M. Studeny, J. Vomlel:
On open questions in the geometric approach to structural
learning Bayesian nets.
International Journal of Approximate Reasoning
52 (2011), n. 5, pp. 627640.
 Abstract

The 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 the standard imset. In a recent paper
(Studeny, Vomlel, Hemmecke 2010), it was shown that the set S of standard imsets is the set of vertices (= extreme points) of a certain polytope P 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 P.
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 (= vectors having integers as components) within P
are standard imsets. Third, we give a partial analysis of the geometric
neighborhood in the case of 4 variables.
 AMS classification 68T30, 62H05
 Keywords
 structural learning Bayesian nets
 standard imset
 polytope
 geometric neighborhood
 differential imset
 A
pdf version of the published paper (262kB) is already openaccess available.
The paper builds on resuls from the following publications: