Bibliography

(eng): Basic idea of an algebraic approach to learning Bayesian network (BN) structures is 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 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 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. The geometric neighborhood in the case of three variables is described and shown to differ 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.