Matrix t-distribution

${\displaystyle \mathbf {M} }$ location (real ${\displaystyle n\times p}$ matrix)
${\displaystyle {\boldsymbol {\Omega }}}$ scale (positive-definite real ${\displaystyle p\times p}$ matrix)
${\displaystyle {\boldsymbol {\Sigma }}}$ scale (positive-definite real ${\displaystyle n\times n}$ matrix)

${\displaystyle \nu }$

${\displaystyle {\frac {\Gamma _{p}\left({\frac {\nu +n+p-1}{2}}\right)}{(\pi )^{\frac {np}{2}}\Gamma _{p}\left({\frac {\nu +p-1}{2}}\right)}}|{\boldsymbol {\Omega }}|^{-{\frac {n}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {p}{2}}}}$

${\displaystyle \times \left|\mathbf {I} _{n}+{\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}\right|^{-{\frac {\nu +n+p-1}{2}}}}$

${\displaystyle \mathbf {M} }$ location (real ${\displaystyle n\times p}$ matrix)
${\displaystyle {\boldsymbol {\Omega }}}$ scale (positive-definite real ${\displaystyle p\times p}$ matrix)
${\displaystyle {\boldsymbol {\Sigma }}}$ scale (positive-definite real ${\displaystyle n\times n}$ matrix)
${\displaystyle \alpha >(p-1)/2}$ shape parameter

${\displaystyle \beta >0}$

${\displaystyle {\frac {\Gamma _{p}(\alpha +n/2)}{(2\pi /\beta )^{\frac {np}{2}}\Gamma _{p}(\alpha )}}|{\boldsymbol {\Omega }}|^{-{\frac {n}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {p}{2}}}}$

${\displaystyle \times \left|\mathbf {I} _{n}+{\frac {\beta }{2}}{\boldsymbol {\Sigma }}^{-1}(\mathbf {X} -\mathbf {M} ){\boldsymbol {\Omega }}^{-1}(\mathbf {X} -\mathbf {M} )^{\rm {T}}\right|^{-(\alpha +n/2)}}$

• ${\displaystyle \Gamma _{p}}$ is the multivariate gamma function.