Home > Information theory

These two wings of information theory are joined together and mutually justified by the information transmission theorem s, or source-channel separation theorem s that justify the use of bits as the universal currency for information in many contexts.

It is generally believed that the modern discipline of information theory began with the publication of Shannon's article "The Mathematical Theory of Communication" in the Bell System Technical JournalBell System Technical Journal was the in-house journal of Bell Laboratories. It is most famous for Claude Shannon's paper A mathematical theory of communication which founded the field of information theory. The journal has since been through several name in JulyJuly is the seventh month of the year in the Gregorian Calendar, with 31 days. July was renamed for Julius Caesar; previously, it was called Quintilis in Latin, since it was the fifth month in the Roman calendar which started in March. Because of its orig and October of 1948. This work drew on earlier publications by Harry Nyquist and Ralph Hartley. In the process of working out a theory of communications that could be applied by electrical engineers to design better telecommunications systems, Shannon defined a measure of entropy:

(where p_i is the probability of i) that, when applied to an information source, could determine the capacity of the channel required to transmit the source as encoded binary digits. If the logarithm in the formula is taken to base 2, then it gives a measure of entropy in bits. Shannon's measure of entropy came to be taken as a measure of the information contained in a message, as opposed to the portion of the message that is strictly determined (hence predictable) by inherent structures, like for instance redundancy in the structure of languages or the statistical properties of a language relating to the frequencies of occurrence of different letter or word pairs, triplets etc. See Markov chains.

Recently however, it has emerged that entropy was defined and used during the Second World War by Alan Turing at Bletchley Park. Turing named it 'weight of evidence' and measured it in units called bans and decibans. Turing and Shannon collaborated during the war but it appears that they independently created the concept. (References are given in Alan Turing: The Enigma by Andrew Hodges .)

Entropy as defined by Shannon is closely related to entropy as defined by physicists. Boltzmann and Gibbs did considerable work on statistical thermodynamics. This work was the inspiration for adopting the term entropy in information theory. There are deep relationships between entropy in the thermodynamic and informational senses. For instance, Maxwell's demon needs information to reverse thermodynamic entropy and getting that information exactly balances out the thermodynamic gain that the demon would otherwise achieve.

Among other useful measures of information is mutual information, a measure of the correlation between two random variables. Mutual information is defined for two events and as

where is the joint entropy, defined as

and is the conditional entropy of conditioned on observing . As such, the mutual information can be intuitively considered the amount of uncertainty in that is eliminated by observations of and vice versa.

Mutual information is closely related to the log-likelihood ratio test for multinomial s and to Pearson's χ² test.

Shannon information is appropriate for measuring uncertainty over an unordered space. An alternative measure of information was created by Fisher for measuring uncertainty over an ordered space. For example, Shannon information is used over the space of alphabetic letters, as letters do not have 'distances' between them. For information about the value of a continuous parameter such as a person's height, Fisher information is used, as estimated heights do have a well-defined distance.

Differences in Shannon information correspond to a special case of the Kullback-Leibler divergence of Bayesian statistics, a measure of the distance between the prior and posterior probability distributions.