J. Vomlel, M. Studeny:
Graphical and algebraic representatives of conditional independence models.
In Advances in Bayesian Networks (P. Lucas, J. A. Gamez, A. Salmeron eds.),
Studies in Fuzziness and Soft Computing 213, Springer 2007, pages 5580.
 Abstract

The topic of this chapter is conditional independence models.
We review mathematical objects that are used to generate conditional
independence models in the area of probabilistic reasoning.
More specifically, we mention undirected graphs, acyclic directed graphs, chain graphs,
and an alternative algebraic approach that uses certain integervalued vectors, named imsets.
We compare the expressive power of these objects and discuss the problem of their uniqueness.
In learning Bayesian networks one meets the problem of nonunique
graphical description of the respective statistical model. One way
to avoid this problem is to use special chain graphs, named
essential graphs. An alternative algebraic approach uses certain
imsets, named standard imsets, instead. We present algorithms that
make it possible to transform graphical representatives into
algebraic ones and conversely. The algorithms were implemented in
the R language.
 AMS classification 68T30
 Keywords
 conditional independence structures
 graphical description
 algebraic description

A
pdf version of a preprint (268kB) is available.
The paper is an extended and revised version of the conference paper:
 M. Studeny, J. Vomlel:
Transition between graphical and algebraic representatives of Bayesian network models.
In Proceedings of the 2nd European Workshop on Probabilistic Graphical Models
(P. Lucas ed.), University of Nijmegen 2004, pp. 193200.
For an
abstract click here.
The paper partially builds on the book:
 M. Studeny:
Probabilistic Conditional Independence Structures. SpringerVerlag, London, 2005.