dist.Matrix.Gamma | R Documentation |
This function provides the density for the matrix gamma distribution.
dmatrixgamma(X, alpha, beta, Sigma, log=FALSE)
X |
This is a |
alpha |
This is a scalar shape parameter (the degrees of freedom),
|
beta |
This is a scalar, positive-only scale parameter,
|
Sigma |
This is a |
log |
Logical. If |
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
.
dmatrixgamma
gives the density.
Statisticat, LLC. software@bayesian-inference.com
dgamma
dmatrixnorm
,
dmvnp
, and
dwishart
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)
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