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

## Description

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

## Usage

 `1` ```dinvmatrixgamma(X, alpha, beta, Psi, log=FALSE) ```

## Arguments

 `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.

## Details

• 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.

## Value

`dinvmatrixgamma` gives the density.

## Author(s)

Statisticat, LLC. [email protected]

`dinvgamma` `dmatrixnorm`, `dmvn`, and `dinvwishart`
 ```1 2 3 4``` ```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) ```