Structural Learning of Bayesian Networks by Using Variable Neighbourhood Search Based on the Space of Orderings

Structural learning of Bayesian networks (BNs) is an NP-hard problem generally addressed by means of heuristic search algorithms. Although these techniques do not guarantee an optimal result, they allow obtaining good solutions with a relatively low computational effort. Many proposals are based on searching the space of directed acyclic graphs. However, there are alternatives consisting of exploring the space of equivalence classes of BNs, which yields more complex and difficult to implement algorithms, or the space of the orderings among variables. In practice, ordering-based methods allow reaching good results, but, they are costly in terms of computation. In this paper, we prove the correctness of the method used to evaluate each permutation when exploring the space of orderings, and we propose two simple and efficient learning algorithms based on this approach. The first one is a Hill climbing method which uses an improved neighbourhood definition, whereas the second algorithm is its natural extension based on the well-known variable neighbourhood search metaheuristic. The algorithms have been tested over a set of different domains in order to study their behaviour in practice.