Notation M
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{\displaystyle {\rm {MG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Sigma }})} Parameters shape parameter (real)
β
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0
{\displaystyle \beta >0}
scale parameter
Σ
{\displaystyle {\boldsymbol {\Sigma }}}
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 |
Σ
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−
α
β
p
α
Γ
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|
X
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−
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/
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exp
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−
1
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X
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{\displaystyle {\frac {|{\boldsymbol {\Sigma }}|^{-\alpha }}{\beta ^{p\alpha }\Gamma _{p}(\alpha )}}|\mathbf {X} |^{\alpha -(p+1)/2}\exp \left({\rm {tr}}\left(-{\frac {1}{\beta }}{\boldsymbol {\Sigma }}^{-1}\mathbf {X} \right)\right)}
Γ
p
{\displaystyle \Gamma _{p}}
is the multivariate gamma function. | ||
In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices. It is a more general version of the Wishart distribution, and is used similarly, e.g. as the conjugate prior of the precision matrix of a multivariate normal distribution and matrix normal distribution. The compound distribution resulting from compounding a matrix normal with a matrix gamma prior over the precision matrix is a generalized matrix t-distribution.
This reduces to the Wishart distribution with
References
Matrix gamma distribution Wikipedia(Text) CC BY-SA