The spatial map of discrete class labels is , where is the class label at spatial location . Assuming conditional independence of the likelihood, the full posterior distribution of the unknown parameters given the observed spatial map is:

where is a specific configuration (spatial map) of the class labels, and are any hyperparameters required to describe the prior on spatial map of class labels .

We consider three different mixture models, which are distinguished by their priors on ( ). These are non-spatial, with and without global class proportion parameters, and spatial mixture models.