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).
|[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)|
Kullback–Leibler information. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Kullback%E2%80%93Leibler_information&oldid=22682