On the Conditional Independence Implication Problem: A Lattice-Theoretic Approach

Abstract

A lattice-theoretic framework is introduced that permits the study of the conditional in- dependence (CI) implication problem relative to the class of discrete probability measures. Semi-lattices are associated with CI state- ments and a finite, sound and complete in- ference system relative to semi-lattice inclu- sions is presented. This system is shown to be (1) sound and complete for saturated CI statements, (2) complete for general CI state- ments, and (3) sound and complete for stable CI statements. These results yield a criterion that can be used to falsify instances of the implication problem and several heuristics are derived that approximate this “lattice- exclusion” criterion in polynomial time. Fi- nally, we provide experimental results that relate our work to results obtained from other existing inference algorithms.