|
Normal-Wishart Distribution
In probability theory and statistics, the normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix (the inverse of the covariance matrix).Bishop, Christopher M. (2006). ''Pattern Recognition and Machine Learning.'' Springer Science+Business Media. Page 690. Definition Suppose : \boldsymbol\mu, \boldsymbol\mu_0,\lambda,\boldsymbol\Lambda \sim \mathcal(\boldsymbol\mu_0,(\lambda\boldsymbol\Lambda)^) has a multivariate normal distribution with mean \boldsymbol\mu_0 and covariance matrix (\lambda\boldsymbol\Lambda)^, where :\boldsymbol\Lambda, \mathbf,\nu \sim \mathcal(\boldsymbol\Lambda, \mathbf,\nu) has a Wishart distribution. Then (\boldsymbol\mu,\boldsymbol\Lambda) has a normal-Wishart distribution, denoted as : (\boldsymbol\mu,\boldsymbol\Lambda) \sim \mathrm(\boldsymbol\mu_0,\lambda,\mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Location Parameter
In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ambiguous boundary, relying more on human or social attributes of place identity and sense of place than on geometry. Types Locality A suburb, locality, human settlement, settlement, or populated place is likely to have a well-defined name but a boundary that is not well defined varies by context. London, for instance, has a legal boundary, but this is unlikely to completely match with general usage. An area within a town, such as Covent Garden in London, also almost always has some ambiguity as to its extent. In geography, location is considered to be more precise than "place". Relative location A relative location, or situation, is described as a displacement from another site. An example is "3 miles northwest of Seattle". Absolute lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Precision Matrix
In statistics, the precision matrix or concentration matrix is the matrix inverse of the covariance matrix or dispersion matrix, P = \Sigma^. For univariate distributions, the precision matrix degenerates into a scalar precision, defined as the reciprocal of the variance, p = \frac. Other summary statistics of statistical dispersion also called ''precision'' (or ''imprecision'') include the reciprocal of the standard deviation, p = \frac; the standard deviation itself and the relative standard deviation; as well as the standard error and the confidence interval (or its half-width, the margin of error). Usage One particular use of the precision matrix is in the context of Bayesian analysis of the multivariate normal distribution: for example, Bernardo & Smith prefer to parameterise the multivariate normal distribution in terms of the precision matrix, rather than the covariance matrix, because of certain simplifications that then arise. For instance, if both the prior and the lik ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Continuous Distributions
{{disambiguation ...
Multivariate may refer to: In mathematics * Multivariable calculus * Multivariate function * Multivariate polynomial In computing * Multivariate cryptography * Multivariate division algorithm * Multivariate interpolation * Multivariate optical computing * Multivariate optimization, used for the design of heat exchangers, see In statistics * Multivariate analysis * Multivariate random variable * Multivariate statistics See also * Univariate * Bivariate (other) Bivariate may refer to: Mathematics * Bivariate function, a function of two variables * Bivariate polynomial, a polynomial of two indeterminates Statistics * Bivariate data, that shows the relationship between two variables * Bivariate analys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal-gamma Distribution
In probability theory and statistics, the normal-gamma distribution (or Gaussian-gamma distribution) is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and precision. Definition For a pair of random variables, (''X'',''T''), suppose that the conditional distribution of ''X'' given ''T'' is given by : X\mid T \sim N(\mu,1 /(\lambda T)) \,\! , meaning that the conditional distribution is a normal distribution with mean \mu and precision \lambda T — equivalently, with variance 1 / (\lambda T) . Suppose also that the marginal distribution of ''T'' is given by :T \mid \alpha, \beta \sim \operatorname(\alpha,\beta), where this means that ''T'' has a gamma distribution. Here ''λ'', ''α'' and ''β'' are parameters of the joint distribution. Then (''X'',''T'') has a normal-gamma distribution, and this is denoted by : (X,T) \sim \operatorname(\mu,\lambda,\alpha,\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal-inverse Wishart Distribution
In probability theory and statistics, the normal-inverse-Wishart distribution (or Gaussian-inverse-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and covariance matrix (the inverse of the precision matrix).Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution./ref> Definition Suppose : \boldsymbol\mu, \boldsymbol\mu_0,\lambda,\boldsymbol\Sigma \sim \mathcal\left(\boldsymbol\mu\Big, \boldsymbol\mu_0,\frac\boldsymbol\Sigma\right) has a multivariate normal distribution with mean \boldsymbol\mu_0 and covariance matrix \tfrac\boldsymbol\Sigma, where :\boldsymbol\Sigma, \boldsymbol\Psi,\nu \sim \mathcal^(\boldsymbol\Sigma, \boldsymbol\Psi,\nu) has an inverse Wishart distribution. Then (\boldsymbol\mu,\boldsymbol\Sigma) has a normal-inverse-Wishart distribution, denoted as : (\boldsymbol\mu,\boldsymbol\Sigma) \sim \mathr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate T-distribution
In statistics, the multivariate ''t''-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's ''t''-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix ''t''-distribution is distinct and makes particular use of the matrix structure. Definition One common method of construction of a multivariate ''t''-distribution, for the case of p dimensions, is based on the observation that if \mathbf y and u are independent and distributed as N(,) and \chi^2_\nu (i.e. multivariate normal and chi-squared distributions) respectively, the matrix \mathbf\, is a ''p'' × ''p'' matrix, and /\sqrt = -, then has the density : \frac\left +\frac(-)^T^(-)\right and is said to be distributed as a multivariate ''t''-distribution with parameters ,,\nu. Note that \mathbf\Sigma is ... [...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] |
Marginal Distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables. This contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables. Marginal variables are those variables in the subset of variables being retained. These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing (that is, focusing on the sums in the margin) over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out. The context here is that the theoretical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wishart Distribution
In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928. It is a family of probability distributions defined over symmetric, nonnegative-definite random matrices (i.e. matrix-valued random variables). In random matrix theory, the space of Wishart matrices is called the ''Wishart ensemble''. These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. In Bayesian statistics, the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random-vector. Definition Suppose is a matrix, each column of which is independently drawn from a -variate normal distribution with zero mean: :G_ = (g_i^1,\dots,g_i^p)^T\sim \mathcal_p(0,V). Then the Wishart distribution is the probability distribution of the random matrix :S= G G^T = \sum_^n G_G_^T kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithmetic mean'', also known as "arithmetic average", is a measure of central tendency of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers ''x''1, ''x''2, ..., x''n'' is typically denoted using an overhead bar, \bar. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the ''sample mean'' (\bar) to distinguish it from the mean, or expected value, of the underlying distribution, the ''population mean'' (denoted \mu or \mu_x).Underhill, L.G.; Bradfield d. (1998) ''Introstat'', Juta and Company Ltd.p. 181/ref> Outside probability and statistics, a wide range of other notions of mean are o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Normal Distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be ''k''-variate normally distributed if every linear combination of its ''k'' components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. Definitions Notation and parameterization The multivariate normal distribution of a ''k''-dimensional random vector \mathbf = (X_1,\ldots,X_k)^ can be written in the following notation: : \mathbf\ \sim\ \mathcal(\boldsymbol\mu,\, \boldsymbol\Sigma), or to make it explicitly known that ''X'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
