Abstract

The conditional independence structures within a system of four discrete random variables are studied. The problem which structures can occur is transformed into an analysis of a certain cone of set functions, namely polymatroids. Then, the cone is decomposed into two parts with help of Ingleton's inequalities. The first part is completely solved by giving the list of generating examples and the second part is strongly reduced.

The problem above was later completely solved in the subsequent series of papers by Frantisek Matus.

AMS classification 62H05, 52B40

Keywords

conditional independence

functional dependence

polymatroid

Ingleton's inequality

Shannon entropy

Here is the whole series of papers by Frantisek Matus:

• F. Matus, M. Studeny (1995) Conditional independences among four random variables I. Combinatorics, Probability & Computing 4 269-278. A pdf copy (converted postscript version) (194kB) is available.
  F. Matus (1995) Conditional independences among four random variables II. Combinatorics, Probability & Computing 4 407-417. A pdf copy (converted postscript version) (188kB) is available.
  F. Matus (1999) Conditional independences among four random variables III: final conclusion. Combinatorics, Probability & Computing 8 269-276. A pdf copy (converted postscript version) (199kB) is available.