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

## Description

This function provides the density for the matrix gamma distribution.

## Usage

 `1` ```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

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

## Examples

 ```1 2 3 4``` ```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) ```

### Example output

``` -82.3189
 -82.3189
```

LaplacesDemon documentation built on July 9, 2021, 5:07 p.m.