Entropies, cross-entropies and Rényi divergence: sharp three-term inequalities for probability density functions
This work provides new fundamental inequalities for information theory, potentially impacting researchers working with Rényi entropy and divergence by offering tighter bounds.
This paper establishes a new sharp three-term inequality involving differential Rényi entropy, Rényi divergence, and Rényi cross-entropy for probability density functions. The equality holds when one PDF is an escort density of the other, and this framework is used to derive further sharp inequalities for various informational functionals.
A new sharp inequality featuring the differential Rényi entropy, the Rényi divergence and the Rényi cross-entropy of a pair of probability density functions is established. The equality is reached when one of the probability density function is an escort density of the other. This inequality is applied, together with a general framework of a pair of transformations reciprocal to each other, to derive a number of further inequalities involving both classical and new informational functionals. A remarkable fact is that, in all these inequalities, the Rényi divergence of two probability density functions is sharply bounded by quotients of informational functionals of cross-type and single type. More precisely, we derive sharp inequalities composed by relative and cross versions of the absolute moments, or of the Fisher information measures (among others), and involving two and three probability density functions.