In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution.

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Notation ${\displaystyle {\mathcal {W}}^{-1}({\mathbf {\Psi } },\nu )}$ ${\displaystyle \nu >p-1}$ degrees of freedom (real)${\displaystyle \mathbf {\Psi } >0}$, ${\displaystyle p\times p}$ scale matrix (pos. def.) ${\displaystyle \mathbf {X} }$ is p × p positive definite ${\displaystyle {\frac {\left|\mathbf {\Psi } \right|^{\nu /2}}{2^{\nu p/2}\Gamma _{p}({\frac {\nu }{2}})}}\left|\mathbf {x} \right|^{-(\nu +p+1)/2}e^{-{\frac {1}{2}}\operatorname {tr} (\mathbf {\Psi } \mathbf {x} ^{-1})}}$ ${\displaystyle \Gamma _{p}}$ is the multivariate gamma function ${\displaystyle \operatorname {tr} }$ is the trace function ${\displaystyle {\frac {\mathbf {\Psi } }{\nu -p-1}}}$For ${\displaystyle \nu >p+1}$ ${\displaystyle {\frac {\mathbf {\Psi } }{\nu +p+1}}}$[1]: 406 see below
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We say ${\displaystyle \mathbf {X} }$ follows an inverse Wishart distribution, denoted as ${\displaystyle \mathbf {X} \sim {\mathcal {W}}^{-1}(\mathbf {\Psi } ,\nu )}$, if its inverse ${\displaystyle \mathbf {X} ^{-1}}$ has a Wishart distribution ${\displaystyle {\mathcal {W}}(\mathbf {\Psi } ^{-1},\nu )}$. Important identities have been derived for the inverse-Wishart distribution.[2]

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