LaplacesDemon (version 16.1.6)

dist.Matrix.Gamma: 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 $k\times k$ 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 $k\times k$ positive-definite scale matrix.

log

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

Value

dmatrixgamma gives the density.

Details

• Application: Continuous Multivariate Matrix

• Density: $p(\theta )=\frac{|\mathrm{\Sigma }{|}^{-\alpha }}{{\beta }^{k\alpha }{\mathrm{\Gamma }}_k(\alpha )}|\theta {|}^{\alpha -(k+1)/2}\exp (tr(-\frac{1}{\beta }{\mathrm{\Sigma }}^{-1}\theta ))$

• Inventors: Unknown

• Notation 1: $\theta \sim {\mathcal{MG}}_k(\alpha ,\beta ,\mathrm{\Sigma })$

• Notation 2: $p(\theta )={\mathcal{MG}}_k(\theta |\alpha ,\beta ,\mathrm{\Sigma })$

• Parameter 1: shape $\alpha >2$

• Parameter 2: scale $\beta >0$

• Parameter 3: positive-definite $k\times k$ scale matrix $\mathrm{\Sigma }$

• 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 $\beta =2$.

See Also

dgamma dmatrixnorm, dmvnp, and dwishart

Examples

# NOT RUN {
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)
# }