Inverse-Wishart distribution

Inverse-Wishart

Notation	${\displaystyle {\mathcal {W}}^{-1}({\mathbf {\Psi } },\nu )}$
Parameters	${\displaystyle \nu >p-1}$ degrees of freedom (real) ${\displaystyle \mathbf {\Psi } >0}$ scale matrix (pos. def)
Support	${\displaystyle \mathbf {X} }$ is positive definite
PDF	${\displaystyle {\frac {\left|{\mathbf {\Psi } }\right|^{\frac {\nu }{2}}}{2^{\frac {\nu p}{2}}\Gamma _{p}({\frac {\nu }{2}})}}\left|\mathbf {X} \right|^{-{\frac {\nu +p+1}{2}}}e^{-{\frac {1}{2}}\operatorname {tr} ({\mathbf {\Psi } }\mathbf {X} ^{-1})}}$ ${\displaystyle \Gamma _{p}}$ is the multivariate gamma function ${\displaystyle \mathrm {tr} }$ is the trace function
Mean	${\displaystyle {\frac {\mathbf {\Psi } }{\nu -p-1}}}$For ${\displaystyle \nu >p+1}$
Mode	${\displaystyle {\frac {\mathbf {\Psi } }{\nu +p+1}}}$
Variance	see below

${\displaystyle {\frac {\left|{\mathbf {\Psi } }\right|^{\frac {\nu }{2}}}{2^{\frac {\nu p}{2}}\Gamma _{p}({\frac {\nu }{2}})}}\left|\mathbf {X} \right|^{-{\frac {\nu +p+1}{2}}}e^{-{\frac {1}{2}}\operatorname {tr} ({\mathbf {\Psi } }\mathbf {X} ^{-1})}}$

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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