Marginalisation over Areas of No Interest

Marginalisation over voxels wholly in the areas of no interest proceeds exactly as in the case of a flat intensity prior, since the multi-variate prior here does not include these parameters. These parameters therefore split into null, degenerate, and uninteresting. The degenerate and uninteresting parameters integrate in the same way as for the flat prior case and the null parameters integrate as part of $D_{null,u}$ in the previous section.

Specifically, this gives

$$\int \exp\left( \frac{-\beta}{2} (Y- G \alpha)^{\mathrm{\textsf{T... ...extsf{T}}}Q \alpha_{in} \right) d\alpha_{null} \, d\alpha_{un} \, d\alpha_{deg}$$

$$\quad = C_1^{-D_{null,u}-D_{deg}} \left(\frac{\lambda}{\pi}\right... ... - \lambda \alpha_{in}^{\mathrm{\textsf{T}}}Q' \alpha_{in} \right) d\alpha_{un}$$

$$\quad = C_1^{-D_{null,u}-D_{deg}} \left(\frac{\lambda}{\pi}\right... ...t\left( G_{un}^{\mathrm{\textsf{T}}}G_{un} \right) \right\vert^{-1/2} \; \times$$

$$\qquad \qquad \qquad \exp\left( \frac{-\beta}{2} (Y- G \alpha)^{\... ...Y - G \alpha) - \lambda \alpha_{in}^{\mathrm{\textsf{T}}}Q' \alpha_{in} \right)$$ (10)

where $Q_{null}$ stands for the square part of $Q$ associated with the shapes of interest that are currently in the null space (totally outside the field of view), and $R_{un} = I - G_{un} ( G_{un}^{\mathrm{\textsf{T}}}G_{un} )^{-1} G_{un}^{\mathrm{\textsf{T}}}$.

$$p(T\vert Y,S,Q,\beta,\lambda) \propto p(T) \, C_1^{D_{un}} \, \vert\det(Q)\vert^{1/2} \vert\det(Q_{null... ...ht)^{(N-D_{in}-D_{un})/2} \left(\frac{\lambda}{\pi}\right)^{D_{in}/2} \, \times$$

$$\qquad \left\vert\det\left( G_{un}^{\mathrm{\textsf{T}}}G_{un} \r... ...\; \exp\left( \frac{-\beta Y^{\mathrm{\textsf{T}}}R_{un} R R_{un} Y}{2} \right)$$

where $Q' = Q_{in} - Q_{cross}^{\mathrm{\textsf{T}}}Q_{null}^{-1} Q_{cross}$, $R_{un} = I - G_{un} ( G_{un}^{\mathrm{\textsf{T}}}G_{un} )^{-1} G_{un}^{\mathrm{\textsf{T}}}$ and
$R = I - R_{un} G_{in} \left[G_{in}^{\mathrm{\textsf{T}}}R_{un} G_{in} + \frac{2\lambda}{\beta} Q'\right]^{-1} G_{in}^{\mathrm{\textsf{T}}}R_{un}$.