dist.Inverse.Matrix.Gamma | R Documentation |
This function provides the density for the inverse matrix gamma distribution.
dinvmatrixgamma(X, alpha, beta, Psi, 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,
|
Psi |
This is a |
log |
Logical. If |
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
.
dinvmatrixgamma
gives the density.
Statisticat, LLC. software@bayesian-inference.com
dinvgamma
dmatrixnorm
,
dmvn
, and
dinvwishart
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)
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