next up previous
Next: for MRF prior Up: Noise Model MCMC Sampling Previous: for ARD prior

$ \beta _{ij}$ (Spatially non-stationary spatial model) with MRF prior

$\displaystyle \beta_{ij}\vert.\sim N(B_{\beta_{ij}}/A_{\beta_{ij}},1/A_{\beta_{ij}})$ (37)

where:
$\displaystyle A_{\beta_{ij}}$ $\displaystyle =$ $\displaystyle \sum_{t}(\phi_{\epsilon_i}q_{j(t-1)}^2+\phi_{\epsilon_j}q_{i(t-1)}^2)
+\phi_{\beta}\sum_{L_{k\ell}\in{\cal N}_{L_{ij}}}1$  
$\displaystyle B_{\beta_{ij}}$ $\displaystyle =$ $\displaystyle \sum_{t}
\left[\phi_{\epsilon_i}q_{j(t-1)}
\left(
q_{it}-\sum_{p=1}^P \alpha_{ip} q_{i(t-p)}
\right.
\right.$  
    $\displaystyle \left.
\left.
-\sum_{k\in{\cal N}_i:k\neq j} \beta_{ik} q_{k(t-1)}
\right)
\right.$  
    $\displaystyle \left.+ \phi_{\epsilon_j}q_{i(t-1)}
\left(
q_{jt}-\sum_{p=1}^P \alpha_{jp} q_{j(t-p)}
\right.
\right.$  
    $\displaystyle \left.
\left.
-\sum_{k\in{\cal N}_j:k\neq i} \beta_{jk} q_{k(t-1)}
\right)
\right]$  
    $\displaystyle + \phi_{\beta}\sum_{L_{k\ell}\in{\cal N}_{L_{ij}}} \beta_{k\ell}$ (38)

where $ L_{ij}$ is the link between voxel $ i$ and voxel $ j$.