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In this paper we introduce a new generalisation of the relative FisherInformation for Markov jump processes on a finite or countable state space, andprove an inequality which connects this object with the relative entropy and alarge deviation rate functional. In addition to possessing various favourableproperties, we show that this generalised Fisher Information converges to theclassical Fisher Information in an appropriate limit. We then use thisgeneralised Fisher Information and the aforementioned inequality toqualitatively study coarse-graining problems for jump processes on discretespaces.

An inequality connecting entropy distance, Fisher Information and large deviations