Kullback-Leibler information

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Kullback–Leibler quantity of information, Kullback–Leibler information quantity, directed divergence

For discrete distributions (cf. Discrete distribution) given by probability vectors , , the Kullback–Leibler (quantity of) information of with respect to is:

where is the natural logarithm (cf. also Logarithm of a number).

More generally, one has:

for probability distributions and with densities and (cf. Density of a probability distribution).

The negative of is the conditional entropy (or relative entropy) of with respect to ; see Entropy.

Various notions of (asymmetric and symmetric) information distances are based on the Kullback–Leibler information.

The quantity is also called the informational divergence (see Huffman code).

See also Information distance; Kullback–Leibler-type distance measures.


[a1] S. Kullback, "Information theory and statistics" , Wiley (1959)
[a2] S. Kullback, R.A. Leibler, "On information and sufficiency" Ann. Math. Stat. , 22 (1951) pp. 79–86
[a3] J. Sakamoto, M. Ishiguro, G. Kitagawa, "Akaike information criterion statistics" , Reidel (1986)
How to Cite This Entry:
Kullback-Leibler information. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article