2 Characterization of inclusion neighbourhood 3 in terms of the essential graph q 4

10 The question of efficient characterization of inclusion neighbourhood is crucial in some 11 methods for learning (equivalence classes of) Bayesian networks. In this paper, neighbouring 12 equivalence classes of a given equivalence class of Bayesian networks are characterized effi13 ciently in terms of the respective essential graph. One can distinguish two kinds of inclusion 14 neighbours: upper and lower ones. This paper reveals the hidded internal structure of both 15 parts of the inclusion neighbourhood. 16 It is shown here that each inclusion neighbour is uniquely described by a pair ([a,b],C) 17 where [a,b] is an unordered pair of distinct nodes and C Nn{a,b} is a disjoint set of nodes 18 in the essential graph. Upper neighbours correspond to edges in the essential graph, while 19 lower neighbours correspond to pairs of nodes that are not edges in the essential graph. Given 20 a pair [a,b] of distinct nodes in the essential graph, the class of those sets C that ([a,b],C) 21 encodes an inclusion neighbour is characterized. The class has a special form; it is uniquely 22 determined by certain distinguished sets. These distinguished sets of the class can be read 23 directly from the essential graph. 24 2004 Published by Elsevier Inc.