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An Integral Representation of the Relative Entropy

1
Department of Mathematics, Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan
2
Department of Information Sciences, Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan
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Author to whom correspondence should be addressed.
Entropy 2012, 14(8), 1469-1477; https://doi.org/10.3390/e14081469
Received: 15 June 2012 / Revised: 28 July 2012 / Accepted: 2 August 2012 / Published: 8 August 2012
Recently the identity of de Bruijn type between the relative entropy and the relative Fisher information with the reference moving has been unveiled by Verdú via MMSE in estimation theory. In this paper, we shall give another proof of this identity in more direct way that the derivative is calculated by applying integrations by part with the heat equation. We shall also derive an integral representation of the relative entropy, as one of the applications of which the logarithmic Sobolev inequality for centered Gaussian measures will be given. View Full-Text
Keywords: relative entropy; relative Fisher information; de Bruijn identity; logarithmic Sobolev inequality; Stam inequality relative entropy; relative Fisher information; de Bruijn identity; logarithmic Sobolev inequality; Stam inequality
MDPI and ACS Style

Hirata, M.; Nemoto, A.; Yoshida, H. An Integral Representation of the Relative Entropy. Entropy 2012, 14, 1469-1477.

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