A generalization of the Kullback-Leibler divergence and its properties

TL;DR

This paper introduces a generalized form of the Kullback-Leibler divergence based on the symmetric Jackson derivative, exploring its fundamental properties and connections to information theory and dynamics.

Contribution

It presents a new generalized divergence measure that maintains key properties of the original Kullback-Leibler divergence, expanding its theoretical framework.

Findings

Retains positivity, metricity, and concavity

Establishes bounds and stability properties

Connects to shift information and Liouville dynamics

Abstract

A generalized Kullback-Leibler relative entropy is introduced starting with the symmetric Jackson derivative of the generalized overlap between two probability distributions. The generalization retains much of the structure possessed by the original formulation. We present the fundamental properties including positivity, metricity, concavity, bounds and stability. In addition, a connection to shift information and behavior under Liouville dynamics are discussed.