Polymatroidal Dependence Structure of a Set of Random Variables

Given a finite set E of random variables, the entropy function h on E is a mapping from the set of all subsets of E into the set of all nonnegative real numbers such that for each A ⊆ E h(A) is the entropy of A . The present paper points out that the entropy function h is a β -function, i.e., a monotone non-decreasing and submodular function with h(O) = 0 and that the pair ( E, h ) is a polymatroid. The polymatroidal structure of a set of random variables induced by the entropy function is fundamental when we deal with the interdependence analysis of random variables such as the information-theoretic correlative analysis, the analysis of multiple-user communication networks, etc. Also, we introduce the notion of the principal partition of a set of random variables by transferring some results in the theory of matroids.