Sharp Bounds on Arimoto's Conditional Rényi Entropies Between Two Distinct Orders

Abstract

This study examines sharp bounds on Arimoto's conditional Rényi entropy of order $\beta$ with a fixed another one of distinct order $\alpha \ne \beta$. Arimoto inspired the relation between the Rényi entropy and the ${\ell }_r$-norm of probability distributions, and he introduced a conditional version of the Rényi entropy. From this perspective, we analyze the ${\ell }_r$-norms of particular distributions. As results, we identify specific probability distributions whose achieve our sharp bounds on the conditional Rényi entropy. The sharp bounds derived in this study can be applicable to other information measures, e.g., the minimum average probability of error, the Bhattacharyya parameter, Gallager's reliability function $E_0$, and Sibson's $\alpha$-mutual information, whose are strictly monotone functions of the conditional Rényi entropy.

61 pages, 6 figures; typos added, references added for Section V; A brief version of this paper was submitted to the 2017 IEEE International Symposium on Information Theory (ISIT2017)

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