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. 3240.
 Abstract

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
semigraphoid 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 settheoretic framework related to "variation
independence"; and a variety of graphical frameworks.
 AMS classification 68T30
 Keywords
 directed graph independence
 probabilistic independence
 projection
 semigraphoid
 undirected graph independence
 Variation independence
 A
pdf copy (converted postscript version) (205kB) is available.
The paper partially builds on the following papers:
 A. P. Dawid: Conditional independence is statistical
theory (with discussion).
Journal of Royal Statistical Society B 41 (1979), pp. 131.
 M. Studeny:
Formal properties of conditional independence in different calculi of AI.
In Symbolic and Quantitative Approaches to Reasoning and Uncertainty
(M. Clarke, R. Kruse, S. Moral eds.), Lecture Notes in Computer Science 747,
SpringerVerlag, Berlin  Heidelberg 1993, pp. 341348.