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 \times 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 \times 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) = \frac{|\Psi|^\alpha}{\beta^{k \alpha} \Gamma_k(\alpha)} |\theta|^{-\alpha-(k+1)/2}\exp(tr(-\frac{1}{\beta}\Psi\theta^{-1}))

• Inventors: Unknown

• Notation 1: \theta \sim \mathcal{IMG}_k(\alpha, \beta, \Psi)

• Notation 2: p(\theta) = \mathcal{IMG}_k(\theta | \alpha, \beta, \Psi)

• Parameter 1: shape \alpha > 2

• Parameter 2: scale \beta > 0

• Parameter 3: positive-definite k \times 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

dinvgamma dmatrixnorm, dmvn, and dinvwishart
library(LaplacesDemon)