mutual information

Let (Ω,,μ) be a discrete probability spaceMathworldPlanetmath, and let X and Y be discrete random variables on Ω.

The mutual informationMathworldPlanetmath I[X;Y], read as “the mutual information of X and Y,” is defined as

I[X;Y] =xΩyΩμ(X=x,Y=y)logμ(X=x,Y=y)μ(X=x)μ(Y=y)

where D denotes the relative entropyMathworldPlanetmath.

Mutual information, or just information, is measured in bits if the logarithm is to the base 2, and in “nats” when using the natural logarithm.


The most obvious characteristic of mutual information is that it depends on both X and Y. There is no information in a vacuum—information is always about something. In this case, I[X;Y] is the information in X about Y. As its name suggests, mutual information is symmetricMathworldPlanetmathPlanetmathPlanetmath, I[X;Y]=I[Y;X], so any information X carries about Y, Y also carries about X.

The definition in terms of relative entropy gives a useful interpretationMathworldPlanetmathPlanetmath of I[X;Y] as a kind of “distance” between the joint distributionPlanetmathPlanetmath μ(x,y) and the product distribution μ(x)μ(y). Recall, however, that relative entropy is not a true distance, so this is just a conceptual tool. However, it does capture another intuitive notion of information. Remember that for X,Y independentPlanetmathPlanetmath, μ(x,y)=μ(x)μ(y). Thus the relative entropy “distance” goes to zero, and we have I[X;Y]=0 as one would expect for independent random variables.

A number of useful expressions, most apparent from the definition, relate mutual information to the entropyMathworldPlanetmath H:

0I[X;Y] H[X] (1)
I[X;Y] =H[X]-H[X|Y] (2)
I[X;Y] =H[X]+H[Y]-H[X,Y] (3)
I[X;X] =H[X] (4)

Recall that the entropy H[X] quantifies our uncertainty about X. The last line justifies the description of entropy as “self-information.”

Historical Notes

Mutual information, or simply information, was introduced by Shannon in his landmark 1948 paper “A Mathematical Theory of Communication.”

Title mutual information
Canonical name MutualInformation
Date of creation 2013-03-22 12:37:35
Last modified on 2013-03-22 12:37:35
Owner drummond (72)
Last modified by drummond (72)
Numerical id 4
Author drummond (72)
Entry type Definition
Classification msc 94A17
Synonym information
Related topic RelativeEntropy
Related topic Entropy
Related topic ShannonsTheoremEntropy
Related topic DynamicStream
Defines information
Defines mutual information