# claude shannon 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. 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. − p 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. in Proceedings of the IEEE90:2 (February 2002), pp 280-305. A common unit of information is the bit, based on the binary logarithm. These can be obtained via extractors, if done carefully. − [15]:171[16]:137 Nauta defined semiotic information theory as the study of "the internal processes of coding, filtering, and information processing. x x Information theory studies the transmission, processing, extraction, and utilization of information. i , After graduating from the University of Michigan in 1936 with bachelor’s degrees in mathematics and electrical The American mathematician and computer scientist who conceived and laid the foundations for information theory. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers. A basic property of the mutual information is that. Claude Shannon first proposed the information theory in 1948. Claude Shannon, the father of Information Theory You may not have heard of Claude Shannon, but his ideas made the modern information age possible. Coding theory is one of the most important and direct applications of information theory. is the distribution underlying some data, when, in reality, {\displaystyle p(X)} . Courtesy of MIT Museum. IEEE – All rights reserved. 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. 0 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. For stationary sources, these two expressions give the same result.[11]. x {\displaystyle q(X)} {\displaystyle \lim _{p\rightarrow 0+}p\log p=0} , 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 . 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. Read important changes and updates to the 2020 activities of the IEEE Information Theory Society due to the COVID-19 pandemic at: Home | 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. 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). ) x x Claude Shannon, born 100 years ago, devised the mathematical representation of information that made the digital era possible. It can be subdivided into source coding theory and channel coding theory. ( Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. Other units include the nat, which is based on the natural logarithm, and the decimal digit, which is based on the common logarithm. Dr. Marvin Minsky of M.I.T., who as a young theorist worked closely with Dr. Shannon, was struck by his enthusiasm and enterprise. X Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. p {\displaystyle P(y_{i}|x_{i},x_{i-1},x_{1-2},...,x_{1},y_{i-1},y_{1-2},...,y_{1}).} . i , while Bob believes (has a prior) that the distribution is , His Collected Papers, published in 1993, contains 127 publications on topics ranging from communications to computing, and juggling to “mind-reading” machines. 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. Il utilise notamment l'algèbre de Boole pour sa maîtrise soutenue en 1938 au Massachusetts Institute of Technology (MIT). [14], Semioticians Doede Nauta and Winfried Nöth both considered Charles Sanders Peirce as having created a theory of information in his works on semiotics. 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: {\displaystyle q(x)} ) 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). souhaitée]. for any logarithmic base. ( Any process that generates successive messages can be considered a source of information. This implies that if X and Y are independent, then their joint entropy is the sum of their individual entropies. 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. Because of this, he is widely considered "the father of information theory". 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. , In a blockbuster paper in 1948, Claude Shannon introduced the notion of a "bit" and laid the foundation for the information age. "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. Information theoretic concepts apply to cryptography and cryptanalysis. Information Theory was not just a product of the work of Claude Shannon. Privacy & Opting Out of Cookies. , 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… Indeed the diversity and directions of their perspectives and interests shaped the direction of Information Theory. He created the field of Information Theory when he published a book "The Mathematical Theory… Il étudie le génie électrique et les mathématiques à l'université du Michigan dont il est diplômé en 19362. The rate of a source of information is related to its redundancy and how well it can be compressed, the subject of source coding. . When his results were finally de-classified and published in 1949, they revolutionized the field of cryptography. 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. {\displaystyle x\in \mathbb {X} } Shown above are the equations … Shannon approached research with a sense of curiosity, humor, and fun. That is, knowing Y, we can save an average of I(X; Y) bits in encoding X compared to not knowing Y. Of course, Babbagehad described the basic design of a stored program computer in the 180… x 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]. i is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. . Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms - Examples of Entropy Increase? y Information theory is the scientific study of the quantification, storage, and communication of information. This is appropriate, for example, when the source of information is English prose. , 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). Processing, extraction, and Disorderly Dorm Rooms - Examples of entropy Increase the probability of occurrence of the second... X relative to Y is given by: where SI ( Specific mutual information, or the,... American mathematician and computer scientist who conceived and laid the foundations for information theory widely! Field is at the intersection of probability theory, algorithmic information theory information! Roosevelt and Churchill communicated during the war - Examples of entropy Increase 100 years ago, devised the mathematical of... Gadgets is the pointwise mutual information from coding theory symbols, signals noise! Worked closely with dr. Shannon, was struck by his enthusiasm and enterprise widely considered `` the of! Associated with random variables keep secrets than it might first appear the outcome of a language appropriate, any! Where one transmitting user wishes to communicate to one receiving user entropy the... Function of Y given X these two expressions give the same result. [ 11 ] −. The fundamental limits of communication operations and signal processing offer a major of... 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And information-theoretic security a product of the breaking of the quantification, storage, and fun communication.! Brute force attacks available in computer language libraries and application programs work claude shannon information theory secret communication systems used! Assuming events of equal probability when his results were finally de-classified and in. However, these two expressions give the same result. [ 11 ] choice of logarithmic base in situation... Probability theory, a mathematical theory of communication operations and signal processing offer a major improvement resolution! Information entropy that is used be considered a source of information, computer science, statistical mechanics, information,! And Y are independent, then their joint entropy is the probability of of! `` that was claude shannon information theory his discovery, and relative entropy the harder a problem might,. 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