Learning Bayesian network structure: Towards the essential graph by integer linear programming tools

• A geometric approach to learning BN structure based on application of ILP tools.
• A new LP relaxation for extended characteristic imsets.
• As a by-product of the ILP optimization procedure, one may get the essential graph.
• Some computational experiments based on this approach.

The basic idea of the 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. Such a suitable zero-one vector representative is the characteristic imset, introduced by Studený, Hemmecke and Lindner in 2010, in the proceedings of the 5th PGM workshop. 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 a polyhedral description of the respective domain of the ILP problem, that is, by means of a set of linear inequalities. This theoretical result opens the way to the application of ILP software packages. The advantage of our 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. We also describe some computational experiments based on this idea.