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Concept# Inverse-Wishart distribution

Summary

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.
We say \mathbf{X} follows an inverse Wishart distribution, denoted as \mathbf{X}\sim \mathcal{W}^{-1}(\mathbf\Psi,\nu), if its inverse \mathbf{X}^{-1} has a Wishart distribution \mathcal{W}(\mathbf \Psi^{-1}, \nu) . Important identities have been derived for the inverse-Wishart distribution.
Density
The probability density function of the inverse Wishart is:
:
f_{\mathbf X}({\mathbf X}; {\mathbf \Psi}, \nu) = \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})}
where \mathbf{X}

Official source

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