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Bernstein Inequalities (probability Theory)
In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let ''X''1, ..., ''X''''n'' be independent Bernoulli random variables taking values +1 and −1 with probability 1/2 (this distribution is also known as the Rademacher distribution), then for every positive \varepsilon, :\mathbb\left (\left, \frac\sum_^n X_i\ > \varepsilon \right ) \leq 2\exp \left (-\frac \right). Bernstein inequalities were proved and published by Sergei Bernstein in the 1920s and 1930s.J.V.Uspensky, "Introduction to Mathematical Probability", McGraw-Hill Book Company, 1937 Later, these inequalities were rediscovered several times in various forms. Thus, special cases of the Bernstein inequalities are also known as the Chernoff bound, Hoeffding's inequality and Azuma's inequality. Some of the inequalities 1. Let X_1, \ldots, X_n be independent zero-mean random variables. Suppose that , X_i, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Trial
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in his ''Ars Conjectandi'' (1713). The mathematical formalisation of the Bernoulli trial is known as the Bernoulli process. This article offers an elementary introduction to the concept, whereas the article on the Bernoulli process offers a more advanced treatment. Since a Bernoulli trial has only two possible outcomes, it can be framed as some "yes or no" question. For example: *Is the top card of a shuffled deck an ace? *Was the newborn child a girl? (See human sex ratio.) Therefore, success and failure are merely labels for the two outcomes, and should not be construed literally. The term "success" in this sense consists in the result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rademacher Distribution
In probability theory and statistics, the Rademacher distribution (which is named after Hans Rademacher) is a discrete probability distribution where a random variate ''X'' has a 50% chance of being +1 and a 50% chance of being -1. A series (that is, a sum) of Rademacher distributed variables can be regarded as a simple symmetrical random walk where the step size is 1. Mathematical formulation The probability mass function of this distribution is : f(k) = \left\{\begin{matrix} 1/2 & \mbox {if }k=-1, \\ 1/2 & \mbox {if }k=+1, \\ 0 & \mbox {otherwise.}\end{matrix}\right. In terms of the Dirac delta function, as : f( k ) = \frac{ 1 }{ 2 } \left( \delta \left( k - 1 \right) + \delta \left( k + 1 \right) \right). Bounds on sums of independent Rademacher variables There are various results in probability theory around analyzing the sum of i.i.d. Rademacher variables, including concentration inequalities such as Bernstein inequalities as well as anti-concentration inequalit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergei Bernstein
Sergei Natanovich Bernstein (russian: Серге́й Ната́нович Бернште́йн, sometimes Romanized as ; 5 March 1880 – 26 October 1968) was a Ukrainian and Russian mathematician of Jewish origin known for contributions to partial differential equations, differential geometry, probability theory, and approximation theory. Work Partial differential equations In his doctoral dissertation, submitted in 1904 to Sorbonne, Bernstein solved Hilbert's nineteenth problem on the analytic solution of elliptic differential equations. His later work was devoted to Dirichlet's boundary problem for non-linear equations of elliptic type, where, in particular, he introduced a priori estimates. Probability theory In 1917, Bernstein suggested the first axiomatic foundation of probability theory, based on the underlying algebraic structure. It was later superseded by the measure-theoretic approach of Kolmogorov. In the 1920s, he introduced a method for proving limit theorems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doklady Akademii Nauk SSSR
The ''Proceedings of the USSR Academy of Sciences'' (russian: Доклады Академии Наук СССР, ''Doklady Akademii Nauk SSSR'' (''DAN SSSR''), french: Comptes Rendus de l'Académie des Sciences de l'URSS) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology. It was first published in 1933 and ended in 1992 with volume 322, issue 3. Today, it is continued by ''Doklady Akademii Nauk'' (russian: Доклады Академии Наук), which began publication in 1992. The journal is also known as the ''Proceedings of the Russian Academy of Sciences (RAS)''. ''Doklady'' has had a complicated publication and translation history. A number of translation journals exist which publish selected articles from the original by subject section; these are listed below. History The Russian Academy of Sciences dates from 1724, with a continuous series of variously named publications dat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chernoff Bound
In probability theory, the Chernoff bound gives exponentially decreasing bounds on tail distributions of sums of independent random variables. Despite being named after Herman Chernoff, the author of the paper it first appeared in, the result is due to Herman Rubin. It is a sharper bound than the first- or second-moment-based tail bounds such as Markov's inequality or Chebyshev's inequality, which only yield power-law bounds on tail decay. However, the Chernoff bound requires the variates to be independent, a condition that is not required by either Markov's inequality or Chebyshev's inequality (although Chebyshev's inequality does require the variates to be pairwise independent). The Chernoff bound is related to the Bernstein inequalities, which were developed earlier, and to Hoeffding's inequality. The generic bound The generic Chernoff bound for a random variable is attained by applying Markov's inequality to . This gives a bound in terms of the moment-generating function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hoeffding's Inequality
In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Wassily Hoeffding in 1963. Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's inequality. It is similar to the Chernoff bound, but tends to be less sharp, in particular when the variance of the random variables is small. It is similar to, but incomparable with, one of Bernstein's inequalities. Statement Let be independent random variables such that a_i \leq X_i \leq b_i almost surely. Consider the sum of these random variables, :S_n = X_1 + \cdots + X_n. Then Hoeffding's theorem states that, for all , :\begin \operatorname \left(S_n - \mathrm\left _n \right\geq t \right) &\leq \exp \left(-\frac \right) \\ \operatorname \left(\left , S_n - \mathrm\left _n \right\right , \geq t \right) &\leq 2\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Azuma's Inequality
In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration inequality, concentration result for the values of martingale (probability theory), martingales that have bounded differences. Suppose \ is a Martingale (probability theory), martingale (or Martingale_(probability_theory)#Submartingales.2C_supermartingales.2C_and_relationship_to_harmonic_functions, super-martingale) and :, X_k - X_, \leq c_k, \, almost surely. Then for all positive integers ''N'' and all positive real number, reals ''\epsilon'', :\text(X_N - X_0 \geq \epsilon) \leq \exp\left ( \right). And symmetrically (when ''X''''k'' is a sub-martingale): :\text(X_N - X_0 \leq -\epsilon) \leq \exp\left ( \right). If ''X'' is a martingale, using both inequalities above and applying the union bound allows one to obtain a two-sided bound: :\text(, X_N - X_0, \geq \epsilon) \leq 2\exp\left ( \right). Proof The proof shares similar idea of the pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov's Inequality
In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function (mathematics), function of a random variable is greater than or equal to some positive Constant (mathematics), constant. It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev (Markov's teacher), and many sources, especially in Mathematical analysis, analysis, refer to it as Chebyshev's inequality (sometimes, calling it the first Chebyshev inequality, while referring to Chebyshev's inequality as the second Chebyshev inequality) or Irénée-Jules Bienaymé, Bienaymé's inequality. Markov's inequality (and other similar inequalities) relate probabilities to expected value, expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable. Statement If is a nonnegative random variable and , then the probability that is at least is at most th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vectorization
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Vectorization may refer to: Computing * Array programming, a style of computer programming where operations are applied to whole arrays instead of individual elements * Automatic vectorization, a compiler optimization that transforms loops to vector operations * Image tracing, the creation of vector from raster graphics * Word embedding, mapping words to vectors, in natural language processing Other uses * Vectorization (mathematics), a linear transformation which converts a matrix into a column vector * Drug vectorization, to (intra)cellular targeting See also * Vector (other) * Vector graphics (other) Vector graphics are a form of computer graphics. Vector graphic may also refer to: *Vector Graphic, a computer company *Vektor Grafix, United Kingdom, UK based computer game development company See also * Raster graphics * Vector (disambigu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concentration Inequality
In probability theory, concentration inequalities provide bounds on how a random variable deviates from some value (typically, its expected value). The law of large numbers of classical probability theory states that sums of independent random variables are, under very mild conditions, close to their expectation with a large probability. Such sums are the most basic examples of random variables concentrated around their mean. Recent results show that such behavior is shared by other functions of independent random variables. Concentration inequalities can be sorted according to how much information about the random variable is needed in order to use them. Markov's inequality Let X be a random variable that is non-negative (almost surely). Then, for every constant a > 0, : \Pr(X \geq a) \leq \frac. Note the following extension to Markov's inequality: if \Phi is a strictly increasing and non-negative function, then :\Pr(X \geq a) = \Pr(\Phi (X) \geq \Phi (a)) \leq \frac. Cheb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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