Probabilistic Conditional Independence under Schema Certainty and Uncertainty

Conditional independence provides an essential framework to deal with knowledge and uncertainty in Artificial Intelligence, and is fundamental in probability and multivariate statistics. Its associated implication problem is paramount for building Bayesian networks. Saturated conditional independencies form an important subclass of conditional independencies. Under schema certainty, the implication problem of this subclass is finitely axiomatizable and decidable in almost linear time. We study the implication problem of saturated conditional independencies under both schema certainty and uncertainty. Under schema certainty, we establish a finite axiomatization with the following property: every independency whose implication is dependent on the underlying schema can be inferred by a single application of the so-called symmetry rule to some independency whose implication is independent from the underlying schema. Removing the symmetry rule from the axiomatization under schema certainty results in an axiomatization for a notion of implication that leaves the underlying schema undetermined. Hence, the symmetry rule is just a means to infer saturated conditional independencies whose implication is truly dependent on the schema.