Class distribution parameters, $\vec{\theta}$

This is sampled from using Metropolis-Hastings. With Metropolis-Hastings, a parameter change is proposed and then accepted or rejected according to the standard Metropolis-Hastings rule. This requires that we recalculate the terms in the joint posterior that change when we change $\vec{\theta}$. These terms are p($\hbox{{\protect\boldmath ${\theta}$}}$) for all three models plus for model 1:

$$\prod_i^N \sum_{k=1}^K \{ \pi_k w_{ik}p(y_i\vert x_i=k,\theta_k)\}$$ (29)

or for models 2 and 3:

$$\prod_i^N \sum_{k=1}^K \{ w_{ik}p(y_i\vert x_i=k,\theta_k)\}$$ (30)