deBruijn identities: from Shannon, Kullback–Leibler and Fisher to generalized φ -entropies, φ -divergences and φ -Fisher informations

In this paper we propose a generalization of the usual deBruijn identity that links the Shannon differential entropy (or the Kullback–Leibler divergence) and the Fisher information (or the Fisher divergence) of the output of a Gaussian channel. The generalization makes use of φ -entropies on the one hand, and of φ -divergences (of the Csizàr class) on the other hand, as generalizations of the Shannon entropy and of the Kullback–Leibler divergence respectively. The generalized deBruijn identities induce the definition of generalized Fisher informations and generalized Fisher divergences; some of such generalizations exist in the literature. Moreover, we provide results that go beyond the Gaussian channel: we are then able to characterize a noisy channel using general measures of mutual information, both for Gaussian and non-Gaussian channels.