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