Axioms and algorithms for inferences involving probabilistic independence

Abstract

This paper offers an axiomatic characterization of the probabilistic relation “X is independent of Y (written (X, Y))”, where X and Y are two disjoint sets of variables. Four axioms for (X, Y) are presented and shown to be complete. Based on these axioms, a polynomial membership algorithm is developed to decide whether any given independence statement (X, Y) logically follows from a set Σ of such statements, i.e., whether (X, Y) holds in every probability distribution that satisfies Σ. The complexity of the algorithm is O(|Σ| · k2 + |Σ| · n), where |Σ| is the number of given statements, n is the number of variables in Σ ∪ {(X, Y)}, and k is the number of variables in (X, Y).