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Not to be confused with information science.
Information theory is a branch of applied mathematics, electrical engineering, and computer science involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data. Since its inception it has broadened to find applications in many other areas, including statistical inference, natural language processing, cryptography, neurobiology,^{[1]} the evolution^{[2]} and function^{[3]} of molecular codes, model selection in ecology,^{[4]} thermal physics,^{[5]} quantum computing, linguistics, plagiarism detection,^{[6]} pattern recognition, anomaly detection and other forms of data analysis.^{[7]}A key measure of information is entropy, which is usually expressed by the average number of bits needed to store or communicate one symbol in a message. Entropy quantifies the uncertainty involved in predicting the value of a random variable. For example, specifying the outcome of a fair coin flip (two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a die (six equally likely outcomes).
Applications of fundamental topics of information theory include lossless data compression (e.g. ZIP files), lossy data compression (e.g. MP3s and JPEGs), and channel coding (e.g. for Digital Subscriber Line (DSL)). The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, and electrical engineering. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields. Important subfields of information theory are source coding, channel coding, algorithmic complexity theory, algorithmic information theory, informationtheoretic security, and measures of information.
Contents
[hide]Overview[edit]
The main concepts of information theory are, first, that a "message" conveys a quantifiable amount of information to the recipient, and, second, that a "channel" (over which messages are transmitted) has a quantifiable information "capacity" defining the limiting rate at which it can convey information. An underlying concept is that information must be news to the recipient; in particular, telling the recipient something the recipient already knows conveys no information. The object of coding theory is to encode messages so as to send them over a channel at as high a rate as possible (within the limits of the channel capacity).These concepts can be grasped by considering the most widespread means of human communication: language. Two important aspects of a concise language are as follows: First, the most common words (e.g., "a", "the", "I") should be shorter than less common words (e.g., "roundabout", "generation", "mediocre"), so that sentences will not be too long. Such a tradeoff in word length is analogous to data compression and is the essential aspect of source coding. Second, if part of a sentence is unheard or misheard due to noise — e.g., a passing car — the listener should still be able to glean the meaning of the underlying message. Such robustness is as essential for an electronic communication system as it is for a language; properly building such robustness into communications is done by channel coding. Source coding and channel coding are the fundamental concerns of information theory.
Note that these concerns have nothing to do with the importance of messages. For example, a platitude such as "Thank you; come again" takes about as long to say or write as the urgent plea, "Call an ambulance!" while the latter may be more important and more meaningful in many contexts. Information theory, however, does not consider message importance or meaning, as these are matters of the quality of data rather than the quantity and readability of data, the latter of which is determined solely by probabilities. "Thank you, come again" conveys less information than "Call an ambulance!" not because it is less important or less urgent, but because it is said more often.
Information theory is generally considered to have been founded in 1948 by Claude Shannon in his seminal work, "A Mathematical Theory of Communication". The central paradigm of classical 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 its entropy; and Shannon's noisychannel 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 can be approached in practice by using appropriate encoding and decoding systems.
Information theory is closely associated with a collection of pure and applied disciplines that have been investigated and reduced to engineering practice under a variety of rubrics throughout the world over the past half century or more: adaptive systems, anticipatory systems, artificial intelligence, complex systems, complexity science, cybernetics, informatics, machine learning, along with systems sciences of many descriptions. Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of coding theory.
Coding theory is concerned with finding explicit methods, called codes, for increasing the efficiency and reducing the net error rate of data communication over a noisy channel to near the limit that Shannon proved is the maximum possible for that channel. These codes can be roughly subdivided into data compression (source coding) and errorcorrection (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible. A third class of information theory codes are cryptographic algorithms (both codes and ciphers). Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis. See the article ban (unit) for a historical application.
Information theory is also used in information retrieval, intelligence gathering, gambling, statistics, and even in musical composition.
Historical background[edit]
Main article: History of information theory
The landmark event that established the discipline of information theory, and brought it to immediate worldwide attention, was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October 1948.Prior to this paper, limited informationtheoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. Harry Nyquist's 1924 paper, Certain Factors Affecting Telegraph Speed, contains a theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation (recalling Boltzmann's constant), 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. Ralph Hartley's 1928 paper, Transmission of Information, uses the word information as a measurable quantity, reflecting the receiver's ability to distinguish one sequence of symbols from any other, thus quantifying information as , where S was the number of possible symbols, and n the number of symbols in a transmission. The 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. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers.
Much of the mathematics behind information theory with events of different probabilities were developed for the field of thermodynamics by Ludwig Boltzmann and J. Willard Gibbs. Connections between informationtheoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored in Entropy in thermodynamics and information theory.
In Shannon's revolutionary and groundbreaking paper, the work for which had been 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 underlying information theory, opening with the assertion that
 "The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."
 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, including the promise of perfect lossfree communication given by the noisychannel coding theorem;
 the practical result of the Shannon–Hartley law for the channel capacity of a Gaussian channel; as well as
 the bit—a new way of seeing the most fundamental unit of information.
Quantities of information[edit]
Main article: Quantities of information
Information theory is based on probability theory and statistics. The most important quantities of information are entropy, the information in a random variable, and mutual information, the amount of information in common between two random variables. The former quantity gives the limit on how far message data can be compressed, while the latter can be used to find the communication rate across a channel.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, based on the binary logarithm. Other units include the nat, which is based on the natural logarithm, and the hartley, which is based on the common logarithm.
In what follows, an expression of the form is considered by convention to be equal to zero whenever This is justified because for any logarithmic base.
Entropy[edit]
The entropy, , of a discrete random variable is a measure of the amount of uncertainty associated with the value of .Suppose one transmits 1000 bits (0s and 1s). If the value of each these bits is known to the receiver (has a specific value with certainty) ahead of transmission, it is clear that no information is transmitted. If, however, each bit is independently equally likely to be 0 or 1, 1000 shannons of information (also often called bits, in the information theoretic sense) have been transmitted. Between these two extremes, information can be quantified as follows. If is the set of all messages that could be, and is the probability of some , then the entropy, , of is defined:^{[8]}
The special case of information entropy for a random variable with two outcomes is the binary entropy function, usually taken to the logarithmic base 2, thus having the shannon (Sh) as unit:
Joint entropy[edit]
The joint entropy of two discrete random variables and is merely the entropy of their pairing: . This implies that if and are independent, then their joint entropy is the sum of their individual entropies.For example, if represents the position of a chess piece — the row and the column, then the joint entropy of the row of the piece and the column of the piece will be the entropy of the position of the piece.
Conditional entropy (equivocation)[edit]
The conditional entropy or conditional uncertainty of given random variable (also called the equivocation of about ) is the average conditional entropy over :^{[9]}Mutual information (transinformation)[edit]
Mutual information measures the amount of information that can be obtained about one random variable by observing another. It is important in communication where it can be used to maximize the amount of information shared between sent and received signals. The mutual information of relative to is given by:A basic property of the mutual information is that
Mutual information is symmetric:
Kullback–Leibler divergence (information gain)[edit]
The Kullback–Leibler divergence (or information divergence, information gain, or relative entropy) is a way of comparing two distributions: a "true" probability distribution p(X), and an arbitrary probability distribution q(X). If we compress data in a manner that assumes q(X) is the distribution underlying some data, when, in reality, p(X) is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. It is thus definedKullback–Leibler divergence of a prior from the truth[edit]
Another interpretation of KL divergence is this: suppose a number X is about to be drawn randomly from a discrete set with probability distribution p(x). If Alice knows the true distribution p(x), while Bob believes (has a prior) that the distribution is q(x), then Bob will be more surprised than Alice, on average, upon seeing the value of X. The KL divergence is the (objective) expected value of Bob's (subjective) surprisal minus Alice's surprisal, measured in bits if the log is in base 2. In this way, the extent to which Bob's prior is "wrong" can be quantified in terms of how "unnecessarily surprised" it's expected to make him.Other quantities[edit]
Other important information theoretic quantities include Rényi entropy (a generalization of entropy), differential entropy (a generalization of quantities of information to continuous distributions), and the conditional mutual information.Coding theory[edit]
Main article: Coding theory
Coding theory is one of the most important and direct applications of information theory. It can be subdivided into source coding theory and channel coding theory. Using a statistical description for data, information theory quantifies the number of bits needed to describe the data, which is the information entropy of the source. Data compression (source coding): There are two formulations for the compression problem:
 lossless data compression: the data must be reconstructed exactly;
 lossy data compression: allocates bits needed to reconstruct the data, within a specified fidelity level measured by a distortion function. This subset of Information theory is called rate–distortion theory.
 Errorcorrecting codes (channel coding): While data compression removes as much redundancy as possible, an error correcting code adds just the right kind of redundancy (i.e., error correction) needed to transmit the data efficiently and faithfully across a noisy channel.
Source theory[edit]
Any process that generates successive messages can be considered a source of information. A memoryless source is one in which each message is an independent identically distributed random variable, whereas the properties of ergodicity and stationarity impose less restrictive constraints. All such sources are stochastic. These terms are well studied in their own right outside information theory.Rate[edit]
Information rate is the average entropy per symbol. For memoryless sources, this is merely the entropy of each symbol, while, in the case of a stationary stochastic process, it isIt 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 source of information is related to its redundancy and how well it can be compressed, the subject of source coding.
Channel capacity[edit]
Main article: Channel capacity
Communications over a channel—such as an ethernet cable—is the primary motivation of information theory. As anyone who's ever used a telephone (mobile or landline) knows, however, such channels often fail to produce exact reconstruction of a signal; noise, periods of silence, and other forms of signal corruption often degrade quality. How much information can one hope to communicate over a noisy (or otherwise imperfect) channel?Consider the communications process over a discrete channel. A simple model of the process is shown below:
Here X represents the space of messages transmitted, and Y the space of messages received during a unit time over our channel. Let be the conditional probability distribution function of Y given X. We will consider 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 , the marginal distribution of messages we choose to send over the channel. Under these constraints, we would like to maximize the rate of information, or the signal, we can communicate over the channel. The appropriate measure for this is the mutual information, and this maximum mutual information is called the channel capacity and is given by:
Channel coding is concerned with finding such nearly optimal codes that can be used to transmit data over a noisy channel with a small coding error at a rate near the channel capacity.
Capacity of particular channel models[edit]
 A continuoustime analog communications channel subject to Gaussian noise — see Shannon–Hartley theorem.
 A binary symmetric channel (BSC) with crossover probability p is a binary input, binary output channel that flips the input bit with probability p. The BSC has a capacity of bits per channel use, where is the binary entropy function to the base 2 logarithm:
 A binary erasure channel (BEC) with erasure probability p is a binary input, ternary output channel. The possible channel outputs are 0, 1, and a third symbol 'e' called an erasure. The erasure represents complete loss of information about an input bit. The capacity of the BEC is 1  p bits per channel use.
Applications to other fields[edit]
Intelligence uses and secrecy applications[edit]
Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the ban, was used in the Ultra project, breaking the German Enigma machine code and hastening the end of World War II in Europe. Shannon himself defined an important concept now called the unicity distance. Based on the redundancy of the plaintext, it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability.Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. A brute force attack can break systems based on asymmetric key algorithms or on most commonly used methods of symmetric key algorithms (sometimes called secret key algorithms), such as block ciphers. The security of all such methods currently comes from the assumption that no known attack can break them in a practical amount of time.
Information theoretic security refers to methods such as the onetime pad that are not vulnerable to such brute force attacks. In such cases, the positive conditional mutual information between the plaintext and ciphertext (conditioned on the key) can ensure proper transmission, while the unconditional mutual information between the plaintext and ciphertext remains zero, resulting in absolutely secure communications. In other words, an eavesdropper would not be able to improve his or her guess of the plaintext by gaining knowledge of the ciphertext but not of the key. However, as in any other cryptographic system, care must be used to correctly apply even informationtheoretically secure methods; the Venona project was able to crack the onetime pads of the Soviet Union due to their improper reuse of key material.
Pseudorandom number generation[edit]
Pseudorandom number generators are widely available in computer language libraries and application programs. They are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern computer equipment and software. A class of improved random number generators is termed cryptographically secure pseudorandom number generators, but even they require random seeds external to the software to work as intended. These can be obtained via extractors, if done carefully. The measure of sufficient randomness in extractors is minentropy, a value related to Shannon entropy through Rényi entropy; Rényi entropy is also used in evaluating randomness in cryptographic systems. Although related, the distinctions among these measures mean that a random variable with high Shannon entropy is not necessarily satisfactory for use in an extractor and so for cryptography uses.Seismic exploration[edit]
One early commercial application of information theory was in the field of seismic oil exploration. Work in this field made it possible to strip off and separate the unwanted noise from the desired seismic signal. Information theory and digital signal processing offer a major improvement of resolution and image clarity over previous analog methods.^{[11]}Semiotics[edit]
Concepts from information theory such as redundancy and code control have been used by semioticians such as Umberto Eco and RossiLandi to explain ideology as a form of message transmission whereby a dominant social class emits its message by using signs that exhibit a high degree of redundancy such that only one message is decoded among a selection of competing ones.^{[12]}Miscellaneous applications[edit]
Information theory also has applications in gambling and investing, black holes, bioinformatics, and music.See also[edit]
 Algorithmic probability
 Algorithmic information theory
 Bayesian inference
 Communication theory
 Constructor theory  a generalization of information theory that includes quantum information
 Inductive probability
 Minimum message length
 Minimum description length
 List of important publications
 Philosophy of information
Applications[edit]
History[edit]
Theory[edit]
 Coding theory
 Detection theory
 Estimation theory
 Fisher information
 Information algebra
 Information asymmetry
 Information field theory
 Information geometry
 Information theory and measure theory
 Kolmogorov complexity
 Logic of information
 Network coding
 Philosophy of Information
 Quantum information science
 Semiotic information theory
 Source coding
 Unsolved Problems
Concepts[edit]
 Ban (unit)
 Channel capacity
 Channel (communications)
 Communication source
 Conditional entropy
 Covert channel
 Decoder
 Differential entropy
 Encoder
 Information entropy
 Joint entropy
 Kullback–Leibler divergence
 Mutual information
 Pointwise mutual information (PMI)
 Receiver (information theory)
 Redundancy
 Rényi entropy
 Selfinformation
 Unicity distance
 Variety
References[edit]
 Jump up ^ F. Rieke, D. Warland, R Ruyter van Steveninck, W Bialek (1997). Spikes: Exploring the Neural Code. The MIT press. ISBN 9780262681087.
 Jump up ^ cf. Huelsenbeck, J. P., F. Ronquist, R. Nielsen and J. P. Bollback (2001) Bayesian inference of phylogeny and its impact on evolutionary biology, Science 294:23102314
 Jump up ^ Rando Allikmets, Wyeth W. Wasserman, Amy Hutchinson, Philip Smallwood, Jeremy Nathans, Peter K. Rogan, Thomas D. Schneider, Michael Dean (1998) Organization of the ABCR gene: analysis of promoter and splice junction sequences, Gene 215:1, 111122
 Jump up ^ Burnham, K. P. and Anderson D. R. (2002) Model Selection and Multimodel Inference: A Practical InformationTheoretic Approach, Second Edition (Springer Science, New York) ISBN 9780387953649.
 Jump up ^ Jaynes, E. T. (1957) Information Theory and Statistical Mechanics, Phys. Rev. 106:620
 Jump up ^ Charles H. Bennett, Ming Li, and Bin Ma (2003) Chain Letters and Evolutionary Histories, Scientific American 288:6, 7681
 Jump up ^ David R. Anderson (November 1, 2003). "Some background on why people in the empirical sciences may want to better understand the informationtheoretic methods" (PDF). Retrieved 20100623.
 Jump up ^ Fazlollah M. Reza (1994) [1961]. An Introduction to Information Theory. Dover Publications, Inc., New York. ISBN 0486682102.
 Jump up ^ Robert B. Ash (1990) [1965]. Information Theory. Dover Publications, Inc. ISBN 0486665216.
 Jump up ^ Jerry D. Gibson (1998). Digital Compression for Multimedia: Principles and Standards. Morgan Kaufmann. ISBN 1558603697.
 Jump up ^ The Corporation and Innovation, Haggerty, Patrick, Strategic Management Journal, Vol. 2, 97118 (1981)
 Jump up ^ Semiotics of Ideology, Noth, Winfried, Semiotica, Issue 148,(1981)
The classic work[edit]
 Shannon, C.E. (1948), "A Mathematical Theory of Communication", Bell System Technical Journal, 27, pp. 379–423 & 623–656, July & October, 1948. PDF.
Notes and other formats.  R.V.L. Hartley, "Transmission of Information", Bell System Technical Journal, July 1928
 Andrey Kolmogorov (1968), "Three approaches to the quantitative definition of information" in International Journal of Computer Mathematics.
Other journal articles[edit]
 J. L. Kelly, Jr., Saratoga.ny.us, "A New Interpretation of Information Rate" Bell System Technical Journal, Vol. 35, July 1956, pp. 917–26.
 R. Landauer, IEEE.org, "Information is Physical" Proc. Workshop on Physics and Computation PhysComp'92 (IEEE Comp. Sci.Press, Los Alamitos, 1993) pp. 1–4.
 R. Landauer, IBM.com, "Irreversibility and Heat Generation in the Computing Process" IBM J. Res. Develop. Vol. 5, No. 3, 1961
Textbooks on information theory[edit]
 Arndt, C. Information Measures, Information and its Description in Science and Engineering (Springer Series: Signals and Communication Technology), 2004, ISBN 9783540408550
 Ash, RB. Information Theory. New York: Interscience, 1965. ISBN 0470034459. New York: Dover 1990. ISBN 0486665216
 Gallager, R. Information Theory and Reliable Communication. New York: John Wiley and Sons, 1968. ISBN 0471290483
 Goldman, S. Information Theory. New York: Prentice Hall, 1953. New York: Dover 1968 ISBN 0486622096, 2005 ISBN 0486442713
 Cover, TM, Thomas, JA. Elements of information theory, 1st Edition. New York: WileyInterscience, 1991. ISBN 0471062596.
 2nd Edition. New York: WileyInterscience, 2006. ISBN 0471241954.
 Csiszar, I, Korner, J. Information Theory: Coding Theorems for Discrete Memoryless Systems Akademiai Kiado: 2nd edition, 1997. ISBN 9630574403
 MacKay, DJC. Information Theory, Inference, and Learning Algorithms Cambridge: Cambridge University Press, 2003. ISBN 0521642981
 Mansuripur, M. Introduction to Information Theory. New York: Prentice Hall, 1987. ISBN 0134846680
 Pierce, JR. "An introduction to information theory: symbols, signals and noise". Dover (2nd Edition). 1961 (reprinted by Dover 1980).
 Reza, F. An Introduction to Information Theory. New York: McGrawHill 1961. New York: Dover 1994. ISBN 0486682102
 Shannon, CE. Warren Weaver. The Mathematical Theory of Communication. Univ of Illinois Press, 1949. ISBN 0252725484
 Stone, JV. Chapter 1 of book "Information Theory: A Tutorial Introduction", University of Sheffield, England, 2014. ISBN 9780956372857.
 Yeung, RW. A First Course in Information Theory Kluwer Academic/Plenum Publishers, 2002. ISBN 0306467917.
 Yeung, RW. Information Theory and Network Coding Springer 2008, 2002. ISBN 9780387792330
Other books[edit]
 Leon Brillouin, Science and Information Theory, Mineola, N.Y.: Dover, [1956, 1962] 2004. ISBN 0486439186
 James Gleick, The Information: A History, a Theory, a Flood, New York: Pantheon, 2011. ISBN 9780375423727
 A. I. Khinchin, Mathematical Foundations of Information Theory, New York: Dover, 1957. ISBN 0486604349
 H. S. Leff and A. F. Rex, Editors, Maxwell's Demon: Entropy, Information, Computing, Princeton University Press, Princeton, New Jersey (1990). ISBN 069108727X
 Tom Siegfried, The Bit and the Pendulum, Wiley, 2000. ISBN 0471321745
 Charles Seife, Decoding The Universe, Viking, 2006. ISBN 067003441X
 Jeremy Campbell, Grammatical Man, Touchstone/Simon & Schuster, 1982, ISBN 0671440624
 Henri Theil, Economics and Information Theory, Rand McNally & Company  Chicago, 1967.
 Escolano, Suau, Bonev, Information Theory in Computer Vision and Pattern Recognition, Springer, 2009. ISBN 9781848822962
External links[edit]
Wikiquote has quotations related to: Information theory 
Library resources about Information theory 
 Erill I. (2012), "A gentle introduction to information content in transcription factor binding sites" (University of Maryland, Baltimore County)
 Hazewinkel, Michiel, ed. (2001), "Information", Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Lambert F. L. (1999), "Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms  Examples of Entropy Increase? Nonsense!", Journal of Chemical Education
 Schneider T. D. (2014), "Information Theory Primer"
 Srinivasa, S., "A Review on Multivariate Mutual Information"
 IEEE Information Theory Society and ITSoc review articles




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