Constrained Entropy Vectors: Some Direct and Converse Results

Properties of the Shannon entropy function are instrumental for analyzing the limits of reliable communication for various communication system models. However, the set of all Shannon entropy functions, also called entropy vectors, is not known yet for three random variables. For entropy vectors for four or more random variables, there exist information inequalities other than the basic inequalities. For three random variables, the non-entropy vectors exist only when they satisfy certain basic equality constraints. This paper presents direct results for such constrained entropy vectors. In particular, the looseness of known inner bounds for generic entropy vectors and the special class of quasi-uniform entropy vectors constrained by certain basic equalities is shown. A converse result in the form of information equality is also presented for quasi-uniform entropy vectors constrained by certain basic equalities.