Characterization of Inclusion Neighbourhood in Terms of the Essential Graph: Upper Neighbours ? 1 Motivation 1.1 Learning Bayesian Networks

The problem of eecient 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 eeciently 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 n fa; bg a disjoint set of nodes. Second, if a; b] is xed, 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. Several approaches to learning Bayesian networks use the method of maximiza-tion of a quality criterion, named also 'quality measure' 3] and 'score metric' 4]. Quality criterion is a function, designed by a statistician, which ascribes to data and a network a real number which 'evaluates' how the statistical model determined by the network is suitable to explain the occurence of data. Since the actual aim of the learning procedure is to get a statistical model (deened by a network) reasonable quality criteria do not distinguish between equivalent Bayesian networks, that is, between networks which deene the same statistical model. Therefore, from operational point of view, the goal is to learn an equivalence class of Bayesian networks, that is, a class of acyclic directed graphs (over a xed set of nodes N). ? This is an extended version with the Appendix which contains the proofs.