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An Integral Representation of the Relative Entropy
Department of Mathematics, Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan
Department of Information Sciences, Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan
* Author to whom correspondence should be addressed.
Received: 15 June 2012; in revised form: 28 July 2012 / Accepted: 2 August 2012 / Published: 8 August 2012
Abstract: 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.
Keywords: relative entropy; relative Fisher information; de Bruijn identity; logarithmic Sobolev inequality; Stam inequality
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Cite This Article
MDPI and ACS Style
Hirata, M.; Nemoto, A.; Yoshida, H. An Integral Representation of the Relative Entropy. Entropy 2012, 14, 1469-1477.
Hirata M, Nemoto A, Yoshida H. An Integral Representation of the Relative Entropy. Entropy. 2012; 14(8):1469-1477.
Hirata, Miku; Nemoto, Aya; Yoshida, Hiroaki. 2012. "An Integral Representation of the Relative Entropy." Entropy 14, no. 8: 1469-1477.