The Cauchy-Schwarz Divergence for Poisson Point Processes

In this paper, we extend the notion of Cauchy-Schwarz divergence to point processes and establish that the Cauchy-Schwarz divergence between the probability densities of two Poisson point processes is half the squared L2-distance between their intensity functions. Extension of this result to mixture...

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Bibliographic Details
Published in:IEEE transactions on information theory 2015-08, Vol.61 (8), p.4475-4485
Main Authors: Hung Gia Hoang, Ba-Ngu Vo, Ba-Tuong Vo, Mahler, Ronald
Format: Article
Language:English
Subjects:
Atmospheric measurements
Density measurement
information divergence
Information theory
Measurement units
Normal distribution
Particle measurements
Poisson distribution
Poisson point process
Probability distribution
random finite sets
Random variables
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Summary:In this paper, we extend the notion of Cauchy-Schwarz divergence to point processes and establish that the Cauchy-Schwarz divergence between the probability densities of two Poisson point processes is half the squared L2-distance between their intensity functions. Extension of this result to mixtures of Poisson point processes and, in the case where the intensity functions are Gaussian mixtures, closed form expressions for the Cauchy-Schwarz divergence are presented. Our result also implies that the Bhattacharyya distance between the probability distributions of two Poisson point processes is equal to the square of the Hellinger distance between their intensity measures. We illustrate the result via a sensor management application where the system states are modeled as point processes.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2015.2441709