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.

中文翻译：

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连接熵距离、Fisher 信息和大偏差的不等式

在本文中，我们介绍了有限或可数状态空间上马尔可夫跳跃过程的相对 Fisher 信息的新概括，并证明了将这个对象与相对熵和大偏差率函数联系起来的不等式。除了拥有各种有利的特性外，我们还表明，这种广义的 Fisher 信息在适当的限制下会收敛到经典的 Fisher 信息。然后我们使用这个广义的 Fisher 信息和前面提到的不等式来定性地研究离散空间上跳跃过程的粗粒度问题。