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A Representation of the Relative Entropy with Respect to a Diffusion Process in Terms of Its Infinitesimal Generator

INRIA Sophia Antipolis Mediterannee, 2004 Route Des Lucioles, Sophia Antipolis, France
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Entropy 2014, 16(12), 6705-6721; https://doi.org/10.3390/e16126705
Received: 23 October 2014 / Revised: 17 December 2014 / Accepted: 18 December 2014 / Published: 22 December 2014
(This article belongs to the Section Information Theory, Probability and Statistics)
In this paper we derive an integral (with respect to time) representation of the relative entropy (or Kullback–Leibler Divergence) R(μ||P), where μ and P are measures on C([0,T];Rd). The underlying measure P is a weak solution to a martingale problem with continuous coefficients. Our representation is in the form of an integral with respect to its infinitesimal generator. This representation is of use in statistical inference (particularly involving medical imaging). Since R(μ||P) governs the exponential rate of convergence of the empirical measure (according to Sanov’s theorem), this representation is also of use in the numerical and analytical investigation of finite-size effects in systems of interacting diffusions. View Full-Text
Keywords: relative entropy; Kullback–Leibler; diffusion; martingale formulation relative entropy; Kullback–Leibler; diffusion; martingale formulation
MDPI and ACS Style

Faugeras, O.; MacLaurin, J. A Representation of the Relative Entropy with Respect to a Diffusion Process in Terms of Its Infinitesimal Generator. Entropy 2014, 16, 6705-6721.

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