Abstract

We recall the basic idea of an algebraic approach to learning 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. Then we show it always includes the inclusion neighborhood, which was introduced earlier in connection with the greedy equivalence search (GES) algorithm. The third result is that the global optimum of an affine function over the polytope coincides with the local optimum relative to the geometric neighborhood. To illustrate the new concept by an example, we describe the geometric neighborhood in the case of three variables and show it differs from the inclusion neighborhood. This leads to a simple 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 search technique is based on the geometric neighborhood instead. We also found out what is the geomteric neigborhood in the case of four and five variables.

AMS classification 68T30, 62H05

Keywords

learning Bayesian networks

standard imset

inclusion neighborhood

geometric neighborhood

GES algorithm

A pdf version of the published paper (402kB) is already open-access available.

The paper comes from (and extends substantially) the observations in the conference paper:

• 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.

Thus, like the original conference paper, the paper refers to results from the following publications:

• M. Studeny (2005). Probabilistic Conditional Independence Structures. London: Springer-Verlag.
• D. M. Chickering: Optimal structure indentification with greedy search. Journal of Machine Learning Research 3 (2002), pp. 507-554.