Marginalising over Null Parameters

As long as the null parameters are associated with areas of no interest, then they can be assigned a flat prior and integrated as shown in the previous section. However, when a shape of interest moves out of the field of view and its parameters become null parameters, then this requires special attention as the prior is no longer separable into null and non-null terms.

Specifically, consider the prior term, $\exp( - \lambda \alpha^{\mathrm{\textsf{T}}}Q \alpha )$, and rewrite the prior as:

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

This gives

$$\int \exp\left( - \lambda \alpha^{\mathrm{\textsf{T}}}Q \alpha \right) \; d\alpha_{null} = \int \exp\left( - \lambda (\alpha_{in}^{\mathrm{\textsf{T}}}Q_{in... ..._{null}^{\mathrm{\textsf{T}}}Q_{null} \alpha_{null} ) \right) \; d\alpha_{null}$$

$$= \int \exp\left( - \lambda (\alpha_{null} + Q_{null}^{-1} Q_{cross... ...xtsf{T}}}Q_{null} (\alpha_{null} + Q_{null}^{-1} Q_{cross} \alpha_{in}) \right)$$

$$\qquad \qquad \exp\left( - \lambda (\alpha_{in}^{\mathrm{\textsf{... ...thrm{\textsf{T}}}Q_{null}^{-1} Q_{cross} \alpha_{in}) \right) \; d\alpha_{null}$$

$$= C_1^{-D_{null,u}} \left(\frac{\lambda}{\pi}\right)^{-D_{null,i}/2... ... - Q_{cross}^{\mathrm{\textsf{T}}}Q_{null}^{-1} Q_{cross} ) \alpha_{in} \right)$$

where $D_{null,i}$ are the number of null parameters associated with shapes of interest, and $D_{null,u}$ are the number of null parameters associated with areas of no interest, such that $D_{null} = D_{null,i} + D_{null,u}$.

The resulting form is still a multi-variate Gaussian, and the effect on the previous formula is to modify $Q$ by replacing it with its reduced form, $Q' = Q_{in} - Q_{cross}^{\mathrm{\textsf{T}}}Q_{null}^{-1} Q_{cross}$ and to pre-multiply the posterior by the factor

$$C_1^{-D_{null,u}} \left(\frac{\lambda}{\pi}\right)^{-D_{null,i}/2} \vert\det(Q_{null})\vert^{-1/2}.$$

where $Q_{null} = Q_{null}$ is the submatrix of $Q$ associated with the null parameters.

Note that because the null space of $G$ depends on both the underlying shape models, $S_k$, and the transformation, $T$, both $D_{null}$ and $Q_{null}$ depend on these and so this factor will not be a constant in the similarity function.

The posterior is now in the form

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

$$\qquad \qquad \left\vert\det\left(G_{in}^{\mathrm{\textsf{T}}}G_{... ...t\vert^{-1/2} \; \exp\left( \frac{-\beta Y^{\mathrm{\textsf{T}}}R Y}{2} \right)$$

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

The factor depending on $C_1$ does not appear, as it is cancelled by the prior, $p(\alpha_{null,u})=C_1^{D_{null,u}}$, and the normalisation of the prior $p(\alpha \vert \lambda , Q)$ includes $(\lambda/\pi)^{(D_{in} + D_{null,i})/2}$ which cancels the other term.