In this work, we first introduce a discrete version of generalized Fisher information measure and develop some new results for it. We then propose Jensen-generalized discrete Fisher (Jensen-GDF) information as a generalized measure, based on the convexity property of generalized discrete Fisher information measure. We further introduce generating functions for generalized discrete Fisher information and Jensen-GDF information measures and use them to develop some results. We also propose a new correlation coefficient in terms of the generalized discrete Fisher information and discuss some of its properties. Finally, to demonstrate the usefulness of the Jensen-generalized discrete Fisher information measure and the proposed correlation coefficient, we apply them to two real-world examples in image processing and forms of contaminated data, and present corresponding numerical results. Our findings show that the Jensen-GDF information measure and the correlation coefficient introduced here are effective criteria for quantifying similarity between two images in image processing settings and for analyzing contaminated data.

In this work, we propose cumulative residual Fisher information and relative cumulative residual Fisher information measures and establish some of their properties. We first show that these cumulative Fisher measures can be expressed based on the hazard function. We then define extended versions of cumulative residual entropy and cumulative residual Fisher information measures based on the Jensen inequality. We also discuss connections between these information measures based on a new version of de Bruijn's identity for survival functions.

This paper presents the Bayes Fisher information measures, defined by the expected Fisher information under a distribution for the parameter, for the arithmetic, geometric, and generalized mixtures of two probability density functions. The Fisher information of the arithmetic mixture about the mixing parameter is related to chi-square divergence, Shannon entropy, and the Jensen-Shannon divergence. The Bayes Fisher measures of the three mixture models are related to the Kullback-Leibler, Jeffreys, Jensen-Shannon, Rényi, and Tsallis divergences. These measures indicate that the farther away are the components from each other, the more informative are data about the mixing parameter. We also unify three different relative entropy derivations of the geometric mixture scattered in statistics and physics literatures. Extensions of two of the formulations to the minimization of Tsallis divergence give the generalized mixture as the solution.