On Polyhedral Approximations of Polytopes for Learning Bayesian Networks

Abstract

The motivation for this paper is the geometric approach to statistical learning Bayesiannetwork (BN) structures. We review three vector encodings of BN structures. The first one hasbeen used by Jaakkola et al. [9] and also by Cussens [4], the other two use special integral vectorsformerly introduced, called imsets [18, 20]. The topic is the comparison of outer polyhedral approximationsof the corresponding polytopes. We show how to transform the inequalities suggested byJaakkola et al. [9] into the framework of imsets. The result of our comparison is the observationthat the implicit polyhedral approximation of the standard imset polytope suggested in [21] givesa tighter approximation than the (transformed) explicit polyhedral approximation from [9]. Asa consequence, we confirm a conjecture from [21] that the above-mentioned implicit polyhedralapproximation of the standard imset polytope is an LP relaxation of that polytope. In the end,we review recent attempts to apply the methods of integer programming to learning BN structuresand discuss the task of finding suitable explicit LP relaxation in the imset-based approach.