R: Inverse Matrix Gamma Distribution

dist.Inverse.Matrix.Gamma {LaplacesDemon} | R Documentation

## Inverse Matrix Gamma Distribution

### Description

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

### Usage

    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. software@bayesian-inference.com

### See Also

`dinvgamma` `dmatrixnorm`, `dmvn`, and `dinvwishart`

### Examples

    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)

[Package LaplacesDemon version 16.1.1 Index]