A geometric view on learning Bayesian network structures

We recall the basic idea of an algebraic approach to learning Bayesian network (BN) structures, namely to represent every BN structure by a certain (uniquely determined) vector, called a 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 that 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 ...

• Type: Journal
• Year: 2010
• Where: IJAR
• Authors: Milan Studený, Jirí Vomlel, Raymond Hemmecke
• Keywords: Geometric Neighborhood, IJAR 2010, Inclusion Neighborhood, Standard Imset