Polyhedral aspects of score equivalence in Bayesian network structure learning

The motivation for this paper is the integer linear programming approach to learning the structure of a decomposable graphical model. We have chosen to represent decomposable models by means of special zero-one vectors, named characteristic imsets. Our approach leads to the study of a special polytope, defined as the convex hull of all characteristic imsets for chordal graphs, named the chordal graph polytope. In this theoretical paper, we introduce a class of clutter inequalities (valid for the vectors in the polytope) and show that all of them are facet-defining for the polytope. We dare to conjecture that they lead to a complete polyhedral description of the polytope. Finally, we propose a linear programming method to solve the separation problem with these inequalities for the use in a cutting plane approach. Cover letter for 2016 IJAR submission Towards using the chordal graph polytope in learning decomposable models by M. Studený and J. Cussens Prague, November 22, 2016. Dear editors, we were pleased by an offer to submit an extended version of our PGM 2016 contribution The chordal graph polytope for learning decomposable models, JMLR Workshop and Conference Proceedings 52: PGM 2016, (A. Antonucci, G. Corani, and C.P. de Campos eds.), pp. 499–510, to a special issue of the IJAR journal following PGM’16. In comparison with the original proceedings paper, this extended journal version contains the proof of the main result that all clutter inequalities are facet-defining for the polytope. This technical proof makes the paper relatively long, but we hope it still fits in usual page limits. Note that to make this theoretical paper reader-friendly we moved the substantial proofs to the appendix. We hope that the paper will be found suitable for the special issue. All the best Milan Studený and James Cussens *Cover Letter