An Integral Representation of the Relative Entropy

被引：5

作者：

Hirata, Miku ^{[2
]}

Nemoto, Aya ^{[2
]}

Yoshida, Hiroaki ^{[1
]}

机构：

[1] Ochanomizu Univ, Dept Informat Sci, Bunkyo Ku, Tokyo 1128610, Japan

[2] Ochanomizu Univ, Dept Math, Bunkyo Ku, Tokyo 1128610, Japan

来源：

ENTROPY | 2012年 / 14卷 / 08期

关键词：

relative entropy; relative Fisher information; de Bruijn identity; logarithmic Sobolev inequality; Stam inequality; LOGARITHMIC SOBOLEV INEQUALITIES; CENTRAL-LIMIT-THEOREM; FISHER INFORMATION; POWER;

D O I：

10.3390/e14081469

中图分类号：

O4 [物理学];

学科分类号：

0702 ;

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 Verdu 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.

引用

页码：1469 / 1477

页数：9

共 17 条

[1]

[Anonymous], 2006, Elements of Information Theory

[2] On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations [J].

Arnold, A ;

Markowich, P ;

Toscani, G ;

Unterreiter, A .

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 2001, 26 (1-2) :43-100

[3]

Bakry Dominique, 1985, SEMINAIRE PROBABILIT, P5, DOI 10.1007/BFb0075847

[4] ENTROPY AND THE CENTRAL-LIMIT-THEOREM [J].

BARRON, AR .

ANNALS OF PROBABILITY, 1986, 14 (01) :336-342

[5]

BLACHMAN NM, 1965, IEEE T INFORM THEORY, V2, P267

[6] SUPERADDITIVITY OF FISHER INFORMATION AND LOGARITHMIC SOBOLEV INEQUALITIES [J].

CARLEN, EA .

JOURNAL OF FUNCTIONAL ANALYSIS, 1991, 101 (01) :194-211

[7] INFORMATION THEORETIC INEQUALITIES [J].

DEMBO, A ;

COVER, TM ;

THOMAS, JA .

IEEE TRANSACTIONS ON INFORMATION THEORY, 1991, 37 (06) :1501-1518

[8] LOGARITHMIC SOBOLEV INEQUALITIES [J].

GROSS, L .

AMERICAN JOURNAL OF MATHEMATICS, 1975, 97 (04) :1061-1083

[9] Mutual information and minimum mean-square error in Gaussian channels [J].

Guo, DN ;

Shamai, S ;

Verdú, S .

IEEE TRANSACTIONS ON INFORMATION THEORY, 2005, 51 (04) :1261-1282

[10] Fisher information inequalities and the central limit theorem [J].

Johnson, O ;

Barron, A .

PROBABILITY THEORY AND RELATED FIELDS, 2004, 129 (03) :391-409

← 1 2 →