\[\mathbf{X} \sim {\rm T}_{n,p}(\alpha,\beta,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega)\]
\[f(\mathbf{X} ; \nu,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega)=\frac{\Gamma_p(\alpha+n/2)}{(2\pi/\beta)^\frac{np}{2} \Gamma_p(\alpha)} |\boldsymbol\Omega|^{-\frac{n}{2}} |\boldsymbol\Sigma|^{-\frac{p}{2}}\times \left|\mathbf{I}_n + \frac{\beta}{2}\boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right|^{-(\alpha+n/2)}\]
\[S\sim W_p(V,n).\]
\[f_{\mathbf X}(\mathbf x) = \frac{|\mathbf{x}|^{(n-p-1)/2} e^{-\operatorname{tr}(\mathbf{V}^{-1}\mathbf{x})/2}}{2^\frac{np}{2}|{\mathbf V}|^{n/2}\Gamma_p(\frac n 2)}\]
\[\mathbf{X}\sim \mathcal{W}^{-1}(\mathbf\Psi,\nu)\]
\[f_{\mathbf x}({\mathbf x}; {\mathbf \Psi}, \nu)=\frac{\left|\mathbf\Psi\right|^{\nu/2}}{2^{\nu p/2}\Gamma_p(\frac{\nu}{2})} \left|\mathbf{x}\right|^{-(\nu+p+1)/2}e^{-\frac{1}{2}\operatorname{tr}(\mathbf\Psi\mathbf{x}^{-1})}\]
\[\mathbf{X} \sim \mathcal{MN}_{n\times p}(\mathbf{M}, \mathbf{U}, \mathbf{V}),\]
if and only if
\[\mathrm{vec}(\mathbf{X}) \sim \mathcal{N}_{np}(\mathrm{vec}(\mathbf{M}), \mathbf{V} \otimes \mathbf{U})\]
\[p(\mathbf{X}\mid\mathbf{M}, \mathbf{U}, \mathbf{V}) = \frac{\exp\left( -\frac{1}{2} \, \mathrm{tr}\left[ \mathbf{V}^{-1} (\mathbf{X} - \mathbf{M})^{T} \mathbf{U}^{-1} (\mathbf{X} - \mathbf{M}) \right] \right)}{(2\pi)^{np/2} |\mathbf{V}|^{n/2} |\mathbf{U}|^{p/2}}\]