Notation I
M
G
p
(
α
,
β
,
Ψ
)
{\displaystyle {\rm {IMG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Psi }})} Parameters α
{\displaystyle \alpha }
shape parameter (real)
β
>
0
{\displaystyle \beta >0}
scale parameter
Ψ
{\displaystyle {\boldsymbol {\Psi }}}
scale (positive-definite real
p
×
p
{\displaystyle p\times p}
matrix) Support X
{\displaystyle \mathbf {X} }
positive-definite real
p
×
p
{\displaystyle p\times p}
matrix PDF |
Ψ
|
α
β
p
α
Γ
p
(
α
)
|
X
|
−
α
−
(
p
+
1
)
/
2
exp
(
t
r
(
−
1
β
Ψ
X
−
1
)
)
{\displaystyle {\frac {|{\boldsymbol {\Psi }}|^{\alpha }}{\beta ^{p\alpha }\Gamma _{p}(\alpha )}}|\mathbf {X} |^{-\alpha -(p+1)/2}\exp \left({\rm {tr}}\left(-{\frac {1}{\beta }}{\boldsymbol {\Psi }}\mathbf {X} ^{-1}\right)\right)}
Γ
p
{\displaystyle \Gamma _{p}}
is the multivariate gamma function. | ||
In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices. It is a more general version of the inverse Wishart distribution, and is used similarly, e.g. as the conjugate prior of the covariance matrix of a multivariate normal distribution or matrix normal distribution. 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.
This reduces to the inverse Wishart distribution with
References
Inverse matrix gamma distribution Wikipedia(Text) CC BY-SA