Matrix Gamma Distribution
This function provides the density for the matrix gamma distribution.
dmatrixgamma(X, alpha, beta, Sigma, log=FALSE)
X |
This is a k x k positive-definite precision matrix. |
alpha |
This is a scalar shape parameter (the degrees of freedom), alpha. |
beta |
This is a scalar, positive-only scale parameter, beta. |
Sigma |
This is a k x k positive-definite scale matrix. |
log |
Logical. If |
Application: Continuous Multivariate Matrix
Density: p(theta) = {|Sigma|^(-alpha) / [beta^(k alpha) Gamma[k](alpha)]} |theta|^[alpha-(k+1)/2] exp(tr(-(1/beta)Sigma^(-1)theta))
Inventors: Unknown
Notation 1: theta ~ MG[k](alpha, beta, Sigma)
Notation 2: p(theta) = MG[k](theta | alpha, beta, Sigma)
Parameter 1: shape alpha > 2
Parameter 2: scale beta > 0
Parameter 3: positive-definite k x k scale matrix Sigma
Mean:
Variance:
Mode:
The matrix gamma (MG), also called the matrix-variate gamma,
distribution is a generalization of the gamma distribution to
positive-definite matrices. It is a more general and flexible version of
the Wishart distribution (dwishart), and is a conjugate
prior of the precision matrix of a multivariate normal distribution
(dmvnp) and matrix normal distribution
(dmatrixnorm).
The compound distribution resulting from compounding a matrix normal with a matrix gamma prior over the precision matrix is a generalized matrix t-distribution.
The matrix gamma distribution is identical to the Wishart distribution when alpha = nu / 2 and beta = 2.
dmatrixgamma gives the density.
Statisticat, LLC. software@bayesian-inference.com
dgamma
dmatrixnorm,
dmvnp, and
dwishart
library(LaplacesDemon) k <- 10 dmatrixgamma(X=diag(k), alpha=(k+1)/2, beta=2, Sigma=diag(k), log=TRUE) dwishart(Omega=diag(k), nu=k+1, S=diag(k), log=TRUE)
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