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Axioms 2017, 6(2), 8; doi:10.3390/axioms6020008

Expansion of the Kullback-Leibler Divergence, and a New Class of Information Metrics

1
Pacific Northwest Research Institute, 720 Broadway, Seattle, WA 98122, USA
2
Albany College of Pharmacy and Health Sciences, 106 New Scotland Avenue, Albany, NY 12208, USA
*
Author to whom correspondence should be addressed.
Academic Editors: Abbe Mowshowitz and Matthias Dehmer
Received: 31 January 2017 / Revised: 25 March 2017 / Accepted: 27 March 2017 / Published: 1 April 2017
(This article belongs to the Special Issue Entropy and Information Theory)
View Full-Text   |   Download PDF [628 KB, uploaded 1 April 2017]   |  

Abstract

Inferring and comparing complex, multivariable probability density functions is fundamental to problems in several fields, including probabilistic learning, network theory, and data analysis. Classification and prediction are the two faces of this class of problem. This study takes an approach that simplifies many aspects of these problems by presenting a structured, series expansion of the Kullback-Leibler divergence—a function central to information theory—and devise a distance metric based on this divergence. Using the Möbius inversion duality between multivariable entropies and multivariable interaction information, we express the divergence as an additive series in the number of interacting variables, which provides a restricted and simplified set of distributions to use as approximation and with which to model data. Truncations of this series yield approximations based on the number of interacting variables. The first few terms of the expansion-truncation are illustrated and shown to lead naturally to familiar approximations, including the well-known Kirkwood superposition approximation. Truncation can also induce a simple relation between the multi-information and the interaction information. A measure of distance between distributions, based on Kullback-Leibler divergence, is then described and shown to be a true metric if properly restricted. The expansion is shown to generate a hierarchy of metrics and connects this work to information geometry formalisms. An example of the application of these metrics to a graph comparison problem is given that shows that the formalism can be applied to a wide range of network problems and provides a general approach for systematic approximations in numbers of interactions or connections, as well as a related quantitative metric. View Full-Text
Keywords: multivariable dependence; interaction information; Kullback-Leibler divergence; information metrics; entropy; graph distances multivariable dependence; interaction information; Kullback-Leibler divergence; information metrics; entropy; graph distances
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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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MDPI and ACS Style

Galas, D.J.; Dewey, G.; Kunert-Graf, J.; Sakhanenko, N.A. Expansion of the Kullback-Leibler Divergence, and a New Class of Information Metrics. Axioms 2017, 6, 8.

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