1 0 X ZIP files), lossy data compression (e.g. Nonsense! After graduating from the University of Michigan in 1936 with bachelor’s degrees in mathematics and electrical i Information theory and digital signal processing offer a major improvement of resolution and image clarity over previous analog methods. He was 84. P A third class of information theory codes are cryptographic algorithms (both codes and ciphers). Information theory is the scientific study of the quantification, storage, and communication of information. https://en.wikipedia.org/w/index.php?title=Information_theory&oldid=1002438403, Short description is different from Wikidata, Articles with too many examples from May 2020, Wikipedia articles with style issues from May 2020, Creative Commons Attribution-ShareAlike License. . 47 (April 1928), pp 617-644; repr. These groundbreaking innovations provided the tools that ushered in the information age. ) Mutual information can be expressed as the average Kullback–Leibler divergence (information gain) between the posterior probability distribution of X given the value of Y and the prior distribution on X: In other words, this is a measure of how much, on the average, the probability distribution on X will change if we are given the value of Y. In a prize-winning masters thesis completed in the Department of Mathematics, Shannon proposed a method for applying a mathematical form of logic called Boolean algebra to the design of relay switching circuits. MP3s and JPEGs), and channel coding (e.g. Shannon's main result, the noisy-channel coding theorem showed that, in the limit of many channel uses, the rate of information that is asymptotically achievable is equal to the channel capacity, a quantity dependent merely on the statistics of the channel over which the messages are sent.[2]. These can be obtained via extractors, if done carefully. To treat them all on equal terms, Shannon decided to forget about exactly how each of these methods transmits a message and simply thought of them as ways of producing strings of symbo… X Coding theory is concerned with finding explicit methods, called codes, for increasing the efficiency and reducing the error rate of data communication over noisy channels to near the channel capacity. where pi is the probability of occurrence of the i-th possible value of the source symbol. The Kullback–Leibler divergence (or information divergence, information gain, or relative entropy) is a way of comparing two distributions: a "true" probability distribution ( For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε; that is, it is always possible to transmit with arbitrarily small block error. 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. ) , Information theory studies the quantification, storage, and communication of information. "Shannon was the person who saw that the binary digit was the fundamental element in all of communication," said Dr. Robert G. Gallager, a professor of electrical engineering who worked with Dr. Shannon at the Massachusetts Institute of Technology. Based on the redundancy of the plaintext, it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability. ) Abstractly, information can be thought of as the resolution of uncertainty. For example, if (X, Y) represents the position of a chess piece—X the row and Y 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. Claude Shannon, the father of Information Theory You may not have heard of Claude Shannon, but his ideas made the modern information age possible. (Here, I(x) is the self-information, which is the entropy contribution of an individual message, and X is the expected value.) See the article ban (unit) for a historical application. The American mathematician and computer scientist who conceived and laid the foundations for information theory. Read important changes and updates to the 2020 activities of the IEEE Information Theory Society due to the COVID-19 pandemic at: Home | Il étudie le génie électrique et les mathématiques à l'université du Michigan dont il est diplômé en 19362. Il obtient un PhD en mathématiques au MIT en 19402. ( After graduation, Shannon moved to the Massachusetts Institute of Technology (MIT) to pursue his graduate studies. is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. , − Work in this field made it possible to strip off and separate the unwanted noise from the desired seismic signal. = Information Theory was not just a product of the work of Claude Shannon. Another interpretation of the KL divergence is the "unnecessary surprise" introduced by a prior from the truth: suppose a number X is about to be drawn randomly from a discrete set with probability distribution La théorie de l'information, sans précision, est le nom usuel désignant la théorie de l'information de Shannon, qui est une théorie probabiliste permettant de quantifier le contenu moyen en information d'un ensemble de messages, dont le codage informatique satisfait une distribution statistique précise. . Between these two extremes, information can be quantified as follows. . A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. → All such sources are stochastic. Claude Shannon is quite correctly described as a mathematician. Claude Shannon: Claude Elwood Shannon, a mathematician born in Gaylord, Michigan (U.S.) in 1916, is credited with two important contributions to information technology: the application of Boolean theory to electronic switching, thus laying the groundwork for the digital computer, and developing the new field called information theory . i Next, Shannon set … X 1 ⁡ 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. q Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored in Entropy in thermodynamics and information theory. 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. Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. p 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. An updated version entitled "A brief introduction to Shannon's information theory" is available on arXiv (2018). IEEE – All rights reserved. For example, a logarithm of base 28 = 256 will produce a measurement in bytes per symbol, and a logarithm of base 10 will produce a measurement in decimal digits (or hartleys) per symbol. Claude Shannon, American mathematician and electrical engineer who laid the theoretical foundations for digital circuits and information theory, a mathematical communication model. While at M.I.T., he worked with Dr. Vannevar Bush on one of the early calculating machines, the "differential analyzer," which used a precisely honed system of shafts, gears, wheels and disks to solve equations in calculus. The entropy of a source that emits a sequence of N symbols that are independent and identically distributed (iid) is N ⋅ H bits (per message of N symbols). p A continuous-time analog communications channel subject to. Indeed the diversity and directions of their perspectives and interests shaped the direction of Information Theory. In 1941, Shannon took a position at Bell Labs, where he had spent several prior summers. . If Alice knows the true distribution It's interesting how Information Theory, Las Vegas and Wall Street have been intertwined over the years. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers. ) i In addition, for any rate R > C, it is impossible to transmit with arbitrarily small block error. An accomplished unicyclist, he was famous for cycling the halls of Bell Labs at night, juggling as he went. The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. However, these theorems only hold in the situation where one transmitting user wishes to communicate to one receiving user. 1 Claude Elwood Shannon was born on April 30, 1916 in Petoskey, Michigan. , "[15]:91, Concepts from information theory such as redundancy and code control have been used by semioticians such as Umberto Eco and Ferruccio Rossi-Landi 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.[17]. Shannon’s most important paper, ‘A mathematical theory of communication,’ was published in 1948. i These codes can be roughly subdivided into data compression (source coding) and error-correction (channel coding) techniques. In what follows, an expression of the form p log p is considered by convention to be equal to zero whenever p = 0. This fundamental treatise both defined a mathematical notion by which information could be quantified and demonstrated that information could be delivered reliably over imperfect communication channels like phone lines or wireless connections. x {\displaystyle q(x)} p 1 1 x Claude Shannon wrote a master’s thesis that jump-started digital circuit design, and a decade later he wrote his seminal paper on information theory, “A Mathematical Theory of Communication.”. Harry Nyquist, "Certain Topics in Telegraph Transmission Theory", Transactions of AIEE, Vol. This is justified because i x x Because of this, he is widely considered "the father of information theory". Privacy & Opting Out of Cookies. p Slides of the corresponding talk are also available. ) 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. ) This is often recalculated as the divergence from the product of the marginal distributions to the actual joint distribution: Mutual information is closely related to the log-likelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson's χ2 test: mutual information can be considered a statistic for assessing independence between a pair of variables, and has a well-specified asymptotic distribution. − For the more general case of a process that is not necessarily stationary, the average rate is, that is, the limit of the joint entropy per symbol. {\displaystyle q(X)} p Courtesy of MIT Museum. i + Because entropy can be conditioned on a random variable or on that random variable being a certain value, care should be taken not to confuse these two definitions of conditional entropy, the former of which is in more common use. ) Dover (2nd Edition). . In a blockbuster paper in 1948, Claude Shannon introduced the notion of a "bit" and laid the foundation for the information age. p If, however, each bit is independently equally likely to be 0 or 1, 1000 shannons of information (more often called bits) have been transmitted. 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. Be the conditional probability distribution function of Y given X Churchill communicated during the war in addition, for rate..., Michigan struck by his enthusiasm and enterprise successive messages can be subdivided into data compression ( source theory. 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