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Kullback–Leibler Divergence
In mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted D_\text(P \parallel Q), is a type of statistical distance: a measure of how one probability distribution ''P'' is different from a second, reference probability distribution ''Q''. A simple interpretation of the KL divergence of ''P'' from ''Q'' is the expected excess surprise from using ''Q'' as a model when the actual distribution is ''P''. While it is a distance, it is not a metric, the most familiar type of distance: it is not symmetric in the two distributions (in contrast to variation of information), and does not satisfy the triangle inequality. Instead, in terms of information geometry, it is a type of divergence, a generalization of squared distance, and for certain classes of distributions (notably an exponential family), it satisfies a generalized Pythagorean theorem (which applies to squared distances). In the simple case, a relative entropy of 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory. Introduction Statistical data collection is concerned with the planning of studies, especially with the design of randomized experiments and with the planning of surveys using random sampling. The initial analysis of the data often follows the study protocol specified prior to the study being conducted. The data from a study can also be analyzed to consider secondary hypotheses inspired by the initial results, or to suggest new studies. A secondary analysis of the data from a planned study uses tools from data analysis, and the process of doing this is mathematical statistics. Data analysis is divided into: * descriptive statistics - the part of st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle (300s BCE). Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular evidence to a universal conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce, contradistinguishing abduction from induction. Various fields study how inference is done in practice. Human inference (i.e. how humans draw conclusions) is traditionally studied within the fields of logic, argumentation studies, and cognitive psychology; artificial intelligence researchers develop automated inference systems to emulate human inference. Statistical inference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Continuity
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus— differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the '' Radon–Nikodym derivative'', or ''density'', of a measure. We have the following chains of inclusions for functions over a compact subset of the real line: : ''absolutely continuous'' ⊆ ''uniformly continuous'' = ''con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expectation (statistics)
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements:Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press. # A sample space, \Omega, which is the set of all possible outcomes. # An event space, which is a set of events \mathcal, an event being a set of outcomes in the sample space. # A probability function, which assigns each event in the event space a probability, which is a number between 0 and 1. In order to provide a sensible model of probability, these elements must satisfy a number of axioms, detailed in this article. In the example of the throw of a standard die, we would take the sample space to be \. For the event space, we could simply use the set of all subsets of the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The American Statistician
''The American Statistician'' is a quarterly peer-reviewed scientific journal covering statistics published by Taylor & Francis on behalf of the American Statistical Association. It was established in 1947. The editor-in-chief is Daniel R. Jeske, a professor at the University of California, Riverside The University of California, Riverside (UCR or UC Riverside) is a public land-grant research university in Riverside, California. It is one of the ten campuses of the University of California system. The main campus sits on in a suburban distr .... External links * Taylor & Francis academic journals Statistics journals Publications established in 1947 English-language journals Quarterly journals 1947 establishments in the United States Academic journals associated with learned and professional societies of the United States {{math-journal-stub ... [...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 ''Methods ... [...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. He was inducted into ... [...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 and an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evidence Lower Bound
In variational Bayesian methods, the evidence lower bound (often abbreviated ELBO, also sometimes called the variational lower bound or negative variational free energy) is a useful lower bound on the log-likelihood of some observed data. Terminology and notation Let X and Z be random variables, jointly-distributed with distribution p_\theta. For example, p_\theta( X) is the marginal distribution of X, and p_\theta( Z \mid X) is the conditional distribution of Z given X. Then, for any sample x\sim p_\theta, and any distribution q_\phi , we have\ln p_\theta(x) \ge \mathbb \mathbb E_\left \ln\frac \rightThe left-hand side is called the ''evidence'' for x, and the right-hand side is called the ''evidence lower bound for x'', or ''ELBO''. We refer to the above inequality as the ''ELBO inequality''. In the terminology of variational Bayesian methods, the distribution p_\theta( X) is called the ''evidence''. Some authors use the term ''evidence'' to mean \ln p_\theta( X), and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expectation–maximization Algorithm
In statistics, an expectation–maximization (EM) algorithm is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the ''E'' step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step. History The EM algorithm was explained and given its name in a classic 1977 paper by Arthur Dempster, Nan Laird, and Donald Rubin. They pointed out that the method had been "proposed many times in special circumstances" by earlier authors. One of the earliest is the gene-counting method for estimating allele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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