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Inverse Dirichlet Distribution
In statistics, the inverse Dirichlet distribution is a derivation of the matrix variate Dirichlet distribution. It is related to the inverse Wishart distribution. Suppose U_1,\ldots,U_r are p\times p positive definite matrices with a matrix variate Dirichlet distribution, \left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_r;a_\right). Then X_i=^,i=1,\ldots,r have an inverse Dirichlet distribution, written \left(X_1,\ldots,X_r\right)\sim \operatorname\left(a_1,\ldots,a_r;a_\right). Their joint probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ... is given by : \left\^ \prod_^r \det\left(X_i\right)^\det\left(I_p-\sum_^r^\right)^ References A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall. {{statistics-stub Prob ... [...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] |
Matrix Variate Dirichlet Distribution
In statistics, the matrix variate Dirichlet distribution is a generalization of the matrix variate beta distribution and of the Dirichlet distribution. Suppose U_1,\ldots,U_r are p\times p positive definite matrices with I_p-\sum_^rU_i also positive-definite, where I_p is the p\times p identity matrix. Then we say that the U_i have a matrix variate Dirichlet distribution, \left(U_1,\ldots,U_r\right)\sim D_p\left(a_1,\ldots,a_r;a_\right), if their joint probability density function is : \left\^\prod_^\det\left(U_i\right)^\det\left(I_p-\sum_^rU_i\right)^ where a_i>(p-1)/2,i=1,\ldots,r+1 and \beta_p\left(\cdots\right) is the multivariate beta function. If we write U_=I_p-\sum_^r U_i then the PDF takes the simpler form : \left\^\prod_^\det\left(U_i\right)^, on the understanding that \sum_^U_i=I_p. Theorems generalization of chi square-Dirichlet result Suppose S_i\sim W_p\left(n_i,\Sigma\right),i=1,\ldots,r+1 are independently distributed Wishart p\times p positive defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Wishart Distribution
In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution. We say \mathbf follows an inverse Wishart distribution, denoted as \mathbf\sim \mathcal^(\mathbf\Psi,\nu), if its inverse \mathbf^ has a Wishart distribution \mathcal(\mathbf \Psi^, \nu) . Important identities have been derived for the inverse-Wishart distribution. Density The probability density function of the inverse Wishart is: : f_(; , \nu) = \frac \left, \mathbf\^ e^ where \mathbf and are p\times p positive definite matrices, , \cdot , is the determinant, and Γ''p''(·) is the multivariate gamma function. Theorems Distribution of the inverse of a Wishart-distributed matrix If \sim \mathcal(,\nu) and is of size p \times p, then \mathbf=^ has an inverse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
