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Information Geometry
Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to probability distributions. Introduction Historically, information geometry can be traced back to the work of C. R. Rao, who was the first to treat the Fisher matrix as a Riemannian metric. The modern theory is largely due to Shun'ichi Amari, whose work has been greatly influential on the development of the field. Classically, information geometry considered a parametrized statistical model as a Riemannian manifold, Riemannian, conjugate connection, statistical, and dually flat manifolds. Unlike usual smooth manifolds with tensor metric and Levi-Civita connection, these take into account conjugate connection, torsion, and Amari-Chentsov metric. All presented above geometric structures find application in information theory and machine lea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution PDF
Normal(s) or The Normal(s) may refer to: Film and television * Normal (2003 film), ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * Normal (2007 film), ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * Normal (2009 film), ''Normal'' (2009 film), an adaptation of Anthony Neilson's 1991 play ''Normal: The Düsseldorf Ripper'' * ''Normal!'', a 2011 Algerian film * The Normals (film), ''The Normals'' (film), a 2012 American comedy film * Normal (New Girl), "Normal" (''New Girl''), an episode of the TV series Mathematics * Normal (geometry), an object such as a line or vector that is perpendicular to a given object * Normal basis (of a Galois extension), used heavily in cryptography * Normal bundle * Normal cone, of a subscheme in algebraic geometry * Normal coordinates, in differential geometry, local coordinates obtained from the exponential map (Riemannian geometry) * Normal distribution, the Gaussian continuo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonparametric Statistics
Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as in parametric statistics. Nonparametric statistics can be used for descriptive statistics or statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated. Definitions The term "nonparametric statistics" has been defined imprecisely in the following two ways, among others: The first meaning of ''nonparametric'' involves techniques that do not rely on data belonging to any particular parametric family of probability distributions. These include, among others: * Methods which are ''distribution-free'', which do not rely on assumptions that the data are drawn from a given parametric family of probability distributions. * Statistics defined to be a function on a sample, without dependency on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bradley Efron
Bradley Efron (; born May 24, 1938) is an American statistician. Efron has been president of the American Statistical Association (2004) and of the Institute of Mathematical Statistics (1987–1988).Cochran, J. (1 September 2015), "ASA Leaders Reminisce: Brad Efron", ''Amstat News''. He is a past editor (for theory and methods) of the ''Journal of the American Statistical Association'', and he is the founding editor of the '' Annals of Applied Statistics''. Efron is also the recipient of many awards (see below). Efron is especially known for proposing the bootstrap resampling technique, which has had a major impact in the field of statistics and virtually every area of statistical application. The bootstrap was one of the first computer-intensive statistical techniques, replacing traditional algebraic derivations with data-based computer simulations. Life and career Efron was born in St. Paul, Minnesota in May 1938, the son of Russian Jewish immigrants Esther and Miles Ef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolai Chentsov
Nikolai Nikolaevich Chentsov (19 February 1930 – 5 July 1992), written also as Nikolaj Nikolajevič Čencov, or Nikolai Chentsov, N. N. Čencov for short, was a Soviet mathematician who made important contributions to stochastic processes, convergence theory and information geometry. Education and career Chentsov was born in Moscow and showed an early interest in mathematics. In the eighth grade (1944), he joined a school mathematics club for high school students who had just returned from evacuation at the Faculty of Mechanics and Mathematics of the Moscow State University, which was led by Alexander Kronrod and Olga Ladyzhenskaya, who was at that time a graduate student. Chentsov continued his studies in the circle under the guidance of a graduate student Eugene Dynkin, who later became his thesis advisor. In 1947, Chentsov entered the Faculty of Mechanics and Mathematics of Moscow State University and was very actively involved in leading a mathematical club for schoolchild ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imre Csiszár
Imre Csiszár () is a Hungarian mathematician with contributions to information theory and probability theory. In 1996 he won the Claude E. Shannon Award, the highest annual award given in the field of information theory. He was born on 7 February 1938 in Miskolc, Hungary. He became interested in mathematics in middle school. He was inspired by his father who was a forest engineer and was among the first to use mathematical techniques in his area. He studied mathematics at the Eötvös Loránd University, Budapest, and received his Diploma in 1961. He got his PhD in 1967 and the scientific degree Doctor of Mathematical Science in 1977. Later, he was influenced by Alfréd Rényi, who was very active in the area of probability theory. In 1990 he was elected Corresponding Member of the Hungarian Academy of Sciences, and in 1995 he became Full Member. Professor Csiszar has been with the Mathematical Institute of the Hungarian Academy of Sciences since 1961. He has been Head of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claude Shannon
Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, computer scientist, cryptographer and inventor known as the "father of information theory" and the man who laid the foundations of the Information Age. Shannon was the first to describe the use of Boolean algebra—essential to all digital electronic circuits—and helped found artificial intelligence (AI). Roboticist Rodney Brooks declared Shannon the 20th century engineer who contributed the most to 21st century technologies, and mathematician Solomon W. Golomb described his intellectual achievement as "one of the greatest of the twentieth century". At the University of Michigan, Shannon dual degreed, graduating with a Bachelor of Science in electrical engineering and another in mathematics, both in 1936. A 21-year-old master's degree student in electrical engineering at MIT, his thesis, "A Symbolic Analysis of Relay and Switching Circuits", demonstrated that electric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Leibler
Richard A. Leibler (March 18, 1914, Chicago, Illinois – October 25, 2003, Reston, Virginia) was an American mathematician and cryptanalyst. Richard Leibler was born in March 1914. He received his A.M. in mathematics from Northwestern University and his Ph.D. from the University of Illinois in 1939. While working at the National Security Agency, he and Solomon Kullback formulated the Kullback–Leibler divergence, a measure of similarity between probability distributions which has found important applications in information theory and cryptology. Leibler is also credited by the NSA as having opened up "new methods of attack" in the celebrated VENONA code-breaking project during 1949-1950; this may be a reference to his joint paper with Kullback, which was published in the open literature in 1951 and was immediately noted by Soviet cryptologists. He was director of the Communications Research Division at the Institute for Defense Analyses from 1962 to 1977, during which he was t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Louis Koszul
Jean-Louis Koszul (; 3 January 1921 – 12 January 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex. He was a second generation member of Bourbaki. Biography Koszul was educated at the in Strasbourg before studying at the Faculty of Science University of Strasbourg and the Faculty of Science of the University of Paris. His Ph.D. thesis, titled ''Homologie et cohomologie des algèbres de Lie'', was written in 1950 under the direction of Henri Cartan. He lectured at many universities and was appointed in 1963 professor in the Faculty of Science at the University of Grenoble. He was a member of the French Academy of Sciences. Koszul was the cousin of the French composer Henri Dutilleux, and the grandchild of the composer Julien Koszul. Koszul married Denise Reyss-Brion on 17 July 1948. They had three children: Michel, Bertrand, and Anne. He died on 12 January 2018, at the age of 97, nine days after his 97th birthday. See a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solomon Kullback
Solomon Kullback (April 3, 1907August 5, 1994) was an American cryptanalyst and mathematician, who was one of the first three employees hired by William F. Friedman at the US Army's Signal Intelligence Service (SIS) in the 1930s, along with Frank Rowlett and Abraham Sinkov. He went on to a long and distinguished career at SIS and its eventual successor, the National Security Agency (NSA). Kullback was the Chief Scientist at the NSA until his retirement in 1962, whereupon he took a position at the George Washington University. The Kullback–Leibler divergence is named after Kullback and Richard Leibler. Life and career Kullback was born to Jewish parents in Brooklyn, New York. His father Nathan had been born in Vilna, Russian Empire, (now Vilnius, Lithuania) and had immigrated to the US as a young man circa 1905, and became a naturalized American in 1911. Kullback attended Boys High School in Brooklyn. He then went to City College of New York, graduating with a BA in 1927 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Jeffreys
Sir Harold Jeffreys, FRS (22 April 1891 – 18 March 1989) was a British geophysicist who made significant contributions to mathematics and statistics. His book, ''Theory of Probability'', which was first published in 1939, played an important role in the revival of the objective Bayesian view of probability. Education Jeffreys was born in Fatfield, County Durham, England, the son of Robert Hal Jeffreys, headmaster of Fatfield Church School, and his wife, Elizabeth Mary Sharpe, a school teacher. He was educated at his father's school and at Rutherford Technical College, then studied at Armstrong College in Newcastle upon Tyne (at that time part of the University of Durham) and with the University of London External Programme.Cook, Alan ev.br>"Jeffreys, Sir Harold (1891–1989)" ''Oxford Dictionary of National Biography'', Oxford University Press, 2004; online edition, September 2004. Retrieved 1 January 2023. Jeffreys subsequently won a scholarship to study the Mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calyampudi Radhakrishna Rao
Prof. Calyampudi Radhakrishna Rao (10 September 1920 – 22 August 2023) was an Indian-American mathematician and statistician. He was professor emeritus at Pennsylvania State University and research professor at the University at Buffalo. Rao was honoured by numerous colloquia, honorary degrees, and festschrifts and was awarded the US National Medal of Science in 2002. The American Statistical Association has described him as "a living legend" whose work has influenced not just statistics, but has had far reaching implications for fields as varied as economics, genetics, anthropology, geology, national planning, demography, biometry, and medicine." ''The Times of India'' listed Rao as one of the top 10 Indian scientists of all time. In 2023, Rao was awarded the International Prize in Statistics, an award often touted as the "statistics' equivalent of the Nobel Prize". Rao was also a Senior Policy and Statistics advisor for the Indian Heart Association non-profit focused on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harald Cramér
Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of statistical theory".Kingman 1986, p. 186. Biography Early life Harald Cramér was born in Stockholm, Sweden on 25 September 1893. Cramér remained close to Stockholm for most of his life. He entered the Stockholm University as an undergraduate in 1912, where he studied mathematics and chemistry. During this period, he was a research assistant under the famous chemist, Hans von Euler-Chelpin, with whom he published his first five articles from 1913 to 1914. Following his lab experience, he began to focus solely on mathematics. He eventually began his work on his doctoral studies in mathematics which were supervised by Marcel Riesz at the Stockholm University. Also influenced by G. H. Hardy, Cramér's research led to a PhD in 1917 for his thesi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
