LaplacesDemon (version 16.1.1)

dist.Inverse.Matrix.Gamma: 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.

Value

dinvmatrixgamma gives the density.

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 \(\beta = 2\).

See Also

dinvgamma dmatrixnorm, dmvn, and dinvwishart

Examples

# NOT RUN {
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)
# }