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Next: Spatial Up: Discrete Labels Mixture Model Previous: Non-spatial without Class Proportions

Non-spatial with Class Proportions

Again, we assume spatial independence between the classification labels as in equation 2. However, taking $ \vec{\lambda}=\vec{\pi}=\{\pi_k:k=1 \ldots K\}$, where $ \pi_k$ are the adaptive global class proportion parameters, the prior on $ x_i$ is now:

$\displaystyle p(x_i=k\vert\vec{\pi})=\pi_k$ (4)

The global class proportions, $ \pi_k$, are the relative weighting of each of the distributions in the mixture. The prior on $ \vec{\pi}$ is non-informative (uniform) over the range $ 0<\pi_k<1$, and $ \sum_k^K \pi_k=1$. Using this in equation 1, the posterior becomes:

$\displaystyle p(\vec{x}=\vec{\kappa},\vec{\theta},\vec{\lambda}\vert\vec{y})\pr...
...appa_i} p(y_i\vert x_i=\kappa_i,\theta_{\kappa_i})\}p(\vec{\theta})p(\vec{\pi})$ (5)