History of information theory

Some of the oldest methods of telecommunications implicitly use many of the ideas that would later be quantified in information theory. Modern telegraphy, starting in the 1830s, used Morse code, in which more common letters (like "E", which is expressed as one "dot") are transmitted more quickly than less common letters (like "J", which is expressed by one "dot" followed by three "dashes"). The idea of encoding information in this manner is the cornerstone of lossless data compression. A hundred years later, frequency modulation illustrated that bandwidth can be considered merely another degree of freedom. The vocoder, now largely looked at as an audio engineering curiosity, was originally designed in 1939 to use less bandwidth than that of an original message, in much the same way that mobile phones now trade off voice quality with bandwidth.

The most direct antecedents of Shannon's work were two papers published in the 1920s by Harry Nyquist and Ralph Hartley, who were both still research leaders at Bell Labs when Shannon arrived in the early 1940s.

Nyquist's 1924 paper, "Certain Factors Affecting Telegraph Speed", is mostly concerned with some detailed engineering aspects of telegraph signals. But a more theoretical section discusses quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation

${\displaystyle W=K\log m\,}$

where W is the speed of transmission of intelligence, m is the number of different voltage levels to choose from at each time step, and K is a constant.

Hartley's 1928 paper, called simply "Transmission of Information", went further by using the word information (in a technical sense), and making explicitly clear that information in this context was a measurable quantity, reflecting only the receiver's ability to distinguish that one sequence of symbols had been intended by the sender rather than any other—quite regardless of any associated meaning or other psychological or semantic aspect the symbols might represent. This amount of information he quantified as

${\displaystyle H=\log S^{n}\,}$

where S was the number of possible symbols, and n the number of symbols in a transmission. The natural unit of information was therefore the decimal digit, much later renamed the hartley in his honour as a unit or scale or measure of information. The Hartley information, H₀, is still used as a quantity for the logarithm of the total number of possibilities.

A similar unit of log₁₀ probability, the ban, and its derived unit the deciban (one tenth of a ban), were introduced by Alan Turing in 1940 as part of the statistical analysis of the breaking of the German second world war Enigma cyphers. The decibannage represented the reduction in (the logarithm of) the total number of possibilities (similar to the change in the Hartley information); and also the log-likelihood ratio (or change in the weight of evidence) that could be inferred for one hypothesis over another from a set of observations. The expected change in the weight of evidence is equivalent to what was later called the Kullback discrimination information.

But underlying this notion was still the idea of equal a-priori probabilities, rather than the information content of events of unequal probability; nor yet any underlying picture of questions regarding the communication of such varied outcomes.

One area where unequal probabilities were indeed well known was statistical mechanics, where Ludwig Boltzmann had, in the context of his H-theorem of 1872, first introduced the quantity

${\displaystyle H=-\sum f_{i}\log f_{i}}$

as a measure of the breadth of the spread of states available to a single particle in a gas of like particles, where f represented the relative frequency distribution of each possible state. Boltzmann argued mathematically that the effect of collisions between the particles would cause the H-function to inevitably increase from any initial configuration until equilibrium was reached; and further identified it as an underlying microscopic rationale for the macroscopic thermodynamic entropy of Clausius.

Boltzmann's definition was soon reworked by the American mathematical physicist J. Willard Gibbs into a general formula for statistical-mechanical entropy, no longer requiring identical and non-interacting particles, but instead based on the probability distribution p_i for the complete microstate i of the total system:

${\displaystyle S=-k_{\text{B}}\sum p_{i}\ln p_{i}\,}$

This (Gibbs) entropy, from statistical mechanics, can be found to directly correspond to the Clausius's classical thermodynamic definition.

Shannon himself was apparently not particularly aware of the close similarity between his new measure and earlier work in thermodynamics, but John von Neumann was. It is said that, when Shannon was deciding what to call his new measure and fearing the term 'information' was already over-used, von Neumann told him firmly: "You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage."

(Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored further in the article Entropy in thermodynamics and information theory).