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:

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