Matrix gamma distribution (original) (raw)

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Generalization of gamma distribution

Matrix gamma
Notation M G p ( α , β , Σ ) {\displaystyle {\rm {MG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Sigma }})} {\displaystyle {\rm {MG}}_{p}(\alpha ,\beta ,{\boldsymbol {\Sigma }})}
Parameters α > p − 1 2 {\displaystyle \alpha >{\frac {p-1}{2}}} {\displaystyle \alpha >{\frac {p-1}{2}}} shape parameter (real) β > 0 {\displaystyle \beta >0} {\displaystyle \beta >0} scale parameter Σ {\displaystyle {\boldsymbol {\Sigma }}} {\displaystyle {\boldsymbol {\Sigma }}} scale (positive-definite real p × p {\displaystyle p\times p} {\displaystyle p\times p} matrix)
Support X {\displaystyle \mathbf {X} } {\displaystyle \mathbf {X} } positive-definite real p × p {\displaystyle p\times p} {\displaystyle p\times p} matrix
PDF | Σ − α β p α Γ p ( α ) X α − p + 1 2 exp ⁡ ( t r ( − 1 β Σ − 1 X ) ) {\displaystyle {\frac { {\boldsymbol {\Sigma }} ^{-\alpha }}{\beta ^{p\alpha }\,\Gamma _{p}(\alpha )}}

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices.[1] It is effectively a different parametrization 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.[1]

A matrix gamma distributions is identical to a Wishart distribution with β Σ = 2 V , α = n 2 . {\displaystyle \beta {\boldsymbol {\Sigma }}=2V,\alpha ={\frac {n}{2}}.} {\displaystyle \beta {\boldsymbol {\Sigma }}=2V,\alpha ={\frac {n}{2}}.}

Notice that the parameters β {\displaystyle \beta } {\displaystyle \beta } and Σ {\displaystyle {\boldsymbol {\Sigma }}} {\displaystyle {\boldsymbol {\Sigma }}} are not identified; the density depends on these two parameters through the product β Σ {\displaystyle \beta {\boldsymbol {\Sigma }}} {\displaystyle \beta {\boldsymbol {\Sigma }}}.

  1. ^ a b Iranmanesh, Anis, M. Arashib and S. M. M. Tabatabaey (2010). "On Conditional Applications of Matrix Variate Normal Distribution". Iranian Journal of Mathematical Sciences and Informatics, 5:2, pp. 33–43.