Continuous weights parameters, $\vec{w_i}$

This is sampled from using Metropolis-Hastings. We loop through all voxels sampling from the continuous weights vector a voxel at a time. Recall that $p(\vec{w_i}\vert\vec{\tilde{w}_i},\gamma)$ is the deterministic logistic transform in equation 12 at voxel $i$. The logistic transform can only take us from a vector $\vec{\tilde{w}_i}$ to a vector $\vec{w_i}$. Hence, in practice we propose jumps on $\vec{\tilde{w}_i}$ and then calculate the vector $\vec{w_i}$ this gives us. Then, for model 1 we need to recalculate:

$$\sum_{k=1}^K \{ \pi_k w_{ik}p(y_i\vert x_i=k,\theta_k)\}$$ (32)

or for models 2 and 3:

$$\sum_{k=1}^K \{ w_{ik}p(y_i\vert x_i=k,\theta_k)\}$$ (33)

Plus for models 2 and 3 we also need to update the parts of the MRF prior that change when we change $\vec{w_i}$. These are:

$$\prod_k exp \left( -\frac{\phi_{\tilde{w}}}{2} \sum_{j\in {\cal N}_i} (\tilde{w}_{ik}-\tilde{w}_{jk})^2 \right)$$ (34)

where ${\cal N}_i$ is the set of voxels in the spatial neighbourhood of voxel $i$.