# An inequality connecting entropy distance, Fisher Information and large deviations

## TL;DR

This paper introduces a generalized Fisher Information for Markov jump processes, establishes an inequality linking it with entropy and large deviations, and applies it to coarse-graining problems in discrete spaces.

## Contribution

It presents a novel generalization of Fisher Information, proves a connecting inequality with entropy and large deviations, and applies these results to coarse-graining of jump processes.

## Key findings

- Generalized Fisher Information converges to classical Fisher Information.
- Established an inequality linking entropy, Fisher Information, and large deviations.
- Applied the inequality to analyze coarse-graining in discrete jump processes.

## Abstract

In this paper we introduce a new generalisation of the relative Fisher Information for Markov jump processes on a finite or countable state space, and prove an inequality which connects this object with the relative entropy and a large deviation rate functional. In addition to possessing various favourable properties, we show that this generalised Fisher Information converges to the classical Fisher Information in an appropriate limit. We then use this generalised Fisher Information and the aforementioned inequality to qualitatively study coarse-graining problems for jump processes on discrete spaces.