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Discrete Labels Mixture Model

The spatial map of discrete class labels is $ \vec{x}$, where $ x_i$ is the class label at spatial location $ i$. Assuming conditional independence of the likelihood, the full posterior distribution of the unknown parameters given the observed spatial map is:

$\displaystyle p(\vec{x}=\vec{\kappa},\vec{\theta},\vec{\lambda}\vert\vec{y})\pr...
..._i})\} p(\vec{x}=\vec{\kappa}\vert\vec{\lambda})p(\vec{\lambda})p(\vec{\theta})$ (1)

where $ \vec{\kappa}$ is a specific configuration (spatial map) of the class labels, and $ \vec{\lambda}$ are any hyperparameters required to describe the prior on spatial map of class labels $ \vec{x}$.

We consider three different mixture models, which are distinguished by their priors on $ \vec{x}$ ( $ p(\vec{x}\vert\vec{\lambda})$). These are non-spatial, with and without global class proportion parameters, and spatial mixture models.