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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. For such models, there is a natural choice of Riemannian metric, known as the Fisher information metric. In the special case that the statistical model is an exponential family, it is possible to induce the statistical manifold with a Hessian metric (i.e a Riemannian metric given by the potential of a convex function). In thi ... [...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), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Normal'' (2009 film), an adaptation of Anthony Neilson's 1991 play ''Normal: The Düsseldorf Ripper'' * ''Normal!'', a 2011 Algerian film * ''The Normals'' (film), a 2012 American comedy film * "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 continuous probability distribution * Normal equations, describing the solution of the linear least ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Differential Geometry
Affine differential geometry is a type of differential geometry which studies invariants of volume-preserving affine transformations. The name ''affine differential geometry'' follows from Klein's Erlangen program. The basic difference between affine and Riemannian differential geometry is that affine differential geometry studies manifolds equipped with a volume form rather than a metric. Preliminaries Here we consider the simplest case, i.e. manifolds of codimension one. Let be an ''n''-dimensional manifold, and let ξ be a vector field on transverse to such that for all where ⊕ denotes the direct sum and Span the linear span. For a smooth manifold, say ''N'', let Ψ(''N'') denote the module of smooth vector fields over ''N''. Let be the standard covariant derivative on R''n''+1 where We can decompose ''DXY'' into a component tangent to ''M'' and a transverse component, parallel to ξ. This gives the equation of Gauss: where is the induced connexion on ''M'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Damiano Brigo
Damiano Brigo (born Venice, Italy 1966) is an applied mathematician and Chair in Mathematical Finance at Imperial College London. He is known for research in filtering theory and mathematical finance. Main results Brigo started his work with the development, with Bernard Hanzon and Francois Le Gland (1998), of the projection filters, a family of approximate nonlinear filters based on the differential geometry approach to statistics, also related to information geometry. With Fabio Mercurio (2002–2003), he has shown how to construct stochastic differential equations consistent with mixture models, applying this to volatility smile modeling in the context of local volatility models. With Aurelien Alfonsi (2005), Brigo introduced new families of multivariate distributions in statistics through the periodic copula function concept. Since 2002, Brigo contributed to credit derivatives modeling and counterparty risk valuation, showing with Pallavicini and Torresetti (2007) h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ole Barndorff-Nielsen
Ole Eiler Barndorff-Nielsen (18 March, 1935 – 26 June, 2022) was a Danish statistician who has contributed to many areas of statistical science. Education and career He was born in Copenhagen, and became interested in statistics when, as a student of actuarial mathematics at the University of Copenhagen, he worked part-time at the Department of Biostatistics of the Danish State Serum Institute. He graduated from the University of Aarhus (Denmark) in 1960, where he has spent most of his academic life, and where he became professor of statistics in 1973. However, in 1962-1963 and 1963-1964 he stayed at the University of Minnesota and Stanford University, respectively, and from August 1974 to February 1975 he was an Overseas Fellow at Churchill College, Cambridge, and visitor at Statistical Laboratory, Cambridge University. Barndorff-Nielsen became Professor Emeritus at Aarhus University at the Thiele Centre for Applied Mathematics in Natural Science and affiliated with th ... [...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] |
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 February 7, 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, and cryptographer known as a "father of information theory". As a 21-year-old master's degree student at the Massachusetts Institute of Technology (MIT), he wrote his thesis demonstrating that electrical applications of Boolean algebra could construct any logical numerical relationship. Shannon contributed to the field of cryptanalysis for national defense of the United States during World War II, including his fundamental work on codebreaking and secure telecommunications. Biography Childhood The Shannon family lived in Gaylord, Michigan, and Claude was born in a hospital in nearby Petoskey. His father, Claude Sr. (1862–1934), was a businessman and for a while, a judge of probate in Gaylord. His mother, Mabel Wolf Shannon (1890–1945), was a language teacher, who also served as the principal of Gaylord High School. Claude Sr. was a descendant of New Jer ... [...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 Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver .... 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 pape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Louis Koszul
Jean-Louis Koszul (; January 3, 1921 – January 12, 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 Julien Koszul (4 December 1844 – 15 January 1927) was a French composer and pipe organist from Alsace. Biography Born in Morschwiller-le-Bas, Alsace, Koszul studied at the École Nied ... [...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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Jeffreys
Sir Harold Jeffreys, FRS (22 April 1891 – 18 March 1989) was a British mathematician, statistician, geophysicist, and astronomer. 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 then studied at Armstrong College in Newcastle upon Tyne, then part of the University of Durham, and with the University of London External Programme. Career Jeffreys became a fellow of St John's College, Cambridge in 1914. At the University of Cambridge he taught mathematics, then geophysics and finally became the Plumian Professor of Astronomy. In 1940 he married fellow mathematician and physicist, Bertha Swirles (1903–1999), and together they wrote ''Meth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calyampudi Radhakrishna Rao
Calyampudi Radhakrishna Rao FRS (born 10 September 1920), commonly known as C. R. Rao, is an Indian-American mathematician and statistician. He is currently professor emeritus at Pennsylvania State University and Research Professor at the University at Buffalo. Rao has been 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. Rao is also a Senior Policy and Statistics advisor for the Indian Heart Association non-profit focused on raising South Asian cardiovascular disease awareness. Early life C. R. Rao was the eighth of the ten children b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
