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

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The paper is an extended and revised version of the conference paper:

The paper partially builds on the book: