Local Entropy Statistics for Point Processes

Point processes are often described with functionals, such as the probability generating functional, the Laplace functional, and the factorial cumulant generating functional. These are used to facilitate modelling of different processes and to determine important statistics via functional differentiation. In information theory, generating functions have also been defined for probability densities to determine information quantities such as the Shannon information and Kullback-Leibler divergence, though as yet there are no such analogues for point processes. The purpose of this article is to exploit the advantages of both types of generating function to facilitate the derivation of information statistics for point processes. In particular, a generating functional for point processes is introduced for determining statistics related to entropy and relative entropy based on Golomb's information function and Moyal's probability generating functional. It is shown that the information generating functional permits the derivation of a suite of statistics, including localised Shannon entropy and Kullback-Leibler divergence calculations.