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. − The goal was to find the fundamental limits of communication operations and signal processing through an operation like data compression. Shannon’s Information Theory Claude Shannon may be considered one of the most influential person of the 20th Century, as he laid out the foundation of the revolutionary information theory. where pi is the probability of occurrence of the i-th possible value of the source symbol. 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. In such a case the capacity is given by the Mutual information rate when there is no feedback availble and the Directed information rate in the case that either there is feedback or not [12] [13] (if there is no feedback the dircted informationj equals the mutual information). There are many ways of sending messages: you could produce smoke signals, use Morse code, the telephone, or (in today's world) send an email. We will consider p(y|x) to be an inherent fixed property of our communications channel (representing the nature of the noise of our channel). Network information theory refers to these multi-agent communication models. Considered the founding father of the electronic communication age, Claude Shannon's work ushered in the Digital Revolution. 1 They are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern computer equipment and software. Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. ⁡ 1 Il y explique comment construire des machines à relais en utilisant l'algèbre de Boole pour décrire l'état des relais (1 : fermé, 0 : ouvert)[réf. 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. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. , ( 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 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. − There were a few mechanical analog computers that could be used to calculate trajectories and tide tables, but nothing that could be described as a digital computer. {\displaystyle i} This task will allow us to propose, in Section 10, a formal reading of the concept of Shannon information, according to which the epistemic and the physical views are different possible models of the formalism. P Il étudie le génie électrique et les mathématiques à l'université du Michigan dont il est diplômé en 19362. x The measure of sufficient randomness in extractors is min-entropy, a value related to Shannon entropy through Rényi entropy; Rényi entropy is also used in evaluating randomness in cryptographic systems. Information rate is the average entropy per symbol. x ) Information Theory was not just a product of the work of Claude Shannon. After attending primary and secondary school in his neighboring hometown of Gaylord, he earned bachelors degrees in both electrical engineering and mathematics from the University of Michigan. i In 1973, he recalled, he persuaded Shannon to give the first annual Shannon lecture at the International Information Theory Symposium, but Shannon almost backed out at the last minute. In 1948, he published ‘The Mathematical Theory of Communication’, which is considered the most noted information theory. Claude Shannon, American mathematician and electrical engineer who laid the theoretical foundations for digital circuits and information theory, a mathematical communication model. A 1 Shannon died on Saturday, February 24, 2001 in Medford, Mass., after a long fight with Alzheimer's disease. MP3s and JPEGs), and channel coding (e.g. However, these theorems only hold in the situation where one transmitting user wishes to communicate to one receiving user. 0 = Shannon’s most important paper, ‘A mathematical theory of communication,’ was published in 1948. 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. {\displaystyle p(X)} A third class of information theory codes are cryptographic algorithms (both codes and ciphers). − Any process that generates successive messages can be considered a source of information. Other units include the nat, which is based on the natural logarithm, and the decimal digit, which is based on the common logarithm. All such sources are stochastic. Despite similar notation, joint entropy should not be confused with cross entropy. It is important in communication where it can be used to maximize the amount of information shared between sent and received signals. i , while Bob believes (has a prior) that the distribution is 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. Information theory often concerns itself with measures of information of the distributions associated with random variables. 0 The appropriate measure for this is the mutual information, and this maximum mutual information is called the channel capacity and is given by: This capacity has the following property related to communicating at information rate R (where R is usually bits per symbol). ) Namely, at time For stationary sources, these two expressions give the same result.[11]. Information theoretic security refers to methods such as the one-time pad that are not vulnerable to such brute force attacks. Based on the probability mass function of each source symbol to be communicated, the Shannon entropy H, in units of bits (per symbol), is given by. Claude Shannon, the father of Information Theory You may not have heard of Claude Shannon, but his ideas made the modern information age possible. i 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. Dover (2nd Edition). Claude Elwood Shannon was born on April 30, 1916 in Petoskey, Michigan. . This implies that if X and Y are independent, then their joint entropy is the sum of their individual entropies. His ideas ripple through nearly every aspect of modern life, influencing such diverse fields as communication, computing, cryptography, neuroscience, artificial intelligence, cosmology, linguistics, and genetics. In the late 1940s Claude Shannon, a research mathematician at Bell Telephone Laboratories, invented a mathematical theory of communication that gave the first systematic framework in which to optimally design telephone systems. In addition, for any rate R > C, it is impossible to transmit with arbitrarily small block error. Information theory and digital signal processing offer a major improvement of resolution and image clarity over previous analog methods. A Mathematical Theory of Communication By C. E. SHANNON INTRODUCTION T HE recent development of various methods of modulation such as PCM and PPM which exchange bandwidth for signal-to-noise ratio has intensified the interest in a general theory of communication. A continuous-time analog communications channel subject to. , Claude Elwood Shannon was born on … 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. x Ce domaine trouve son origine scientifique avec Claude Shannon qui en est le … i These terms are well studied in their own right outside information theory. . Information theory studies the transmission, processing, extraction, and utilization of information. His theories laid the groundwork for the electronic communications networks that now lace the earth. , − The former quantity is a property of the probability distribution of a random variable and gives a limit on the rate at which data generated by independent samples with the given distribution can be reliably compressed. , 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. for any logarithmic base. These can be obtained via extractors, if done carefully. . It is common in information theory to speak of the "rate" or "entropy" of a language. 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]. His Collected Papers, published in 1993, contains 127 publications on topics ranging from communications to computing, and juggling to “mind-reading” machines. His later work on chess-playing machines and an electronic mouse that could run a maze helped create the field of artificial intelligence, the effort to make machines that think. The theory has also found applications in other areas, including statistical inference,[1] cryptography, neurobiology,[2] perception,[3] linguistics, the evolution[4] and function[5] of molecular codes (bioinformatics), thermal physics,[6] quantum computing, black holes, information retrieval, intelligence gathering, plagiarism detection,[7] pattern recognition, anomaly detection[8] and even art creation. This is appropriate, for example, when the source of information is English prose. The American mathematician and computer scientist who conceived and laid the foundations for information theory. 1 See the article ban (unit) for a historical application. ( 2 Abstractly, information can be thought of as the resolution of uncertainty. . For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a die (with six equally likely outcomes). 1 Claude Shannon, known as the ‘father of information theory’, was a celebrated American cryptographer, mathematician and electrical engineer. Shannon defined the quantity of information produced by a source--for example, the quantity in a message--by a formula similar to the equation that … Claude Shannon • “The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point.” (Claude Shannon 1948) • Channel Coding Theorem: It is possible to achieve near perfect communication of information over a noisy channel 1916 - 2001 • In this course we will: 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. Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. In 1941, Shannon took a position at Bell Labs, where he had spent several prior summers. , ( In this way, the extent to which Bob's prior is "wrong" can be quantified in terms of how "unnecessarily surprised" it is expected to make him. . X ( y 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. Claude Shannon: Born on the planet Earth (Sol III) in the year 1916 A.D. Generally regarded as the father of the information age, he formulated the notion of channel capacity in 1948 A.D. , No scientist has an impact-to-fame ratio greater than Claude Elwood Shannon, the creator of information theory. p The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. i Because of this, he is widely considered "the father of information theory". 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. His theories laid the groundwork for the electronic communications networks that now lace the earth. Il utilise notamment l'algèbre de Boole pour sa maîtrise soutenue en 1938 au Massachusetts Institute of Technology (MIT). | He was also the first recipient of the Harvey Prize (1972), the Kyoto Prize (1985), and the Shannon Award (1973). ( i In what follows, an expression of the form p log p is considered by convention to be equal to zero whenever p = 0. ) ) Slides of the corresponding talk are also available. It was originally proposed by Claude Shannon in 1948 to find fundamental limits on signal processing and communication operations such as data compression, in a landmark paper titled "A Mathematical Theory of Communication". , x Of course, Babbagehad described the basic design of a stored program computer in the 180… 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. , This is justified because 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. 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. One early commercial application of information theory was in the field of seismic oil exploration. q = 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 and the development of the Internet. If, however, each bit is independently equally likely to be 0 or 1, 1000 shannons of information (more often called bits) have been transmitted. Synopsis. 2 x The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy channel in the limit of long block lengths, when the channel statistics are determined by the joint distribution. Based on the redundancy of the plaintext, it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability. ) Shannon approached research with a sense of curiosity, humor, and fun. , then the entropy, H, of X is defined:[9]. Harry Nyquist, "Certain Topics in Telegraph Transmission Theory", Transactions of AIEE, Vol. ) A common unit of information is the bit, based on the binary logarithm. Claude Shannon, born 100 years ago, devised the mathematical representation of information that made the digital era possible. 1 {\displaystyle q(X)} 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 Il obtient un PhD en mathématiques au MIT en 19402. . ( 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… Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. An accomplished unicyclist, he was famous for cycling the halls of Bell Labs at night, juggling as he went. Communications over a channel—such as an ethernet cable—is the primary motivation of information theory. {\displaystyle q(X)} 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 y Info. 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. When his results were finally de-classified and published in 1949, they revolutionized the field of cryptography. X An updated version entitled "A brief introduction to Shannon's information theory" is available on arXiv (2018). i The Kullback–Leibler divergence (or information divergence, information gain, or relative entropy) is a way of comparing two distributions: a "true" probability distribution 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. "[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]. 1961 (reprinted by Dover 1980). {\displaystyle p(x)} Understanding, before almost anyone, the power that springs from encoding information in a simple language of 1's and 0's, Dr. Shannon as a young scientist at Bell Laboratories wrote two papers that remain monuments in the fields of computer science and information theory. ( Work in this field made it possible to strip off and separate the unwanted noise from the desired seismic signal. This page was last edited on 24 January 2021, at 13:22. 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. Claude Elwood Shannon was an American mathematician, cryptographer, and electrical engineer, who garnered fame when he conceptualised information theory with the landmark paper, ‘Mathematical Theory of Communication’, which he put out in 1948. y ( Let p(y|x) be the conditional probability distribution function of Y given X. x . Shannon, who died in 2001 at … Dr. Marvin Minsky of M.I.T., who as a young theorist worked closely with Dr. Shannon, was struck by his enthusiasm and enterprise. This innovation, credited as the advance that transformed circuit design “from an art to a science,” remains the basis for circuit and chip design to this day. A basic property of this form of conditional entropy is that: Mutual information measures the amount of information that can be obtained about one random variable by observing another. x A property of entropy is that it is maximized when all the messages in the message space are equiprobable p(x) = 1/n; i.e., most unpredictable, in which case H(X) = log n. 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: The joint entropy of two discrete random variables X and Y is merely the entropy of their pairing: (X, Y). x The conditional entropy or conditional uncertainty of X given random variable Y (also called the equivocation of X about Y) is the average conditional entropy over Y:[10]. 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. . It is a theory that has been extrapolated into thermal physics, quantum computing, linguistics, and even plagiarism detection. Shannon said that all information has a "source rate" that can be measured in bits per second and requires a transmission channel with a capacity equal to or greater than the source rate. 2 Shannon wanted to measure the amount of information you could transmit via various media. A basic property of the mutual information is that. x The unit of information was therefore the decimal digit, which has since sometimes been called the hartley in his honor as a unit or scale or measure of information. Other bases are also possible, but less commonly used. Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of coding theory. . is the distribution underlying some data, when, in reality, Information theory is based on probability theory and statistics. For more information about Shannon and his impact, see the article by Michelle Effros and H. Vincent Poor, Claude Shannon: His Work and Its Legacy, Published with the permission of the EMS Newsletter: reprinted from N°103 (March 2017) pp.29-34. The last of these awards, named in his honor, is given by the Information Theory Society of the Institute of Electrical and Electronics Engineers (IEEE) and remains the highest possible honor in the community of researchers dedicated to the field that he invented. 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. X x , then Bob will be more surprised than Alice, on average, upon seeing the value of X. . X , and an arbitrary probability distribution X p {\displaystyle p(x)} "That was really his discovery, and from it the whole communications revolution has sprung.". This is an introduction to Shannon's Information Theory. ∈ Privacy & Opting Out of Cookies. Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms - Examples of Entropy Increase? y If is the set of all messages {x1, ..., xn} that X could be, and p(x) is the probability of some After graduation, Shannon moved to the Massachusetts Institute of Technology (MIT) to pursue his graduate studies. That is, knowing Y, we can save an average of I(X; Y) bits in encoding X compared to not knowing Y. He created the field of Information Theory when he published a book "The Mathematical Theory… Use of this website signifies your agreement to the IEEE Terms and Conditions. "Whatever came up, he engaged it with joy, and he attacked it with some surprising resource — which might be some new kind of technical concept or a hammer and saw with some scraps of wood," Dr. Minsky said. Though analog computers like this turned out to be little more than footnotes in the history of the computer, Dr. Shannon quickly made his mark with digital electronics, a considerably more influential idea. His war-time work on secret communication systems was used to build the system over which Roosevelt and Churchill communicated during the war. 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. . In a blockbuster paper in 1948, Claude Shannon introduced the notion of a "bit" and laid the foundation for the information age. is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. The American mathematician and computer scientist who conceived and laid the foundations for information theory. y Claude Shannon's information theory and Language Models Published on January 15, 2021 January 15, 2021 • 14 Likes • 1 Comments 1 He gained his PhD from MIT in the subject, but he made substantial contributions to the theory and practice of computing. Important quantities of information are entropy, a measure of information in a single random variable, and mutual information, a measure of information in common between two random variables. x Although it is sometimes used as a 'distance metric', KL divergence is not a true metric since it is not symmetric and does not satisfy the triangle inequality (making it a semi-quasimetric). , Nondiscrimination Policy | souhaitée]. Information theory studies the quantification, storage, and communication of information. The mutual information of X relative to Y is given by: where SI (Specific mutual Information) is the pointwise mutual information. i While Shannon worked in a field for which no Nobel prize is offered, his work was richly rewarded by honors including the National Medal of Science (1966) and honorary degrees from Yale (1954), Michigan (1961), Princeton (1962), Edin- burgh (1964), Pittsburgh (1964), Northwestern (1970), Oxford (1978), East Anglia (1982), Carnegie-Mellon (1984), Tufts (1987), and the University of Pennsylvania (1991). , Information Theory Information Theory before Shannon To understand the contributions, motivations and methodology of Claude Shannon, it is important to examine the state of communication engineering before the advent of Shannon™s 1948 paper, fiA Mathematical Theory of Communicationfl. However, as in any other cryptographic system, care must be used to correctly apply even information-theoretically secure methods; the Venona project was able to crack the one-time pads of the Soviet Union due to their improper reuse of key material. the mutual information, and the channel capacity of a noisy channel, including the promise of perfect loss-free communication given by the noisy-channel coding theorem; Data compression (source coding): There are two formulations for the compression problem: Error-correcting 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. Shannon received both a master 's degree in electrical engineering and his Ph.D. in mathematics from M.I.T transmitting! Theory to speak of the breaking of the work of claude Shannon first proposed the theory... 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Shannon set … information theory of Bell Labs, all implicitly assuming events equal! The maze traversing mouse, named ‘ Theseus ’ class of information theory, Las Vegas Wall. Bell Labs, all implicitly assuming events of equal probability that if X and are... Phd from MIT in the field is at the intersection of probability,!

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