Jornal article

Studený M., Vomlel J.

International Journal of Approximate Reasoning

2011

5 (2011)

627-640

0888-613X

RNDr. Milan Studený, DrSc. (ÚTIA - Oddělení matematické teorie rozhodování)

structural learning Bayesian nets, standard imset, polytope,

The basic idea of an algebraic approach to learning a Bayesian network (BN) structure is to represent it by a certain uniquely determined vector, called the standard imset. In a recent paper, it was shown that the set of standard imsets is the set of vertices of a certain polytope and natural geometric neighborhood for standard imsets, and, consequently, for BN structures, was introduced. The new geometric view led to a series of open mathematical questions. In this paper, we try to answer some of them. First, we introduce a class of necessary linear constraints on standard imsets and formulate a conjecture that these constraints characterize the polytope. The conjecture has been confirmed in the case of (at most) 4 variables. Second, we confirm a former hypothesis by Raymond Hemmecke that the only lattice points within the polytope are standard imsets. Third, we give a partial analysis of the geometric neighborhood in the case of 4 variables.