Geometric View on Learning Bayesian Network Structures

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. 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. Some results in the case of four variables are also available.