World Library  
Flag as Inappropriate
Email this Article

Joint entropy

Article Id: WHEBN0000910967
Reproduction Date:

Title: Joint entropy  
Author: World Heritage Encyclopedia
Language: English
Subject: Information theory and measure theory, Information theory, Entropy rate, Chain rule for Kolmogorov complexity, Dual total correlation
Collection: Entropy and Information
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Joint entropy

Venn diagram for various information measures associated with correlated variables X and Y. The area contained by both circles is the joint entropy H(X,Y). The circle on the left (red and violet) is the individual entropy H(X), with the red being the conditional entropy H(X|Y). The circle on the right (blue and violet) is H(Y), with the blue being H(Y|X). The violet is the mutual information I(X;Y).

In information theory, joint entropy is a measure of the uncertainty associated with a set of variables.

Contents

  • Definition 1
  • Properties 2
    • Greater than individual entropies 2.1
    • Less than or equal to the sum of individual entropies 2.2
  • Relations to other entropy measures 3

Definition

The joint Shannon entropy of two variables X and Y is defined as

H(X,Y) = -\sum_{x} \sum_{y} P(x,y) \log_2[P(x,y)] \!

where x and y are particular values of X and Y, respectively, P(x,y) is the joint probability of these values occurring together, and P(x,y) \log_2[P(x,y)] is defined to be 0 if P(x,y)=0.

For more than two variables X_1, ..., X_n this expands to

H(X_1, ..., X_n) = -\sum_{x_1} ... \sum_{x_n} P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)] \!

where x_1,...,x_n are particular values of X_1,...,X_n, respectively, P(x_1, ..., x_n) is the probability of these values occurring together, and P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)] is defined to be 0 if P(x_1, ..., x_n)=0.

Properties

Greater than individual entropies

The joint entropy of a set of variables is greater than or equal to all of the individual entropies of the variables in the set.

H(X,Y) \geq \max[H(X),H(Y)]
H(X_1, ..., X_n) \geq \max[H(X_1), ..., H(X_n)]

Less than or equal to the sum of individual entropies

The joint entropy of a set of variables is less than or equal to the sum of the individual entropies of the variables in the set. This is an example of subadditivity. This inequality is an equality if and only if X and Y are statistically independent.

H(X,Y) \leq H(X) + H(Y)
H(X_1, ..., X_n) \leq H(X_1) + ... + H(X_n)

Relations to other entropy measures

Joint entropy is used in the definition of conditional entropy

H(X|Y) = H(Y,X) - H(Y)\,

and mutual information

I(X;Y) = H(X) + H(Y) - H(X,Y)\,

In quantum information theory, the joint entropy is generalized into the joint quantum entropy.

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 USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov 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.