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Inverse Matrix Gamma Distribution
In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices. It is a more general version of the inverse Wishart distribution, and is used similarly, e.g. as the conjugate prior of the covariance matrix of a multivariate normal distribution or matrix normal distribution. The compound distribution resulting from compounding a matrix normal with an inverse matrix gamma prior over the covariance matrix is a generalized matrix t-distribution. This reduces to the inverse Wishart distribution with \nu degrees of freedom when \beta=2, \alpha=\frac. See also * inverse Wishart distribution. * matrix gamma distribution. * matrix normal distribution. * matrix t-distribution. * 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 fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shape Parameter
In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. that is neither a location parameter nor a scale parameter (nor a function of these, such as a rate parameter). Such a parameter must affect the ''shape'' of a distribution rather than simply shifting it (as a location parameter does) or stretching/shrinking it (as a scale parameter does). For example, "peakedness" refers to how round the main peak is. Estimation Many estimators measure location or scale; however, estimators for shape parameters also exist. Most simply, they can be estimated in terms of the higher moments, using the method of moments, as in the '' skewness'' (3rd moment) or '' kurtosis'' (4th moment), if the higher moments are defined and finite. Estimators of shape often involve higher-order statistics (non-linear fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance Matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the x and y directions contain all of the necessary information; a 2 \times 2 matrix would be necessary to fully characterize the two-dimensional variation. The covariance matrix of a random vector \mathbf is typically denoted by \operatorname_ or \Sigma. Definition Throughout this article, boldfaced unsub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Matrices
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the eigenvalues of a random matrix, and should depend only on the symmetry class of the underlying evolution. In solid-state physics, random matrices model the behaviour of large disordered Hamiltonians in the mean-field approximation. In quantum chaos, the Bohigas–Giannoni–Schmit (BGS) conjecture asserts ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix T-distribution
In statistics, the matrix ''t''-distribution (or matrix variate ''t''-distribution) is the generalization of the multivariate ''t''-distribution from vectors to matrices. The matrix ''t''-distribution shares the same relationship with the multivariate ''t''-distribution that the matrix normal distribution shares with the multivariate normal distribution. For example, the matrix ''t''-distribution is the compound distribution that results from sampling from a matrix normal distribution having sampled the covariance matrix of the matrix normal from an inverse Wishart distribution. In a Bayesian analysis of a multivariate linear regression model based on the matrix normal distribution, the matrix ''t''-distribution is the posterior predictive distribution. Definition For a matrix ''t''-distribution, the probability density function at the point \mathbf of an n\times p space is : f(\mathbf ; \nu,\mathbf,\boldsymbol\Sigma, \boldsymbol\Omega) = K \times \left, \mathbf_n + \boldsy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Gamma Distribution
In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices.Iranmanesh, Anis, M. Arashib and S. M. M. Tabatabaey (2010)"On Conditional Applications of Matrix Variate Normal Distribution" ''Iranian Journal of Mathematical Sciences and Informatics'', 5:2, pp. 33–43. It is a more general version of the Wishart distribution, and is used similarly, e.g. as the conjugate prior of the precision matrix of a multivariate normal distribution and matrix normal distribution. The compound distribution resulting from compounding a matrix normal with a matrix gamma prior over the precision matrix is a generalized matrix t-distribution. This reduces to the Wishart distribution with \beta=2, \alpha=\frac. Notice that in this parametrization, the parameters \beta and \boldsymbol\Sigma are not identified; the density depends on these two parameters through the product \beta\boldsymbol\Sigma. See also * inverse matrix gamma distributi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Matrix T-distribution
In statistics, the matrix ''t''-distribution (or matrix variate ''t''-distribution) is the generalization of the multivariate ''t''-distribution from vectors to matrices. The matrix ''t''-distribution shares the same relationship with the multivariate ''t''-distribution that the matrix normal distribution shares with the multivariate normal distribution. For example, the matrix ''t''-distribution is the compound distribution that results from sampling from a matrix normal distribution having sampled the covariance matrix of the matrix normal from an inverse Wishart distribution. In a Bayesian analysis of a multivariate linear regression model based on the matrix normal distribution, the matrix ''t''-distribution is the posterior predictive distribution. Definition For a matrix ''t''-distribution, the probability density function at the point \mathbf of an n\times p space is : f(\mathbf ; \nu,\mathbf,\boldsymbol\Sigma, \boldsymbol\Omega) = K \times \left, \mathbf_n + \boldsy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound 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 resultin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Normal Distribution
In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables. Definition The probability density function for the random matrix X (''n'' × ''p'') that follows the matrix normal distribution \mathcal_(\mathbf, \mathbf, \mathbf) has the form: : p(\mathbf\mid\mathbf, \mathbf, \mathbf) = \frac where \mathrm denotes trace and M is ''n'' × ''p'', U is ''n'' × ''n'' and V is ''p'' × ''p'', and the density is understood as the probability density function with respect to the standard Lebesgue measure in \mathbb^, i.e.: the measure corresponding to integration with respect to dx_ dx_\dots dx_ dx_\dots dx_\dots dx_. The matrix normal is related to the multivariate normal distribution in the following way: :\mathbf \sim \mathcal_(\mathbf, \mathbf, \mathbf), if and only if :\ ... [...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'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Prior
In Bayesian probability theory, if the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function p(x \mid \theta). A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise, numerical integration may be necessary. Further, conjugate priors may give intuition by more transparently showing how a likelihood function updates a prior distribution. The concept, as well as the term "conjugate prior", were introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory.Howard Raiffa and Robert Schlaifer. ''Applied Statistical Decision Theory''. Division of Research, Graduate School of Business Administration, Harvard University, 1961. A similar concept had been discovered independently by George A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scale Parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family of probability distributions is such that there is a parameter ''s'' (and other parameters ''θ'') for which the cumulative distribution function satisfies :F(x;s,\theta) = F(x/s;1,\theta), \! then ''s'' is called a scale parameter, since its value determines the " scale" or statistical dispersion of the probability distribution. If ''s'' is large, then the distribution will be more spread out; if ''s'' is small then it will be more concentrated. If the probability density exists for all values of the complete parameter set, then the density (as a function of the scale parameter only) satisfies :f_s(x) = f(x/s)/s, \! where ''f'' is the density of a standardized version of the density, i.e. f(x) \equiv f_(x). An estimator of a scale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
