R: Matrix Gamma Distribution

dist.Matrix.Gamma {LaplacesDemon} | R Documentation

## Matrix Gamma Distribution

### Description

This function provides the density for the matrix gamma distribution.

### Usage

    dmatrixgamma(X, alpha, beta, Sigma, log=FALSE)

### Arguments

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

### Details

- 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.

### Value

`dmatrixgamma` gives the density.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

### See Also

`dgamma` `dmatrixnorm`, `dmvnp`, and `dwishart`

### Examples

    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)

[Package LaplacesDemon version 16.1.1 Index]