learning methods.

Main References:

 R.J. Evans, Graphical methods for inequality constraints in marginalized DAGs, 22nd Workshop on Machine Learning and Signal Processing, 2012.

 R.J. Evans, Margins of discrete Bayesian networks, arXiv:1501.02103, 2015

 R.J. Evans and T.S. Richardson.  Smooth, identifiable supermodels of discrete DAG models with latent variables, arXiv:1511.06813, 2015

 R.J. Evans, Graphs for margins of Bayesian networks, Scandinavian Journal of Statistics, 43 (3), pp 625-648, 2016.