Conditions for the existence of a generalization of R\'enyi divergence

TL;DR

This paper establishes necessary and sufficient conditions for generalizing Renyi divergence using deformed exponential functions, highlighting differences between atomic and non-atomic measures.

Contribution

It provides a precise characterization of when deformed exponential functions can be used to generalize Renyi divergence based on the measure type.

Findings

Not all deformed exponential functions are suitable for non-atomic measures.

Any deformed exponential function can be used when the measure is purely atomic.

Conditions involve specific properties of the deformed exponential functions.

Abstract

We give necessary and sufficient conditions for the existence of a generalization of R\'enyi divergence, which is defined in terms of a deformed exponential function. If the underlying measure $\mu$ is non-atomic, we found that not all deformed exponential functions can be used in the generalization of R\'enyi divergence; a condition involving the deformed exponential function is provided. In the case $\mu$ is purely atomic (the counting measure on the set of natural numbers), we show that any deformed exponential function can be used in the generalization.