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 55-80.
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 integer-valued 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 non-unique
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
- conditional independence structures
- graphical description
- algebraic description
pdf version of a preprint (268kB) is available.
The paper is an extended and revised version of the conference paper:
The paper partially builds on the book:
- 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. 193-200.
abstract click here.
- M. Studeny:
Probabilistic Conditional Independence Structures. Springer-Verlag, London, 2005.