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Gibbs Sampling
In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is difficult. This sequence can be used to approximate the joint distribution (e.g., to generate a histogram of the distribution); to approximate the marginal distribution of one of the variables, or some subset of the variables (for example, the unknown parameters or latent variables); or to compute an integral (such as the expected value of one of the variables). Typically, some of the variables correspond to observations whose values are known, and hence do not need to be sampled. Gibbs sampling is commonly used as a means of statistical inference, especially Bayesian inference. It is a randomized algorithm (i.e. an algorithm that makes use of random numbers), and is an alternative to deterministic algorithms for statistical inferenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals. Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with autocovariance. Unit root processes, trend-stationary processes, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joint Distribution
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables. It also encodes the conditional probability distributions, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s). In the formal mathematical setup of measure theory, the joint distribution is given by the pushforward measure, by the map obtained by pairing together the given random variables, of the sample space's probability measure. In the case of real-valued random variables, the joint distribution, as a particular multivariate distribution, may be expressed by a multivariate cumulativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Network
A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. Efficient algorithms can perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (''e.g.'' speech signals or protein sequences) are called dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams. Graphical mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Posterior Probability
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior probability contains everything there is to know about an uncertain proposition (such as a scientific hypothesis, or parameter values), given prior knowledge and a mathematical model describing the observations available at a particular time. After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating. In the context of Bayesian statistics, the posterior probability distribution usually describes the epistemic uncertainty about statistical parameters conditional on a collection of observed data. From a given posterior distribution, various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest posterior density interval (HPD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Distribution
In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value x of X as a parameter. When both X and Y are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable. If the conditional distribution of Y given X is a continuous distribution, then its probability density function is known as the conditional density function. The properties of a conditional distribution, such as the moments, are often referred to by corresponding names such as the conditional mean and conditional variance. Mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Sampling
Slice sampling is a type of Markov chain Monte Carlo algorithm for pseudo-random number sampling, i.e. for drawing random samples from a statistical distribution. The method is based on the observation that to sample a random variable one can sample uniformly from the region under the graph of its density function. Motivation Suppose you want to sample some random variable ''X'' with distribution ''f''(''x''). Suppose that the following is the graph of ''f''(''x''). The height of ''f''(''x'') corresponds to the likelihood at that point. If you were to uniformly sample ''X'', each value would have the same likelihood of being sampled, and your distribution would be of the form ''f''(''x'') = ''y'' for some ''y'' value instead of some non-uniform function ''f''(''x''). Instead of the original black line, your new distribution would look more like the blue line. In order to sample ''X'' in a manner which will retain the distribution ''f''(''x''), some sampling technique must be u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variations And Extensions
Variation or Variations may refer to: Science and mathematics * Variation (astronomy), any perturbation of the mean motion or orbit of a planet or satellite, particularly of the moon * Genetic variation, the difference in DNA among individuals or the differences between populations ** Human genetic variation, genetic differences in and among populations of humans * Magnetic variation, difference between magnetic north and true north, measured as an angle * ''p''-variation in mathematical analysis, a family of seminorms of functions * Coefficient of variation in probability theory and statistics, a standardized measure of dispersion of a probability distribution or frequency distribution * Total variation in mathematical analysis, a way of quantifying the change in a function over a subset of \mathbb^n or a measure space * Calculus of variations in mathematical analysis, a method of finding maxima and minima of functionals Arts * Variation (ballet) or pas seul, solo dance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metropolis–Hastings Algorithm
In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult. This sequence can be used to approximate the distribution (e.g. to generate a histogram) or to compute an integral (e.g. an expected value). Metropolis–Hastings and other MCMC algorithms are generally used for sampling from multi-dimensional distributions, especially when the number of dimensions is high. For single-dimensional distributions, there are usually other methods (e.g. adaptive rejection sampling) that can directly return independent samples from the distribution, and these are free from the problem of autocorrelated samples that is inherent in MCMC methods. History The algorithm was named after Nicholas Metropolis and W.K. Hastings. Metropolis was the first author to appear on the list of authors of the 1953 article ''Equation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IEEE Transactions On Pattern Analysis And Machine Intelligence
''IEEE Transactions on Pattern Analysis and Machine Intelligence'' (sometimes abbreviated as ''IEEE PAMI'' or simply ''PAMI'') is a monthly peer-reviewed scientific journal published by the IEEE Computer Society. Background The journal covers research in computer vision and image understanding, pattern analysis and recognition, machine intelligence, machine learning, search techniques, document and handwriting analysis, medical image analysis, video and image sequence analysis, content-based retrieval of image and video, and face and gesture recognition. The editor-in-chief is Kyoung Mu Lee (Seoul National University). According to the ''Journal Citation Reports'', the journal has a 2021 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 24.314. References Ext ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Geman
Donald Jay Geman (born September 20, 1943) is an American applied mathematician and a leading researcher in the field of machine learning and pattern recognition. He and his brother, Stuart Geman, are very well known for proposing the Gibbs sampler and for the first proof of the convergence of the simulated annealing algorithm, in an article that became a highly cited reference in engineering (over 21K citations according to Google Scholar, as of January 2018). He is a professor at the Johns Hopkins University and simultaneously a visiting professor at École Normale Supérieure de Cachan. Biography Geman was born in Chicago in 1943. He graduated from the University of Illinois at Urbana-Champaign in 1965 with a B.A. degree in English Literature and from Northwestern University in 1970 with a Ph.D. in Mathematics. His dissertation was entitled as "Horizontal-window conditioning and the zeros of stationary processes." He joined University of Massachusetts - Amherst in 1970, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stuart Geman
Stuart Alan Geman (born March 23, 1949) is an American mathematician, known for influential contributions to computer vision, statistics, probability theory, machine learning, and the neurosciences. ikipediaList of important publications in computer science. He and his brother, Donald Geman, are well known for proposing the Gibbs sampler, and for the first proof of convergence of the simulated annealing algorithm. Biography Geman was born and raised in Chicago. He was educated at the University of Michigan (B.S., Physics, 1971), Dartmouth Medical College (MS, Neurophysiology, 1973), and the Massachusetts Institute of Technology (Ph.D, Applied Mathematics, 1977). Since 1977, he has been a member of the faculty at Brown University, where he has worked in the Pattern Theory group, and is currently the James Manning Professor of Applied Mathematics. He has received many honors and awards, including selection as a Presidential Young Investigator and as an ISI Highly Cited research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
