  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.
