The Inclusion-Exclusion Rule and its Application to the Junction Tree Algorithm

In this paper, we consider the inclusion-exclusion rule – a known yet seldom used rule of probabilistic inference. Unlike the widely used sum rule which requires easy access to all joint probability values, the inclusion-exclusion rule requires easy access to several marginal probability values. We therefore develop a new representation of the joint distribution that is amenable to the inclusion-exclusion rule. We compare the relative strengths and weaknesses of the inclusion-exclusion rule with the sum rule and develop a hybrid rule called the inclusionexclusion-sum (IES) rule, which combines their power. We apply the IES rule to junction trees, treating the latter as a target for knowledge compilation and show that in many cases it greatly reduces the time required to answer queries. Our experiments demonstrate the power of our approach. In particular, at query time, on several networks, our new scheme was an order of magnitude faster than the junction tree algorithm.