Multi-variate Gaussian Prior

$$p(T\vert Y,S,Q,\beta,\lambda) \propto p(T) \, C_1^N \, \vert\det(Q)\vert^{1/2} \vert\det(Q_{null})\vert... ...v,u})/2} \left( \frac{2 \lambda}{\beta} \right)^{(D_{in}+D_{pv,i})/2} \, \times$$

$$\qquad \left\vert\det\left( G_{un}^{\mathrm{\textsf{T}}}G_{un} \r... ...}}}R_{un} G_{in} + \frac{2\lambda}{\beta} Q'\right)\right\vert^{-1/2} \; \times$$

$$\qquad \exp\left( \frac{-\beta}{2} Y^{\mathrm{\textsf{T}}}R_{un} ... ... q_j^{-1} w_j \right) \, \left( \prod_{j=1}^{D_{pv,i}} {q'}_j^{-1} w'_j \right)$$

where
$R_{pv,u} = I - R^{1/2} R_{un} G_{pv,u} ( G_{pv,u}^{\mathrm{\textsf{T}}}R_{un} R R_{un} G_{pv,u})^{-1} G_{pv,u}^{\mathrm{\textsf{T}}}R^{1/2} R_{un}$;
$R_{pv} = I - R_{pv,u} R^{1/2} R_{un} G_{pv,i} ( G_{pv,i}^{\mathrm{\textsf{T}}}... .../2} R_{un} G_{pv,i})^{-1} G_{pv,i}^{\mathrm{\textsf{T}}}R_{pv,u} R^{1/2} R_{un}$;
$q_j^2$ is the $j$th diagonal of $( G_{pv,u}^{\mathrm{\textsf{T}}}R_{un} R R_{un} G_{pv,u})$;
${q'}_j^2$ is the $j$th diagonal of $( G_{pv,i}^{\mathrm{\textsf{T}}}R_{un} R^{1/2} R_{pv,u} R^{1/2} R_{un} G_{pv,i} + (2\lambda / \beta) Q'')$;

$$w_j = \frac{1}{2} \ensuremath{\mathrm{erfc}}\left( - \left(\frac{... ...eta}{2}\right)^{1/2} \left(q_j C_1^{-1} - (R^{1/2} R_{un} Y)_j \right) \right)$$

and

$$w'_j = \frac{1}{2} \ensuremath{\mathrm{erfc}}\left( - \left(\frac... ...)^{1/2} \left({q'}_j C_1^{-1} - (R_{pv,u} R^{1/2} R_{un} Y)_j \right) \right).$$