Information, exactness of reproducibility of

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A measure of the quality of information transmission from an information source (cf. Information, source of) to a receiver (addressee) over a communication channel. The criteria relevant to the exactness of reproducibility of information in the theory of information transmission are usually treated statistically, by isolating the class of admissible joint distributions for pairs in the set of all probability measures on the product , where is the measurable space of values of a communication generated by the source, and is the measurable space of values of the communication received. Exactness of reproducibility of information is often defined in terms of a distortion measure , , , which is a non-negative measurable function of and . The set of admissible communications is then specified by the formula


for a given .

In particular, when and , one often uses a componentwise condition for the exactness of reproducibility of information, namely

where , , , , , and where , , , is again a non-negative measurable function. In this case, instead of condition (1) one sometimes uses the following condition:


In the case when and

the conditions (1) and (2) turn into restrictions on the mean or maximal probability of erroneous decoding (cf. Erroneous decoding, probability of) of separate components of the communication, respectively. In the case of sources with continuous spaces (such as a Gaussian source), it is often assumed that .


[1] R. Gallagher, "Information theory and reliable communication" , Wiley (1968)
[2] T. Berger, "Rate distortion theory" , Prentice-Hall (1971)



[a1] I. Csiszar, J. Körner, "Information theory. Coding theorems for discrete memoryless systems" , Akad. Kiado (1981)
How to Cite This Entry:
Information, exactness of reproducibility of. Encyclopedia of Mathematics. URL:,_exactness_of_reproducibility_of&oldid=11964
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