dist.Matrix.Gamma {LaplacesDemon}

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.

Details

• Application: Continuous Multivariate Matrix

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

• Inventors: Unknown

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

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

• Parameter 1: shape \alpha > 2

• Parameter 2: scale \beta > 0

• Parameter 3: positive-definite k \times 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. [e-mail]

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)