Junction Trees Construction: Application to Bayesian Networks

All exact inference algorithms for computing a posterior probability in general Bayesian networks have two conceptual phases: One phase handles operations on the graphical structure itself, and the other performs probabilistic computations. The junction tree is introduced to show the important relationship between graph theory and efficient probabilistic inference through a very important and interesting mathematical property of junction trees, the running intersection property. This paper examines an alternative method for constructing junction trees that is essential for the efficient computations of probabilistic enquiries posed on Bayesian networks. It presents a new method for converting a sequence of subsets that covers the Bayesian network into a junction tree; in other words, it converts a sequence of subsets in a Bayesian network into a proper set of cliques satisfying the running intersection property. We propose an algorithm in two versions (one of which is original) that allows a sequence of cliques possessing the running intersection property to be built, hence can be used with the junction tree algorithm for local computations on graphs. The input is an existing covering and the algorithm sequentially modifies the existing sequence of subsets to produce a new one with the required property. The obtained set of cliques and separators coincide with the junction trees obtained by the moralization and triangulation process, but it has the advantage of adapting to any computational task by adding links to the Bayesian network graph.