An Integral Representation of the Relative Entropy

Entropy

Miku Hirata¹  Aya Nemoto¹ 

[1] Department of Mathematics, Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan; E-Mails:

关键词: relative entropy;    relative Fisher information;    de Bruijn identity;    logarithmic Sobolev inequality;    Stam inequality;   

DOI  :  10.3390/e14081469

来源: mdpi

【 摘 要 】

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.

【 授权许可】

CC BY   
© 2012 by the authors; licensee MDPI, Basel, Switzerland.