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Entropy 2015, 17(7), 4918-4939; doi:10.3390/e17074918

Fisher Information Properties

Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Monseñor Álvaro del Portillo 12.455, Las Condes, Santiago, Chile
Academic Editor: Raúl Alcaraz Martínez
Received: 18 June 2015 / Accepted: 10 July 2015 / Published: 13 July 2015
(This article belongs to the Section Information Theory)
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Abstract

A set of Fisher information properties are presented in order to draw a parallel with similar properties of Shannon differential entropy. Already known properties are presented together with new ones, which include: (i) a generalization of mutual information for Fisher information; (ii) a new proof that Fisher information increases under conditioning; (iii) showing that Fisher information decreases in Markov chains; and (iv) bound estimation error using Fisher information. This last result is especially important, because it completes Fano’s inequality, i.e., a lower bound for estimation error, showing that Fisher information can be used to define an upper bound for this error. In this way, it is shown that Shannon’s differential entropy, which quantifies the behavior of the random variable, and the Fisher information, which quantifies the internal structure of the density function that defines the random variable, can be used to characterize the estimation error. View Full-Text
Keywords: Fisher information; Cramer–Rao bound; Shannon differential entropy; Markov chains Fisher information; Cramer–Rao bound; Shannon differential entropy; Markov chains
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Zegers, P. Fisher Information Properties. Entropy 2015, 17, 4918-4939.

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