M. Studeny, J. Vomlel: A geometric approach to learning BN structures In Proceedings of the 4th European Workshop on Probabilistic Graphical Models (M. Jaeger,T. D. Nielsen eds.), University of Aalborg, 2008, pp. 281-288.

We recall the basic idea of an algebraic approach to learning a Bayesian network (BN) structure, namely to represent every BN structure by a certain (uniquely determined) vector, called standard imset. The main result of the paper is that the set of standard imsets is the set of vertices (= extreme points) of a certain polytope. Motivated by the geometric view, we introduce the concept of the geometric neighborhood for standard imsets, and, consequently, for BN structures. To illustrate this concept by an example, we describe the geometric neighborhood in the case of three variables and show it differs from the inclusion neighborhood, which was introduced earlier in connection with the GES algorithm. This leads to an example of the failure of the GES algorithm if data are not ``generated" from a perfectly Markovian distribution. The point is that one can avoid this failure if the greedy search technique is based on the geometric neighborhood instead.

AMS classification 68T30

standard imset
inclusion neighborhood
geometric neighborhood

A scanned pdf copy (691kB) is available.

The contribution refers to results from the following publications: