Control parameter, $\phi_{\tilde{w}}$

This is only required in model 3. $\phi_{\tilde{w}}$ has a full conditional which can be sampled from. Hence for $\phi_{\tilde{w}}$ we can employ Gibbs sampling. The full conditional for $\phi_{\tilde{w}}$ is given by:

$$\phi_{\tilde{w}}\vert. \sim Ga\left( \frac{N}{2}+\tilde{a}_{\tilde{w}}, \frac{1}{4}\sum_i \su... ...\in{\cal N}_i} (\tilde{w}_{pi}-\tilde{w}_{pj})^2 +\tilde{b}_{\tilde{w}} \right)$$ (28)

$\tilde{a}_{\tilde{w}}$ is set to $10^{-4}$, and $\tilde{b}_{\tilde{w}}$ is set to $10^{-4}$ to give a disperse prior.