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. 627-640.
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
- structural learning Bayesian nets
- standard imset
- geometric neighborhood
- differential imset
pdf version of the published paper (262kB) is already open-access available.
The paper builds on resuls from the following publications: