Markov Random Field (MRF) prior

We would like to consider a model which spatially regularises the AR parameters. To do this we use a Markov Random Field (MRF) pairwise difference Gaussian prior (37) on the field $\alpha_p\equiv(\alpha_{pi})$:

$$p(\alpha_p\vert\phi_{\alpha_p}) \propto \phi_{\alpha_p}^{N/2} \exp \left\{ \frac{-\phi_{\alpha_p}}{4}\sum_i \sum_{j\in{\cal N}_{i}} (\alpha_{pi}-\alpha_{pj})^2 \right\}$$ (14)

where $j\in{\cal N}_{i}$ is the set of neighbouring voxels to voxel $i$ (for this we use 26-connectivity in 3D). Note that the parameter controlling the amount of spatial regularisation, $\phi _{\alpha _p}$, is determined adaptively from the data (see section 3.3.6).