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Normal-inverse Gamma Distribution
In probability theory and statistics, the normal-inverse-gamma distribution (or Gaussian-inverse-gamma distribution) is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance. Definition Suppose : x \mid \sigma^2, \mu, \lambda\sim \mathrm(\mu,\sigma^2 / \lambda) \,\! has a normal distribution with mean \mu and variance \sigma^2 / \lambda, where :\sigma^2\mid\alpha, \beta \sim \Gamma^(\alpha,\beta) \! has an inverse gamma distribution. Then (x,\sigma^2) has a normal-inverse-gamma distribution, denoted as : (x,\sigma^2) \sim \text\Gamma^(\mu,\lambda,\alpha,\beta) \! . (\text is also used instead of \text\Gamma^.) The normal-inverse-Wishart distribution is a generalization of the normal-inverse-gamma distribution that is defined over multivariate random variables. Characterization Probability density function : f(x,\sigma^2\mid\mu,\lambda,\alpha,\beta) = \frac \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar (mathematics)
A scalar is an element of a field which is used to define a ''vector space''. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers). A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Distributions
Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous game, a generalization of games used in game theory ** Law of Continuity, a heuristic principle of Gottfried Leibniz * Continuous function, in particular: ** Continuity (topology), a generalization to functions between topological spaces ** Scott continuity, for functions between posets ** Continuity (set theory), for functions between ordinals ** Continuity (category theory), for functors ** Graph continuity, for payoff functions in game theory * Continuity theorem may refer to one of two results: ** Lévy's continuity theorem, on random variables ** Kolmogorov continuity theorem, on stochastic processes * In geometry: ** Parametric continuity, for parametrised curves ** Geometric continuity, a concept primarily applied to the conic secti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Probability Distribution
In probability and statistics, a compound probability distribution (also known as a mixture distribution or contagious distribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. If the parameter is a scale parameter, the resulting mixture is also called a scale mixture. The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the ''latent'' random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution"). Definition A compound probability distribution is the probability distribution that results from assuming that a random variable X is distributed according to some parametrized distribution F with an unknown parameter \theta that is again distributed according to some other distribution G. The resulting di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Precision (statistics)
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 ... [...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] |
Natural Parameter
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family", or the older term Koopman–Darmois family. The terms "distribution" and "family" are often used loosely: specifically, ''an'' exponential family is a ''set'' of distributions, where the specific distribution varies with the parameter; however, a parametric ''family'' of distributions is often referred to as "''a'' distribution" (like "the normal distribution", meaning "the family of normal distributions"), and the set of all exponential families is sometimes lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Family
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family", or the older term Koopman–Darmois family. The terms "distribution" and "family" are often used loosely: specifically, ''an'' exponential family is a ''set'' of distributions, where the specific distribution varies with the parameter; however, a parametric ''family'' of distributions is often referred to as "''a'' distribution" (like "the normal distribution", meaning "the family of normal distributions"), and the set of all exponential families is sometimes l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Student 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] |
Inverse-gamma Distribution
In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution. Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution, if an uninformative prior is used, and as an analytically tractable conjugate prior, if an informative prior is required. It is common among some Bayesians to consider an alternative parametrization of the normal distribution in terms of the precision, defined as the reciprocal of the variance, which allows the gamma distribution to be used directly as a conjugate prior. Other Bayesians prefer to parametrize the inverse gamma distribution differently, as a scaled inverse chi-squared distribution. Characterizatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Student's T-distribution
In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in situations where the sample size is small and the population's standard deviation is unknown. It was developed by English statistician William Sealy Gosset under the pseudonym "Student". The ''t''-distribution plays a role in a number of widely used statistical analyses, including Student's ''t''-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. Student's ''t''-distribution also arises in the Bayesian analysis of data from a normal family. If we take a sample of n observations from a normal distribution, then the ''t''-distribution with \nu=n-1 degrees of freedom can be de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
