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.

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 integer-valued vectors, named standard imsets, instead. In this paper we present algorithms which make it possible to transform graphical representatives into algebraic ones and conversely. A direct formula is a basis for translation of any chain graph equivalent to a Bayesian network into a standard imset. An inverse algorithm, which gives the essential graph, has two stages. The middle result of this procedure is a certain sequence of sets of variables, which can be turned into a hierarchical junction tree. We present both the mathematical theory and the algorithms, which we implemented in the R language.

essential graph
standard imset
reconstruction algorithm

A scanned pdf copy (612kB) is available. Moreover, A pdf copy of an extended version (a converted postscript) (198kB) is available.

The paper builds partially on Chapter 8 from