Inverse matrix gamma distribution

{{one source |date=April 2024}}

{{Probability distribution|

name =Inverse matrix gamma|

type =density|

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notation =$\{\backslash rm\; IMG\}\_\{p\}(\backslash alpha,\backslash beta,\backslash boldsymbol\backslash Psi)$|

parameters = $\backslash alpha\; >\; (p\; -\; 1)/2$ shape parameter

$\backslash beta\; >\; 0$ scale parameter

$\backslash boldsymbol\backslash Psi$ scale (positive-definite real $p\backslash times\; p$ matrix)

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support =$\backslash mathbf\{X\}$ positive-definite real $p\backslash times\; p$ matrix|

pdf =$\backslash frac\{|\backslash boldsymbol\backslash Psi|^\{\backslash alpha\}\}\{\backslash beta^\{p\backslash alpha\}\backslash Gamma\_p(\backslash alpha)\}\; |\backslash mathbf\{X\}|^\{-\backslash alpha-(p+1)/2\}\backslash exp\backslash left(-\backslash frac\{1\}\{\backslash beta\}\{\backslash rm\; tr\}\backslash left(\backslash boldsymbol\backslash Psi\backslash mathbf\{X\}^\{-1\}\backslash right)\backslash right)$

• $\backslash Gamma\_p$ is the multivariate gamma function.|

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In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices.{{cite journal |last=Iranmanesha |first=Anis |first2=M. |last2=Arashib |first3=S. M. M. |last3=Tabatabaeya |year=2010 |title=On Conditional Applications of Matrix Variate Normal Distribution |journal=Iranian Journal of Mathematical Sciences and Informatics |volume=5 |issue=2 |pages=33–43 |url=https://www.sid.ir/En/Journal/ViewPaper.aspx?ID=220524 }} 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.{{citation needed|date=May 2012}}

This reduces to the inverse Wishart distribution with $\backslash nu$ degrees of freedom when $\backslash beta=2,\; \backslash alpha=\backslash frac\{\backslash nu\}\{2\}$.