These functions provide the density and random number generation for the Wishart distribution.
This is the symmetric, positive-definite k x k matrix Omega.
This is the scalar degrees of freedom nu.
This is the symmetric, positive-semidefinite, k x k scale matrix S.
Application: Continuous Multivariate
Density: p(theta) = (2^(nu*k/2) * pi^(k(k-1)/4) * [Gamma((nu+1-i)/2) * ... * Gamma((nu+1-k)/2)])^(-1) * |S|^(-nu/2) * |Omega|^((nu-k-1)/2) * exp(-(1/2) * tr(S^(-1) Omega))
Inventor: John Wishart (1928)
Notation 1: Omega ~ W[nu](S)
Notation 2: p(Omega) = W[nu](Omega | S)
Parameter 1: degrees of freedom nu >= k
Parameter 2: symmetric, positive-semidefinite k x k scale matrix S
Mean: E(Omega) = nuS
Variance: var(Omega) = nu(S[i,j]^2 + S[i,i]S[j,j])
Mode: mode(Omega) = (nu-k-1)S, for nu >= k + 1
The Wishart distribution is a generalization to multiple dimensions of the chi-square distribution, or, in the case of non-integer degrees of freedom, of the gamma distribution. However, the Wishart distribution is not called the multivariate chi-squared distribution because the marginal distribution of the off-diagonal elements is not chi-squared.
The Wishart is the conjugate prior distribution for the precision matrix Omega, the inverse of which (covariance matrix Sigma) is used in a multivariate normal distribution.
The integral is finite when nu >= k, where nu is the scalar degrees of freedom parameter, and k is the dimension of scale matrix S. The density is finite when nu >= k + 1, which is recommended.
The degrees of freedom, nu, is equivalent to specifying a prior sample size, indicating the confidence in S, where S is a prior guess at the order of covariance matrix Sigma. A flat prior distribution is obtained as nu -> 0.
When applicable, the alternative Cholesky parameterization should be
preferred. For more information, see
The Wishart prior lacks flexibility, having only one parameter, nu, to control the variability for all k(k + 1)/2 elements. Popular choices for the scale matrix S include an identity matrix or sample covariance matrix. When the model sample size is small, the specification of the scale matrix can be influential.
Although the related inverse Wishart distribution has a dependency between variance and correlation, the Wishart distribution does not have this dependency.
The matrix gamma (
dmatrixgamma) distribution is a more
general version of the Wishart distribution, and the Yang-Berger
dyangberger) distribution is an alterative that is a
least informative prior (LIP).
dwishart gives the density and
rwishart generates random deviates.
Wishart, J. (1928). "The Generalised Product Moment Distribution in Samples from a Normal Multivariate Population". Biometrika, 20A(1-2), p. 32–52.
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