Characterization of inclusion neighbourhood in terms of the essential graph: Upper neighbours

Abstract

The problem of efficient characterization of inclusion neighbourhood is crucial for some methods of learning (equivalence classes of) Bayesian networks. In this paper, neighbouring equivalence classes of a given equivalence class of Bayesian networks are characterized efficiently by means of the respective essential graph. The characterization reveals hidded internal structure of the inclusion neighbourhood. More exactly, upper neighbours, that is, those neighbouring equivalence classes which describe more independencies, are completely characterized here. First, every upper neighbour is characterized by a pair ([a, b], C) where [a, b] is an edge in the essential graph and C ⊆ N \{a, b} a disjoint set of nodes. Second, if [a, b] is fixed, the class of sets C which characterize the respective neighbours is a tuft of sets determined by its least set and the list of its maximal sets. These sets can be read directly from the essential graph. An analogous characterization of lower neighbours, which is more complex, is mentioned.