Extensions of fisher information and stam's inequality

Erwin Lutwak; Songjun Lv; Deane Yang; Gaoyong Zhang

doi:10.1109/TIT.2011.2177563

Extensions of fisher information and stam's inequality

Erwin Lutwak, Songjun Lv, Deane Yang, Gaoyong Zhang

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We explain how the classical notions of Fisher information of a random variable and Fisher information matrix of a random vector can be extended to a much broader setting. We also show that Stam's inequality for Fisher information and Shannon entropy, as well as the more generalized versions proved earlier by the authors, are all special cases of more general sharp inequalities satisfied by random vectors. The extremal random vectors, which we call generalized Gaussians, contain Gaussians as a limiting case but are noteworthy because they are heavy-tailed.

Original language	English (US)
Article number	6157091
Pages (from-to)	1319-1327
Number of pages	9
Journal	IEEE Transactions on Information Theory
Volume	58
Issue number	3
DOIs	https://doi.org/10.1109/TIT.2011.2177563
State	Published - Mar 2012

Keywords

Entropy
Fisher information
Rényi entropy
Shannon entropy
Shannon theory
Stam inequality
information measure
information theory

ASJC Scopus subject areas

Information Systems
Computer Science Applications
Library and Information Sciences

Access to Document

10.1109/TIT.2011.2177563

Cite this

@article{6a15d82088cf42a28fe7a13e461ab856,

title = "Extensions of fisher information and stam's inequality",

abstract = "We explain how the classical notions of Fisher information of a random variable and Fisher information matrix of a random vector can be extended to a much broader setting. We also show that Stam's inequality for Fisher information and Shannon entropy, as well as the more generalized versions proved earlier by the authors, are all special cases of more general sharp inequalities satisfied by random vectors. The extremal random vectors, which we call generalized Gaussians, contain Gaussians as a limiting case but are noteworthy because they are heavy-tailed.",

keywords = "Entropy, Fisher information, R{\'e}nyi entropy, Shannon entropy, Shannon theory, Stam inequality, information measure, information theory",

author = "Erwin Lutwak and Songjun Lv and Deane Yang and Gaoyong Zhang",

note = "Funding Information: Manuscript received December 30, 2009; revised September 27, 2011; accepted October 03, 2011. Date of current version February 29, 2012. This work was supported in part by the National Science Foundation (NSF) under Grant DMS-0706859 and the Chinese NSF under Grant 10801140. E. Lutwak, D. Yang, and G. Zhang are with the Department of Mathematics, Polytechnic Institute of New York University, Brooklyn, NY 11201 USA (e-mail: [email protected]; [email protected]; [email protected]). S. Lv is with the College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 401331, China (e-mail: [email protected]). Communicated by I. Kontoyiannis, Associate Editor for Shannon Theory. Digital Object Identifier 10.1109/TIT.2011.2177563",

year = "2012",

month = mar,

doi = "10.1109/TIT.2011.2177563",

language = "English (US)",

volume = "58",

pages = "1319--1327",

journal = "IEEE Transactions on Information Theory",

issn = "0018-9448",

publisher = "Institute of Electrical and Electronics Engineers Inc.",

number = "3",

}

TY - JOUR

T1 - Extensions of fisher information and stam's inequality

AU - Lutwak, Erwin

AU - Lv, Songjun

AU - Yang, Deane

AU - Zhang, Gaoyong

N1 - Funding Information: Manuscript received December 30, 2009; revised September 27, 2011; accepted October 03, 2011. Date of current version February 29, 2012. This work was supported in part by the National Science Foundation (NSF) under Grant DMS-0706859 and the Chinese NSF under Grant 10801140. E. Lutwak, D. Yang, and G. Zhang are with the Department of Mathematics, Polytechnic Institute of New York University, Brooklyn, NY 11201 USA (e-mail: [email protected]; [email protected]; [email protected]). S. Lv is with the College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 401331, China (e-mail: [email protected]). Communicated by I. Kontoyiannis, Associate Editor for Shannon Theory. Digital Object Identifier 10.1109/TIT.2011.2177563

PY - 2012/3

Y1 - 2012/3

N2 - We explain how the classical notions of Fisher information of a random variable and Fisher information matrix of a random vector can be extended to a much broader setting. We also show that Stam's inequality for Fisher information and Shannon entropy, as well as the more generalized versions proved earlier by the authors, are all special cases of more general sharp inequalities satisfied by random vectors. The extremal random vectors, which we call generalized Gaussians, contain Gaussians as a limiting case but are noteworthy because they are heavy-tailed.

AB - We explain how the classical notions of Fisher information of a random variable and Fisher information matrix of a random vector can be extended to a much broader setting. We also show that Stam's inequality for Fisher information and Shannon entropy, as well as the more generalized versions proved earlier by the authors, are all special cases of more general sharp inequalities satisfied by random vectors. The extremal random vectors, which we call generalized Gaussians, contain Gaussians as a limiting case but are noteworthy because they are heavy-tailed.

KW - Entropy

KW - Fisher information

KW - Rényi entropy

KW - Shannon entropy

KW - Shannon theory

KW - Stam inequality

KW - information measure

KW - information theory

UR - http://www.scopus.com/inward/record.url?scp=84863236970&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84863236970&partnerID=8YFLogxK

U2 - 10.1109/TIT.2011.2177563

DO - 10.1109/TIT.2011.2177563

M3 - Article

AN - SCOPUS:84863236970

SN - 0018-9448

VL - 58

SP - 1319

EP - 1327

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 3

M1 - 6157091

ER -

Extensions of fisher information and stam's inequality

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this