Density evaluation and random number generation for the Matrix-Normal Inverse-Wishart (MNIW) distribution, as well as its constituent distributions, i.e., the Matrix-Normal, Matrix-T, Wishart, and Inverse-Wishart distributions.
The Matrix-Normal Inverse-Wishart (MNIW) distribution \((\boldsymbol{X}, \boldsymbol{V}) \sim \textrm{MNIW}(\boldsymbol{\Lambda}, \boldsymbol{\Sigma}, \boldsymbol{\Psi}, \nu)\) on random matrices \(\boldsymbol{X}_{p \times q}\) and symmetric positive-definite \(\boldsymbol{V}_{q\times q}\) is defined as $$ \begin{array}{rcl} \boldsymbol{V} & \sim & \textrm{Inverse-Wishart}(\boldsymbol{\Psi}, \nu) \\ \boldsymbol{X} \mid \boldsymbol{V} & \sim & \textrm{Matrix-Normal}(\boldsymbol{\Lambda}, \boldsymbol{\Sigma}, \boldsymbol{V}), \end{array} $$ where the Matrix-Normal distribution is defined as the multivariate normal $$ \textrm{vec}(\boldsymbol{X}) \sim \mathcal{N}(\textrm{vec}(\boldsymbol{\Lambda}), \boldsymbol{V} \otimes \boldsymbol{\Sigma}), $$ where \(\textrm{vec}(\boldsymbol{X})\) is a vector stacking the columns of \(\boldsymbol{X}\), and \(\boldsymbol{V} \otimes \boldsymbol{\Sigma}\) denotes the Kronecker product.
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