relative entropy

Let p and q be probability distributions with supportsMathworldPlanetmathPlanetmath 𝒳 and 𝒴 respectively, where 𝒳𝒴. The relative entropy or Kullback-Leibler distancePlanetmathPlanetmath between two probability distributions p and q is defined as

D(p||q):=x𝒳p(x)logp(x)q(x). (1)

While D(p||q) is often called a distance, it is not a true metric because it is not symmetricPlanetmathPlanetmath and does not satisfy the triangle inequalityMathworldMathworldPlanetmath. However, we do have D(p||q)0 with equality iff p=q.

-D(p||q) =-x𝒳p(x)logp(x)q(x) (2)
=x𝒳p(x)logq(x)p(x) (3)
log(x𝒳p(x)q(x)p(x)) (4)
=log(x𝒳q(x)) (5)
log(x𝒴q(x)) (6)
=0 (7)

where the first inequalityMathworldPlanetmath follows from the concavity of log(x) and the second from expanding the sum over the support of q rather than p.

Relative entropy also comes in a continuousMathworldPlanetmathPlanetmath version which looks just as one might expect. For continuous distributions f and g, 𝒮 the support of f, we have

D(f||g):=𝒮flogfg. (8)
Title relative entropy
Canonical name RelativeEntropy
Date of creation 2013-03-22 12:19:32
Last modified on 2013-03-22 12:19:32
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 10
Author Mathprof (13753)
Entry type Definition
Classification msc 60E05
Classification msc 94A17
Synonym Kullback-Leibler distance
Related topic Metric
Related topic ConditionalEntropy
Related topic MutualInformation
Related topic ProofOfGaussianMaximizesEntropyForGivenCovariance