$\alpha _{pi}$ with MRF prior

$$\alpha_{pi}\vert.\sim N(B_{\alpha_{pi}}/A_{\alpha_{pi}},1/A_{\alpha_{pi}})$$ (44)

where

$$A_{\alpha_{pi}} = \phi_{\epsilon_i}\sum_{t=1}^T q_{i(t-p)}^2 + \phi_{\alpha_p}\sum_{j\in{\cal N}_{i}}1$$ (45)

$$B_{\alpha_{pi}} = \phi_{\epsilon_i}\sum_{t=1}^T \left[ \left( q_{it}- \sum_{d=1}^D \left( \sum_{j\in{\cal N}_{i}} \beta_{ij} q_{j(t-1)} \right) \right. \right.$$

$$\qquad\qquad \left. \left. -\sum_{p' \neq p} \alpha_{p'i} q_{i(t-p')} \right) q_{i(t-p)} \right]$$

$$+ \phi_{\alpha_p}\sum_{j\in{\cal N}_{i}} \alpha_{pj}$$ (46)