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: ''The topic of this article is distinct from the topics of Library And Information Science and Information Technology .'' Information theory is a field of mathematics concerning the storage and transmission of Data and includes the fundamental concepts of Source Coding and Channel Coding . Source coding involves compression (abbreviation) of data in a manner such that another person can recover either an ''identical'' copy of the uncompressed data ( Lossless Data Compression , which uses the concept of Entropy ) or an ''approximate'' copy of the uncompressed data ( Lossy Data Compression , which uses the theory of Rate Distortion ). Channel coding considers how to transmit such data, asking at how high a Rate data can be communicated to someone else through a noisy medium with an arbitrarily small probability of error. These topics are rigorously addressed using mathematics introduced by Claude Shannon in 1948. His papers spawned the field of information theory, which goes beyond the above questions to extended and combined problems such as those in Network Information Theory and related problems including Portfolio Theory and Cryptography . The impact of information theory has been crucial to the success of the Voyager missions to deep space, the invention of the CD , the feasibility of Mobile Phone s, the development of the Internet and Broadband Internet Access , the analysis of DNA , and numerous other fields. OVERVIEW Information theory is the mathematical theory of data communication and storage, generally considered to have been founded in 1948 by Claude E. Shannon . The central Paradigm of classic information theory is the engineering problem of the transmission of information over a noisy channel. The most fundamental results of this theory are Shannon's Source Coding Theorem , which establishes that on average the number of ''bits'' needed to represent the result of an uncertain event is given by the Entropy ; and Shannon's Noisy-channel Coding Theorem , which states that ''reliable'' communication is possible over ''noisy'' channels provided that the rate of communication is below a certain threshold called the channel capacity. The channel capacity is achieved with appropriate encoding and decoding systems. Information theory is closely associated with a collection of pure and applied disciplines that have been carried out under a variety of banners in different parts of the world over the past half century or more: Adaptive System s, Anticipatory System s, Artificial Intelligence , Complex System s, Complexity Science , Cybernetics , Informatics , Machine Learning , along with Systems Science s of many descriptions. Information theory is a broad and deep mathematical theory, with equally broad and deep applications, chief among them Coding Theory . Coding theory is concerned with finding explicit methods, called ''codes'', of increasing the efficiency and fidelity of data communication over a noisy channel up near the limit that Shannon proved is all but possible. These codes can be roughly subdivided into Data Compression and Error-correction codes. It took many years to find the good codes whose existence Shannon proved. A third class of codes are cryptographic Cipher s; concepts from coding theory and information theory are much used in Cryptography and Cryptanalysis . ''See the article Deciban for an interesting historical application.'' Information theory is also used in Information Retrieval , Intelligence Gathering , Gambling , Statistics , and even Musical Composition and Whale Song s. HISTORY The decisive event which established the subject of information theory, and brought it to immediate worldwide attention, was the publication of Claude E. Shannon ( 1916 – 2001 )'s classic paper " A Mathematical Theory Of Communication " in the '' Bell System Technical Journal '' in July and October of 1948 . In this revolutionary and groundbreaking paper, the work for which Shannon had substantially completed at Bell Labs by the end of 1944 , Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process, which underlies information theory; and with it the ideas of the Information Entropy and Redundancy of a source, and its relevance through the Source Coding Theorem ; the Mutual Information , and the Channel Capacity of a noisy channel, as underwritten by the promise of perfect loss-free communication given by the Noisy-channel Coding Theorem ; the practical result of the Shannon-Hartley Law for the channel capacity of a Gaussian channel; and of course the Bit - a new common currency of information. Before 1948 Quantitative ideas of information 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 there 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 : 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 introducing the word information, and making explicitly clear the idea that information in this context was a measurable quantity, reflecting only that the receiver was able to distinguish that one sequence of symbols had been sent 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 : 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 , ''H''0, is still used as a quantity for the log 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. Entropy in statistical mechanics 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 : 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 identified it as an underlying microscopic rationale for the macroscopic Thermodynamic Entropy of Clausius . (The ''H''-theorem of Boltzmann subsequently led to no end of controversy; and can still cause lively debates to the present day, often aggravated by protagonists not realising that they are arguing at cross-purposes. The theorem relies on a hidden assumption, that useful information is destroyed by the collisions, which can be questioned; also, it relies on a non-equilibrium state being singled out as the initial state (not the final state), which breaks time symmetry; also, strictly it applies only in a statistical sense, namely that an average ''H''-function would be non-decreasing). Boltzmann's definition was soon reworked by the American mathematical physicist J. Willard Gibbs into a general formula for the statistical-mechanical entropy, no longer requiring identical and non-interacting particles, but instead based on the probability distribution ''pi'' for the complete microstate ''i'' of the total system: : This (Gibbs) entropy from statistical mechanics can be found to directly correspond to the Clausius's classical thermodynamical definition, as explored further in the article: '' Thermodynamic Entropy ''. Szilard, Lewis. Shannon himself was apparently not particularly aware of the Close Similarity between his new quantity and the earlier work in thermodynamics; but John Von Neumann was. The story goes that when Shannon was deciding what to call his new quantity, fearing that '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." Development since 1948 The publication of Shannon's 1948 paper, " A Mathematical Theory Of Communication ", in the ''Bell System Technical Journal'' was the founding of information theory as we know it today. Many developments and applications of the theory have taken place since then, which have made many modern devices for data communication and storage such as CD-ROM s and Mobile Phone s possible. MATHEMATICAL THEORY OF INFORMATION : ''For a more thorough discussion of these basic equations, see Information Entropy .'' The abstract idea of what "information" really is must be made more concrete so that mathematicians can analyze it. Self-information Shannon defined a measure of information content called the Self-information or surprisal of a message ''m'': : where is the probability that message ''m'' is chosen from all possible choices in the message space . This equation causes messages with lower probabilities to contribute more to the overall value of ''I(m)''. In other words, infrequently occurring messages are more valuable. (This is a consequence from the property of Logarithm s that is very large when is near 0 for unlikely messages and very small when is near 1 for almost certain messages). For example, if John says "See you later, honey" to his wife every morning before leaving to office, that information holds little "content" or "value". But, if he shouts "Get lost" at his wife one morning, then that message holds more value or content (because, supposedly, the probability of him choosing that message is very low). Entropy The Entropy of a discrete message space is a measure of the amount of uncertainty one has about which message will be chosen. It is defined as the Average self-information of a message from that message space: : | ||
| : <math> H(Xy) | \mathbb{E}_ p(xy) = -\sum_{x \in X} p(xy) \log p(xy)</math> |
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| Where <math>p(xy) | rac{p(x,y)}{p(y)}</math> is the Conditional Probability of <math>x</math> given <math>y</math> |
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| :<math> H(XY) | \mathbb E_Y \{H(Xy)\} = -\sum_{y \in Y} p(y) \sum_{x \in X} p(xy) \log p(xy) = \sum_{x,y} p(x,y) \log rac{p(y)}{p(x,y)}</math> |
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| : <math> H(XY) | H(X,Y) - H(Y) \,</math> |
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| :<math>I(XY) | \sum_{x,y} p(y)\, p(xy) \log rac{p(xy)}{p(x)} = \sum_{x,y} p(x,y) \log rac{p(x,y)}{p(x)\, p(y)}</math> |
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| : <math>I(XY) | H(X) - H(XY)\,</math> |
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| : <math>I(X Y) | D_{KL}\left(p(X,Y) \ p(X)p(Y)
ight)\,</math> |
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| : <math> H(Xy) | -\int_X f(xy) \log f(xy) \,dx </math> |
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| : <math> H(XY) | -\int_Y \int_X f(x,y) \log rac{f(x,y)}{f(y)} \,dx \,dy</math> |
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| Here ''X'' Represents The Space Of Messages Transmitted, And ''Y'' The Space Of Messages Received During A Unit Time Over Our Channel Let <math>p(yx)</math> Be The Conditional Probability Distribution Function Of ''Y'' Given ''X'' We Will Consider <math>p(yx)</math> To Be An Inherent Fixed Property Of Our Communications Channel (representing The Nature Of The '''noise''' Of Our Channel) Then The Joint Distribution Of ''X'' And ''Y'' Is Completely Determined By Our Channel And By Our Choice Of <math>f(x)</math>, The Marginal Distribution Of Messages We Choose To Send Over The Channel Under These Constraints, We Would Like To Maximize The Amount Of Information, Or The '''signal''', We Can Communicate Over The Channel The Appropriate Measure For This Is The Transinformation, And This Maximum Transinformation Is Called The | "http://wwwinformationdelightinfo/encyclopedia/entry/channel_capacity" class="copylinks">Channel Capacity and is given by: |
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| The '''rate''' Of A Source Of Information Is (in The Most General Case) <math>r | \mathbb E H(M_tM_{t-1},M_{t-2},M_{t-3}, \cdots)</math>, the expected, or average, conditional entropy per message (ie per unit time) given all the previous messages generated It is common in information theory to speak of the "rate" or "entropy" of a language This is appropriate, for example, when the source of information is English prose The rate of a memoryless source is simply <math>H(M)</math>, since by definition there is no interdependence of the successive messages of a memoryless source The rate of a source of information is related to its Redundancy and how well it can be Compressed |
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| : <math>H(XY) | \mu( ilde X \,\backslash\, ilde Y),</math> |
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