Information, amount of

From Encyclopedia of Mathematics
Revision as of 16:59, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

An information-theoretical measure of the quantity of information contained in one random variable relative to another random variable. Let and be random variables defined on a probability space and taking values in measurable spaces (cf. Measurable space) and , respectively. Let , , and , , , , be their joint and marginale probability distributions. If is absolutely continuous with respect to the direct product of measures , if is the (Radon–Nikodým) density of with respect to , and if is the information density (the logarithms are usually taken to base 2 or ), then, by definition, the amount of information is given by

If is not absolutely continuous with respect to , then , by definition.

In case the random variables and take only a finite number of values, the expression for takes the form


are the probability functions of , and the pair , respectively. (In particular,

is the entropy of .) In case and are random vectors and the densities , and of , and the pair , respectively, exist, one has

In general,

where the supremum is over all measurable functions and with a finite number of values. The concept of the amount of information is mainly used in the theory of information transmission.

For references, see , ,

to Information, transmission of.

How to Cite This Entry:
Information, amount of. Encyclopedia of Mathematics. URL:,_amount_of&oldid=12464
This article was adapted from an original article by R.L. DobrushinV.V. Prelov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article