Integer Linear Programming Approach to Learning Bayesian Network Structure: towards the Essential Graph

The basic idea of a geometric approach to learning a Bayesian network (BN) structure is to represent every BN structure by a certain vector. If the vector representative is chosen properly, it allows one to re-formulate the task of finding the global maximum of a score over BN structures as an integer linear programming (ILP) problem. Suitable such a zeroone vector representative is the characteristic imset, introduced in (Studený, Hemmecke and Lindner, 2010). In this paper, extensions of characteristic imsets are considered which additionally encode chain graphs without flags equivalent to acyclic directed graphs. The main contribution is the polyhedral description (= in terms of a set of linear inequalities) of the respective domain of the ILP problem. It is just a theoretical result, but it opens the way to the application of ILP software packages in the area of learning a BN structure. The advantage of this approach is that, as a by-product of the ILP optimization procedure, one may get the essential graph, which is a traditional graphical BN representative.