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Non-spatial with Class Proportions

We can approximate the distribution in equation 5 by replacing the discrete labels, $ x_i$, with continuous weights vectors, $ \vec{w_i}$:

$\displaystyle p(\vec{w},\vec{\theta},\vec{\pi}\vert\vec{y}) \propto \prod_i^N \...
... \pi_k w_{ik}p(y_i\vert x_i=k,\theta_k)\} p(\vec{w})p(\vec{\theta})p(\vec{\pi})$ (13)

where $ \pi_k$ are the global class proportion parameters defined in equation 4 and represent the proportion of each class in the mixture model, and $ p(\vec{w})$ is given by equations 10-12. Note that when we infer on the posterior, $ \vec{\pi}$ will depend upon $ \vec{w}$ in same way that $ \vec{\pi}$ depends on $ \vec{x}$ in the discrete labels mixture model.