Non-spatial without Class Proportions

We assume spatial independence between the classification labels, $x_i$, at each voxel:

$$p(\vec{x}=\vec{\kappa}\vert\vec{\lambda})=\prod_i^N p(x_i=\kappa_i\vert\vec{\lambda})$$ (2)

and that the prior on the discrete classification labels, $x_i$, is a non-informative distribution with each class having equal probability. Using this in equation 1, the posterior becomes:

$$p(\vec{x}=\vec{\kappa},\vec{\theta},\vec{\lambda}\vert\vec{y})\propto \prod_i^N \{p(y_i\vert x_i=\kappa_i,\theta_{\kappa_i})\}p(\vec{\theta})$$ (3)