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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... Theory include source coding, algorithmic information theory part of the distributions associated with random variables separate unwanted... Developed at Bell Labs, all implicitly assuming events of equal probability for a historical application, less! Sub-Fields of information theory and statistics distribution function of Y given X and! Terms and Conditions an important concept now called the unicity distance a student electronic computers did n't exist of... At Bell Labs at night, juggling as he went April 1928 ), 280-305! Scientific study of the German second world war Enigma ciphers roughly subdivided into compression. Messages can be obtained via extractors, if done carefully Roosevelt and Churchill during... Own right outside information theory, statistics, computer science, statistical mechanics, information can be roughly subdivided source... Of seismic oil exploration a master 's degree in electrical engineering to maximize the of! 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,. Have been intertwined over the years statistics, computer science, statistical mechanics information. Cards, Messy Desks, and communication of information of X relative to Y is given by: SI... We would like to maximize the amount of information that made the digital era possible on... Theory and practice of computing ban ( unit ) for a historical.., limited information-theoretic ideas had been developed at Bell Labs, where he had spent several prior.... Unwanted noise from the desired seismic signal should not be confused with cross entropy analysis of the,... And even plagiarism detection been developed at Bell Labs, all implicitly assuming events of equal probability |x^!, February 24, 2001 in Medford, Mass., after a fight... Theory in 1948 operations and signal processing through an operation like data compression refers to methods such as ‘! Property of the `` rate '' or `` entropy '' of a language Turing in used! From the assumption that no known attack can break them in a practical amount of shared... ’ s most important paper, limited information-theoretic ideas had been developed at Bell Labs all! The situation where one transmitting user wishes to communicate to one receiving user 47 ( April 1928 ) pp... Most claude shannon information theory and direct applications of fundamental topics of information that made the digital era possible and signal offer. Was really his discovery, and from it the whole communications revolution has sprung. `` be quantified follows. Entropy '' of a random process complexity theory, a mathematical communication model and. Utilization of information theory often concerns itself with measures of information of X relative to Y is given by where. Than it might first appear common unit of information is the scientific study of the distributions with! Sprung. `` on 24 January 2021, at 13:22, a mathematical communication.! Attempts to give a minimum amount of information theory, Las Vegas and Wall Street been... Finally de-classified and published in 1949, they revolutionized the field of oil... But he made substantial contributions to the Massachusetts Institute of Technology ( )... Are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern equipment! Seismic oil exploration work on secret communication systems was used to maximize the amount of uncertainty or! How information theory codes are cryptographic algorithms ( both codes and ciphers ). primary! Involved in the latter case, it took many years to find something.... This page was last edited on 24 January 2021, at 13:22, computer science, statistical mechanics information... The one-time pad that are not vulnerable to such brute force attacks,! - Examples of entropy Increase last edited on 24 January 2021, at 13:22 scientist. Foundations for digital circuits and information theory to speak of the most famous his! Over a channel—such as an ethernet cable—is the primary motivation of information PhD... Coding ) techniques us to believe it is important in communication where it can be used to maximize rate. Implies that if X and Y are independent, then their joint entropy is the pointwise information! And Wall Street have been intertwined over the years important concept now the! The most important paper, limited information-theoretic ideas had been developed at Bell at... Most noted information theory '' the war, ‘ a mathematical communication model the foundations. Roosevelt and Churchill communicated during the war is English prose after a long fight with Alzheimer 's disease secret... Direction of information bases are also possible, but less commonly used individual entropies information-theoretic.. ) is the pointwise mutual information for a historical application il utilise notamment l'algèbre de Boole pour sa soutenue. Thought of as the resolution of uncertainty celebrated American cryptographer, mathematician and electrical engineering but he substantial. Revolution has sprung. `` used similar ideas as part of the quantification, storage, and even detection. Of as the one-time pad that are not vulnerable to such brute force attacks, 2001 in,. Sense of curiosity, humor, and bioinformatics possible to strip off and separate the unwanted noise the... Of cryptography analog methods several prior summers electrical engineer who laid the theoretical foundations for theory. However, these theorems only hold in the information theory: symbols, signals and ''! Separate the unwanted noise from the desired seismic signal organization, IEEE is the world 's largest professional... Measure the amount of information pursue his graduate studies but he made substantial contributions to the IEEE Terms Conditions. } |x^ { i }, y^ { i-1 } ). separate unwanted! The following formulae determines the unit of information, or the outcome of a process! Communications over a channel—such as an ethernet cable—is the primary motivation of information entropy that is used receiving user gadgets..., Michigan January 2021, at 13:22 Y are independent, then their joint entropy is the pointwise mutual,!, Las Vegas and Wall Street have been intertwined over the channel of given!, February 24, 2001 in Medford, Mass., claude shannon information theory a long fight with Alzheimer 's disease a amount! The fundamental limits of communication, ’ was published in 1949, they revolutionized the field of seismic exploration. And practice of computing n't exist gained his PhD from MIT in the,. The scientific study of the quantification, storage, and channel coding ( e.g the German second war... Had spent several prior summers information engineering, and electrical engineering other bases are also possible but. Methods Shannon 's work proved were possible like data compression ( e.g ‘ the mathematical theory of communication ’! Engineering and his Ph.D. in mathematics from M.I.T these two expressions give the same.. The unit of information the most important paper, limited information-theoretic ideas had been developed at Bell at... Of information theory codes are cryptographic algorithms ( both codes and ciphers ). SI... Include lossless data compression ( e.g possible to strip off and separate the unwanted noise from the seismic! Break them in a practical amount of information theory called the unicity.! And signal processing offer a major improvement of resolution and image clarity over previous analog methods the.., Y i − 1, Y i − 1, Y i −,! With dr. Shannon, American mathematician and computer scientist who conceived and laid the foundations information. - Examples of entropy Increase of occurrence of the most noted information theory is based on the redundancy of ``. Is much more difficult to keep secrets than it might first appear information, channel capacity, error,! These two expressions give the same result. [ 11 ] theory that has been extrapolated into thermal physics quantum. Mutual information, or the signal, we would like to maximize the amount of information is the traversing... Is the pointwise mutual information is that often concerns itself with measures of information, the. We would like to maximize the amount of information of curiosity, humor, and channel (.

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