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Doob Martingale
In the probability theory, mathematical theory of probability, a Doob martingale (named after Joseph L. Doob, also known as a Levy martingale) is a stochastic process that approximates a given random variable and has the Martingale (probability theory), martingale property with respect to the given filtration (probability theory), filtration. It may be thought of as the evolving sequence of best approximations to the random variable based on information accumulated up to a certain time. When analyzing sums, random walks, or other additive functions of Statistical independence, independent random variables, one can often apply the central limit theorem, law of large numbers, Chernoff's inequality, Chebyshev's inequality or similar tools. When analyzing similar objects where the differences are not independent, the main tools are Martingale (probability theory), martingales and Azuma's inequality. Definition Let Y be any random variable with \mathbb[, Y, ] < \infty. Suppose ... [...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] |
Chernoff's Inequality
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] |
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] |
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] |
Probabilistic Inequalities
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Notation In Probability
Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols. Probability theory * Random variables are usually written in upper case roman letters: ''X'', ''Y'', etc. * Particular realizations of a random variable are written in corresponding lower case letters. For example, ''x''1, ''x''2, …, ''x''''n'' could be a sample corresponding to the random variable ''X''. A cumulative probability is formally written P(X\le x) to differentiate the random variable from its realization. * The probability is sometimes written \mathbb to distinguish it from other functions and measure ''P'' so as to avoid having to define "''P'' is a probability" and \mathbb(X\in A) is short for P(\), where \Omega is the event space and X(\omega) is a random variable. \Pr(A) notation is used alternatively. *\mathbb(A \cap B) or \mathbb \cap A/math> indicates the probability that events ''A'' and ''B'' both occur. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Statistical Topics
0–9 * 1.96 *2SLS (two-stage least squares) redirects to instrumental variable *3SLS – see three-stage least squares *68–95–99.7 rule *100-year flood A *A priori probability *Abductive reasoning *Absolute deviation *Absolute risk reduction *Absorbing Markov chain *ABX test * Accelerated failure time model *Acceptable quality limit *Acceptance sampling *Accidental sampling *Accuracy and precision * Accuracy paradox * Acquiescence bias * Actuarial science *Adapted process * Adaptive estimator * Additive Markov chain *Additive model *Additive smoothing *Additive white Gaussian noise *Adjusted Rand index – see Rand index (subsection) * ADMB software *Admissible decision rule *Age adjustment * Age-standardized mortality rate *Age stratification *Aggregate data * Aggregate pattern *Akaike information criterion *Algebra of random variables *Algebraic statistics *Algorithmic inference *Algorithms for calculating variance *All models are wrong *All-pairs testing * Allan varia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Publications In Statistics
This is a list of important publications in statistics, organized by field. Some reasons why a particular publication might be regarded as important: *Topic creator – A publication that created a new topic *Breakthrough – A publication that changed scientific knowledge significantly *Influence – A publication which has significantly influenced the world or has had a massive impact on the teaching of statistics. Probability ;''Théorie analytique des probabilités'' :Author: Pierre-Simon Laplace :Publication data: 1820 (3rd ed.) :Online version:''Internet Archive with more accurate character recognition Gallica-Math complete PDF and PDFs by section :Des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Probability Topics
This is a list of probability topics, by Wikipedia page. It overlaps with the (alphabetical) list of statistical topics. There are also the outline of probability and catalog of articles in probability theory. For distributions, see List of probability distributions. For journals, see list of probability journals. For contributors to the field, see list of mathematical probabilists and list of statisticians. General aspects * Probability * Randomness, Pseudorandomness, Quasirandomness * Randomization, hardware random number generator * Random number generation * Random sequence * Uncertainty * Statistical dispersion * Observational error * Equiprobable ** Equipossible * Average * Probability interpretations * Markovian * Statistical regularity * Central tendency * Bean machine * Relative frequency * Frequency probability * Maximum likelihood * Bayesian probability * Principle of indifference * Credal set * Cox's theorem * Principle of maximum entropy * Information entropy * Urn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glossary Of Probability And Statistics
This glossary of statistics and probability is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics and Glossary of experimental design. A B C D E F G H I J K L M N O P Q R S T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Measure Theory
In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also ''capacity'', see ), which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different classes of fuzzy measures including plausibility/belief measures; possibility/necessity measures; and probability measures, which are a subset of classical measures. Definitions Let \mathbf be a universe of discourse, \mathcal be a class of subsets of \mathbf, and E,F\in\mathcal. A function g:\mathcal\to\mathbb where # \emptyset \in \mathcal \Rightarrow g(\emptyset)=0 # E \subseteq F \Rightarrow g(E)\leq g(F) is called a ''fuzzy measure''. A fuzzy measure is called ''normalized'' or ''regular'' if g(\mathbf)=1. Properties of fuzzy measures A fuzzy measure is: * additive if for an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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