Marginalisation over Partial Volume Parameters

Pure partial volume parameters can again be split into those associated with shapes of interest, $\alpha_{pv,i}$, and those associated with areas of no interest, $\alpha_{pv,u}$. The former have non-flat priors and the latter have flat priors. Integration of the former requires more care, as they interact with the multi-variate prior. In order to do this, rewrite the appropriate matrices as

$$\alpha = \left[ \begin{array}{c} \alpha_{pv,i} \\ \alpha_{in} \\ ... ...^{\mathrm{\textsf{T}}}\\ Q_{b} & & Q_{cross} & & Q_{null} \end{array} \right].$$

Therefore the posterior integrations take the form

$$\int \exp \left( \frac{-\beta}{2} (Y - G \alpha)^{\mathrm{\textsf... ...Y - G \alpha) - \lambda \alpha^{\mathrm{\textsf{T}}}Q \alpha \right) \, d\alpha$$

$$\qquad \qquad = \int \exp \left( \frac{-\beta}{2} (Y - G_{pv,i} \... ...ambda \alpha_{pv,i}^{\mathrm{\textsf{T}}}Q_{pv} \alpha_{pv,i} \right) \; \times$$

$$\qquad \qquad \qquad \exp \left( \frac{-\beta}{2} (\alpha_{in}^{\... ...f{T}}}+ 2 \alpha_{null}^{\mathrm{\textsf{T}}}Q_c) \alpha_{in} \right) \; \times$$

$$\qquad \qquad \qquad \qquad \exp \left( - \lambda ( 2 \alpha_{pv,... ... ) \alpha_{null} \right) \; \, d\alpha_{pv,i} \; d\alpha_{in} \; d\alpha_{null}$$

Integrating with respect to $\alpha_{in}$ gives

$$\int \exp \left( \frac{-\beta}{2} (\alpha_{in}^{\mathrm{\textsf{T... ...+ 2 \alpha_{null}^{\mathrm{\textsf{T}}}Q_c) \alpha_{in} \right) \, d\alpha_{in}$$

$$\qquad \qquad = \int \exp \left( \frac{-\beta}{2} (\alpha_{in}^{\... ... \alpha_{in} + 2 B_0^{\mathrm{\textsf{T}}}\alpha_{in} ) \right) \, d\alpha_{in}$$

$$\qquad \qquad = \left( \frac{\beta}{2} \right)^{-D_{in}/2} \vert ... ...-1/2} \exp \left( \frac{\beta}{2} B_0^{\mathrm{\textsf{T}}}A_0^{-1} B_0 \right)$$

where $A_0 = G_{in}^{\mathrm{\textsf{T}}}G_{in} + \frac{2 \lambda}{\beta} Q_{in}$ and $B_0 = G_{in}^{\mathrm{\textsf{T}}}G_{pv,i} \alpha_{pv,i} - G_{in}^{\mathrm{\te... ...2\lambda}{\beta} ( Q_a \alpha_{pv,i} + Q_c^{\mathrm{\textsf{T}}}\alpha_{null} )$.

Substituting this back into the previous expression and integrating over $\alpha_{null}$ gives

$$\int \exp \left( \frac{-\beta}{2} (Y - G \alpha)^{\mathrm{\textsf... ...Y - G \alpha) - \lambda \alpha^{\mathrm{\textsf{T}}}Q \alpha \right) \, d\alpha$$

$$\quad = \left( \frac{\beta}{2} \right)^{-D_{in}/2} \vert \det(A_0... ...ambda \alpha_{pv,i}^{\mathrm{\textsf{T}}}Q_{pv} \alpha_{pv,i} \right) \; \times$$

$$\qquad \exp \left( \frac{\beta}{2} (Y^{\mathrm{\textsf{T}}}G_{in}... ...xtsf{T}}}G_{pv,i} + \frac{2\lambda}{\beta} Q_a) \alpha_{pv,i} \right) \; \times$$

$$\qquad \exp \left( - \lambda \left[ 2 Y^{\mathrm{\textsf{T}}}G_{i... ...xtsf{T}}}) \right] \alpha_{null} \right) \; \, d\alpha_{pv,i} \; d\alpha_{null}$$

$$\quad = \left( \frac{\beta}{2} \right)^{-D_{in}/2} \vert \det(A_0... ...thrm{\textsf{T}}}\alpha_{null} ) \right) \; \, d\alpha_{pv,i} \; d\alpha_{null}$$

$$\quad = \left( \frac{\beta}{2} \right)^{-D_{in}/2} \vert \det(A_0... ...eft( \lambda B_1^{\mathrm{\textsf{T}}}A_1^{-1} B_1 \right) \; \, d\alpha_{pv,i}$$

$$\quad = \left( \frac{\beta}{2} \right)^{-D_{in}/2} \lambda^{-D_{n... ...{pv,i} + 2 B_3^{\mathrm{\textsf{T}}}\alpha_{pv,i}) \right) \; \, d\alpha_{pv,i}$$

where

$$R_0 = I - G_{in} A_0^{-1} G_{in}^{\mathrm{\textsf{T}}}$$

$$A_0 = G_{in}^{\mathrm{\textsf{T}}}G_{in} + \frac{2 \lambda}{\beta} Q_{in}$$

$$A_1 = Q_{null} - \frac{2\lambda}{\beta} Q_c A_0^{-1} Q_c^{\mathrm{\textsf{T}}}$$

$$B_1^{\mathrm{\textsf{T}}} = Y^{\mathrm{\textsf{T}}}G_{in} A_0^{-1} Q_c^{\mathrm{\textsf{T}}}+... ...mbda}{\beta} Q_a^{\mathrm{\textsf{T}}}A_0^{-1} Q_c^{\mathrm{\textsf{T}}}\right)$$

$$A_2 = G_{pv,i}^{\mathrm{\textsf{T}}}G_{pv,i} + \frac{2\lambda}{\beta} Q... ...}}A_0^{-1} ( G_{in}^{\mathrm{\textsf{T}}}G_{pv,i} + \frac{2\lambda}{\beta} Q_a)$$

$$B_2^{\mathrm{\textsf{T}}} = Y^{\mathrm{\textsf{T}}}G_{in} A_0^{-1} (G_{in}^{\mathrm{\textsf{T}}}G_{pv,i} + \frac{2\lambda}{\beta} Q_a) - Y^{\mathrm{\textsf{T}}}G_{pv,i}$$

$$A_3 = A_2 - \frac{2\lambda}{\beta} \left( Q_b^{\mathrm{\textsf{T}}}- G_... ...hrm{\textsf{T}}}A_0^{-1} Q_c^{\mathrm{\textsf{T}}}\right)^{\mathrm{\textsf{T}}}$$

$$B_3^{\mathrm{\textsf{T}}} = B_2^{\mathrm{\textsf{T}}}+ Y^{\mathrm{\textsf{T}}}G_{in} A_0^{-1}... ...mbda}{\beta} Q_a^{\mathrm{\textsf{T}}}A_0^{-1} Q_c^{\mathrm{\textsf{T}}}\right)$$

$$R_1 = R_0 - \frac{2\lambda}{\beta} G_{in} A_0^{-1} Q_c^{\mathrm{\textsf{T}}}A_1^{-1} Q_c A_0^{-1} G_{in}^{\mathrm{\textsf{T}}}$$

However, as shown above, if all the previous integrations are performed first, then the remaining posterior takes the form

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

$$\qquad \left\vert\det\left( G_{un}^{\mathrm{\textsf{T}}}G_{un} \r... ...f{T}}}R_{un} G_1 + \frac{2\lambda}{\beta} Q'\right)\right\vert^{-1/2} \; \times$$

$$\qquad \qquad \int \exp\left( \frac{-\beta}{2} (Y - G_{pv} \alpha... ...athrm{\textsf{T}}}Q'' \alpha_{pv,i} \right) \, d\alpha_{pv,i} \, d\alpha_{pv,u}$$

where $R_{un} = I - G_{un} ( G_{un}^{\mathrm{\textsf{T}}}G_{un} )^{-1} G_{un}^{\mathrm{\textsf{T}}}$; $R = I - R_{un} G_1 \left[G_1^{\mathrm{\textsf{T}}}R_{un} G_1 + \frac{2\lambda}{\beta} Q'\right]^{-1} G_1^{\mathrm{\textsf{T}}}R_{un}$; $G_1 = [ \, G_{pv,i} \; \; G_{in} \, ]$; and

$$Q' = \left[ \begin{array}{ccc} Q_{pv} - Q_{b}^{\mathrm{\textsf{T... ...} - Q_{cross}^{\mathrm{\textsf{T}}}Q_{null}^{-1} Q_{cross} \end{array} \right]$$

and $Q'' = (Q_{pv} - Q_{b}^{\mathrm{\textsf{T}}}Q_{null}^{-1} Q_{b}) - (Q_{a} - Q_{... ...1} Q_{cross})^{-1} (Q_{a} - Q_{cross}^{\mathrm{\textsf{T}}}Q_{null}^{-1} Q_{b})$.

Assuming that $G_{pv,u}^{\mathrm{\textsf{T}}}R_{un} R R_{un} G_{pv,u}$ and $(G_{pv,i}^{\mathrm{\textsf{T}}}R_{un} R R_{un} G_{pv,i} + (2\lambda/\beta) {Q'}_{pv})$ both have negligible off-diagonal terms gives

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

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

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

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

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

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

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 $( G_{pv,i}^{\mathrm{\textsf{T}}}R_{un} R^{1/2} R_{pv,u} R^{1/2} R_{un} G_{pv,i} + (2\lambda / \beta) Q'')$;

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

and

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