M. Studeny and P. Bocek: CI-models arising among 4 random variables. In Proceedings of the 3rd workshop WUPES, September 11-15, 1994, Trest, Czech Republic, pp. 268-282.

The paper is a lucid survey (without technicalities) of some results from the (later published) paper:
F. Matus and M. Studeny: Conditional independences among four random variables I. Combinatorics, Probability and Computing 4 (1995), n. 3, pp. 267-278.
Let X(1), X(2), X(3), X(4) be a system of four finitely-valued random variables. By the conditional independence model induced by this system we understand the list of triplets (A,B,C) of disjoint subsets of {1, 2, 3, 4} such that X(A) is conditionally independent of X(B) given X(C) (here X(A) is the random vector composed of random variables X(i) where i belongs to A). The topic of the contribution is the question which lists of such triplets are conditional independence models induced by a system of four finitely-valued random variables. These conditional independence models are almost completely characterized in terms of probabilistically valid inference rules (= axioms), 24 of them are mentioned. A dual characterization is based on irreducible models: every conditional independence model is an intersection of irreducible conditional independency models. The list of all known irreducible conditional independency models over four variables together with the corresponding constructions of probability distributions is given.

AMS classification 68T30

conditional independence model
(probabilistically sound) inference rule
irreducible model
submaximal model

A pdf copy (converted postscript version) (192kB) is available.