Propositional Reasoning about Saturated Conditional Probabilistic Independence

Abstract

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. Unfortunately, the problem does not enjoy a finite ground axiomatization and is already coNP-complete to decide for restricted subclasses. Saturated conditional independencies form an important subclass of conditional independencies whose implication problem is decidable in almost linear time. Geiger and Pearl have established a finite ground axiomatization for this class. We establish a new completeness proof for this axiomatization, utilizing a new sound inference rule. The proof introduces special probability models where two values have probability one half. Special probability models allow us to establish a semantic proof for the equivalence between the implication of saturated conditional independencies and formulae in a Boolean propositional fragment. The equivalence extends the duality between the propositional fragment and multivalued dependencies in relational databases to a trinity involving saturated conditional independencies.