The paper is an overview of the theory developed in the series of three papers

M. Studeny: Description of structures of stochastic conditional independence by means of faces and imsets. International Journal of General Systems 23 (1994/5), n. 2-4, p. 123-137, pp. 201-219, pp. 323-341.

where the proofs of main results are given.

Abstract

A new approach to the mathematical description of structures of stochastic conditional independence by means of imsets and faces developed in the series of papers mentioned above is related to the classical approach by means of the notion of structural semigraphoid. It is shown how to realize the corresponding deductive mechanism to infer probabilistically valid consequences of input information about conditional independence structure, called facial implication. In the case of four attributes (= random variables), structural semigraphoids are characterized in terms of inference rules.

AMS classification 68T30, 62H05

Keywords

conditional independence

dependency model

semigraphoid

structural face

skeletal imset

structural semigraphoid

A scanned pdf copy (698kB) is available. The paper is an overview of the mentioned theory, the proofs of main results are in the above mentioned series of papers.

The topic of the paper was motivated by the works:

• J. Pearl: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufman, San Mateo CA 1988.
• M. Studeny: Conditional independence relations have no finite complete characterization. In Information Theory, Statistical Decision Functions and Random Processes. Transactions of the 11th Prague Conference vol. B (S. Kubik, J.A. Visek eds.), Kluwer, Dordrecht - Boston - London (also Academia, Prague) 1992, pp. 377-396.
• D. Geiger and J. Pearl: Logical and algorithmical properties of conditional independence and graphical models. Annals of Statistics 21 (1993), n. 4, pp. 2001-2021.