Nonnegative Entropy Measures of Multivariate Symmetric Correlations

A study of nonnegativity “in general” in the symmetric (correlative) entropy space as well as discussions of some related problems is presented. The main result is summarized as Theorems 4.1 and 5.3, which give the necessary and sufficient condition for an element of the symmetric (correlative) entropy space to be nonnegative. In particular, Theorem 4.1 may be regarded as establishing a mathematical foundation for information-theoretic analysis of multivariate symmetric correlation. On the basis of these results, we propose a “hierarchical structure” of probabilistic dependence relations where it is shown that any symmetric correlation associated with a nonnegative entropy is decomposed into pairwise conditional and/or nonconditional correlations. A systematic duality existing in the set of nonnegative entropies is also considerably clarified.