Minimal I-map Dags: Necessary and Sufficient Conditions

Underlying Bayesian network modeling and casual discovery is Pearl's result that for every strictly positive distribution there exists a unique minimal undirected independency map (I-map), and, for every total ordering of variables, a unique minimal I-map dag [Pearl, 1988]. Whilst strict positivity is sufficient for uniqueness, we show that it is not necessary. We show that for the existence of a unique minimal I-map dag it is a necessary and sufficient condition that there be no two variables which are equivalent. This allows modeling of a wider class of processes, including those with fully effective causal interventions. Our result is as simple as, but more general than, that of Pearl.