dist.Inverse.Matrix.Gamma: Inverse Matrix Gamma Distribution
In ecbrown/LaplacesDemon: Complete Environment for Bayesian Inference

Description Usage Arguments Details Value Author(s) See Also Examples

This function provides the density for the inverse matrix gamma distribution.

1	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 `log=TRUE`, then the logarithm of the density is returned.

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)