The motivation for this paper is the geometric approach to statistical learning Bayesiannetwork (BN) structures. We review three vector... Show moreThe 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 has been used by Jaakkola et al. [9] and also by Cussens [4], the other two use special integral vectors formerly introduced, called imsets [18, 20]. The topic is the comparison of outer polyhedral approximations of the corresponding polytopes. We show how to transform the inequalities suggested by Jaakkola et al. [9] into the framework of imsets. The result of our comparison is the observation that the implicit polyhedral approximation of the standard imset polytope suggested in [21] gives a tighter approximation than the (transformed) explicit polyhedral approximation from [9]. As a consequence, we confirm a conjecture from [21] that the above-mentioned implicit polyhedral approximation 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 structures and discuss the task of finding suitable explicit LP relaxation in the imset-based approach. Show less

In this note, we propose a new linear-algebraic method for the implication problem among conditional independence statements, which is... Show moreIn this note, we propose a new linear-algebraic method for the implication problem among conditional independence statements, which is inspired by the factorization characterization of conditional independence. First, we give a criterion in the case of a discrete strictly positive density and relate it to an earlier linear-algebraic approach. Then, we extend the method to the case of a discrete density that need not be strictly positive. Finally, we provide a computational result in the case of six variables. Show less