A. P. Dawid, M. Studeny: Conditional products: an alternative approach to conditional cndependence. In Artificial Intelligence and Statistics 99. Proceedings of the 7th Workshop (D. Heckerman, J. Whittaker eds.), Morgan Kaufmann, San Francisco 1999, pp. 32-40.

We introduce a new abstract approach to the study of conditional independence, founded on a concept analogous to the factorization properties of probabilistic independence, rather than the separation properties of a graph. The basic ingredient is the "conditional product", which provides a way of combining the basic objects under consideration while preserving as much independence as possible. We introduce an appropriate axiom system for conditional product, and show how, when these axioms are obeyed, they induce a derived concept of conditional independence which obeys the usual semi-graphoid axioms. The general structure is used to throw light on three specific areas: the familiar probabilistic framework (both the discrete and the general case); a set-theoretic framework related to "variation independence"; and a variety of graphical frameworks.

AMS classification 68T30

directed graph independence
probabilistic independence
undirected graph independence
Variation independence

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The paper partially builds on the following papers: