World Library  
Flag as Inappropriate
Email this Article

Limiting density of discrete points

Article Id: WHEBN0017216567
Reproduction Date:

Title: Limiting density of discrete points  
Author: World Heritage Encyclopedia
Language: English
Subject: Catalog of articles in probability theory, Asymptotic distribution, Probability theory, Entropy (information theory), List of statistics articles
Publisher: World Heritage Encyclopedia

Limiting density of discrete points

In information theory, the limiting density of discrete points is an adjustment to the formula of Claude Shannon for differential entropy.

It was formulated by Edwin Thompson Jaynes to address defects in the initial definition of differential entropy.


Shannon originally wrote down the following formula for the entropy of a continuous distribution, known as differential entropy:

H(X)=-\int p(x)\log p(x)\,dx.

Unlike Shannon's formula for the discrete entropy, however, this is not the result of any derivation (Shannon simply replaced the summation symbol in the discrete version with an integral) and it turns out to lack many of the properties that make the discrete entropy a useful measure of uncertainty. In particular, it is not invariant under a change of variables and can even become negative.

Jaynes (1963, 1968) argued that the formula for the continuous entropy should be derived by taking the limit of increasingly dense discrete distributions.[1][2] Suppose that we have a set of n discrete points \{x_i\}, such that in the limit n \to \infty their density approaches a function m(x) called the "invariant measure".

\lim_{n \to \infty}\frac{1}{n}\,(\mbox{number of points in }a

Jaynes derived from this the following formula for the continuous entropy, which he argued should be taken as the correct formula:

H(X)=-\int p(x)\log\frac{p(x)}{m(x)}\,dx.

It is similar to the (negative of the) KullbackÔÇôLeibler divergence or relative entropy, which is a comparison between two probability distributions, with one difference. In the Kullback-Leibler divergence, m(x) must be a probability density, whereas in Jaynes' formula, m(x) is simply a density, meaning that it does not have to integrate to 1.

Jaynes' continuous entropy formula has the property of being invariant under a change of variables, provided that m(x) and p(x) are transformed in the same way. (This motivates the moniker "invariant measure" for m.) This solves many of the difficulties that come from applying Shannon's continuous entropy formula.


  1. ^
  2. ^

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.