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Category# Quantities of information

Summary

The mathematical theory of information is based on probability theory and statistics, and measures information with several quantities of information. The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. The most common unit of information is the bit, or more correctly the shannon, based on the binary logarithm. Although "bit" is more frequently used in place of "shannon", its name is not distinguished from the bit as used in data-processing to refer to a binary value or stream regardless of its entropy (information content) Other units include the nat, based on the natural logarithm, and the hartley, based on the base 10 or common logarithm.
In what follows, an expression of the form is considered by convention to be equal to zero whenever is zero. This is justified because for any logarithmic base.
Shannon derived a measure of information content called the self-information or "surprisal" of a message :
where is the probability that message is chosen from all possible choices in the message space . The base of the logarithm only affects a scaling factor and, consequently, the units in which the measured information content is expressed. If the logarithm is base 2, the measure of information is expressed in units of shannons or more often simply "bits" (a bit in other contexts is rather defined as a "binary digit", whose average information content is at most 1 shannon).
Information from a source is gained by a recipient only if the recipient did not already have that information to begin with. Messages that convey information over a certain (P=1) event (or one which is known with certainty, for instance, through a back-channel) provide no information, as the above equation indicates. Infrequently occurring messages contain more information than more frequently occurring messages.
It can also be shown that a compound message of two (or more) unrelated messages would have a quantity of information that is the sum of the measures of information of each message individually.

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