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

or for models 2 and 3:
    $\displaystyle \prod_i^N \sum_{k=1}^K \{ w_{ik}p(y_i\vert x_i=k,\theta_k)\}$ (30)