# Information, amount of

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An information-theoretical measure of the quantity of information contained in one random variable relative to another random variable. Let $\xi$ and $\eta$ be random variables defined on a probability space $( \Omega , \mathfrak A , {\mathsf P} )$ and taking values in measurable spaces (cf. Measurable space) $( \mathfrak X , S _ {\mathfrak X } )$ and $( \mathfrak Y , S _ {\mathfrak Y } )$, respectively. Let $p _ {\xi \eta } ( C)$, $C \in S _ {\mathfrak X } \times S _ {\mathfrak Y }$, and $p _ \xi ( A)$, $A \in S _ {\mathfrak X }$, $p _ \eta ( B)$, $B \in S _ {\mathfrak Y }$, be their joint and marginale probability distributions. If $p _ {\xi \eta } ( \cdot )$ is absolutely continuous with respect to the direct product of measures $p _ \xi \times p _ \eta ( \cdot )$, if $a _ {\xi \eta } ( \cdot )$ is the (Radon–Nikodým) density of $p _ {\xi \eta } ( \cdot )$ with respect to $p _ \xi \times p _ \eta ( \cdot )$, and if $i _ {\xi \eta } ( \cdot ) = \mathop{\rm log} a _ {\xi \eta } ( \cdot )$ is the information density (the logarithms are usually taken to base 2 or $e$), then, by definition, the amount of information is given by

$$I ( \xi , \eta ) = \ \int\limits _ {\mathfrak X \times \mathfrak Y } i _ {\xi \eta } ( x , y ) p _ {\xi \eta } ( d x , d y ) =$$

$$= \ \int\limits _ {\mathfrak X \times \mathfrak Y } a _ {\xi \eta } ( x , y ) \mathop{\rm log} \ a _ {\xi \eta } ( x , y ) p _ \xi ( d x ) p _ \eta ( d y ) .$$

If $p _ {\xi \eta } ( \cdot )$ is not absolutely continuous with respect to $p _ \xi \times p _ \eta ( \cdot )$, then $I ( \xi , \eta ) = + \infty$, by definition.

In case the random variables $\xi$ and $\eta$ take only a finite number of values, the expression for $I ( \xi , \eta )$ takes the form

$$I ( \xi , \eta ) = \ \sum _ { i= } 1 ^ { n } \sum _ { j= } 1 ^ { m } p _ {ij} \mathop{\rm log} \ \frac{p _ {ij} }{p _ {i} q _ {i} } ,$$

where

$$\{ p _ {i} \} _ {i=} 1 ^ {n} ,\ \ \{ q _ {j} \} _ {j=} 1 ^ {m} ,\ \ \{ {p _ {ij} } : {i = 1 \dots n ; j = 1 \dots m } \}$$

are the probability functions of $\xi$, $\eta$ and the pair $( \xi , \eta )$, respectively. (In particular,

$$I ( \xi , \xi ) = - \sum _ { i= } 1 ^ { n } p _ {i} \mathop{\rm log} p _ {i} = H ( \xi )$$

is the entropy of $\xi$.) In case $\xi$ and $\eta$ are random vectors and the densities $p _ \xi ( x)$, $p _ \eta ( y)$ and $p _ {\xi \eta } ( x , y )$ of $\xi$, $\eta$ and the pair $( \xi , \eta )$, respectively, exist, one has

$$I ( \xi , \eta ) = \ \int\limits p _ {\xi \eta } ( x , y ) \mathop{\rm log} \frac{p _ {\xi \eta } ( x , y ) }{p _ \xi ( x) p _ \eta ( y) } \ d x d y .$$

In general,

$$I ( \xi , \eta ) = \ \sup I ( \phi ( \xi ) , \psi ( \eta ) ) ,$$

where the supremum is over all measurable functions $\phi ( \cdot )$ and $\psi ( \cdot )$ 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 , ,

How to Cite This Entry:
Information, amount of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Information,_amount_of&oldid=47349
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