\begin{align}
&P(\{X^{(\ell)\mu}_{\beta,b}\}_{b\in P^{-1}(a)}|\{X^{(\ell-1)\mu}_{\beta b}\}_{b\in P^{-1}(a)}, X^{(\ell+1)\mu}_{\beta,a})=P(X^{(\ell+1)\mu}_{\beta a}|\{X^{(\ell)\mu}_{\beta,b}\}_{b\in P^{-1}(a)})\times\\&\times\prod_{b\in P^{-1}(a)}P(X^{(\ell)\mu}_{\beta b}|X^{(\ell-1)\mu}_{\beta b})\propto\exp\left[-\frac{1}{2\Delta^{(\ell)}_X}\sum_{b\in P^{-1}(a)}\left(X^{(\ell)\mu}_{\beta b}-\sigma(X^{(\ell-1)\mu}_{\beta b})\right)^2\right.\\&\left.-\frac{1}{2\Delta_X^{(\ell+1)}}\left(X^{(\ell+1)\mu}_{\beta a}-\frac{1}{|P^{-1}(a)|}\sum_{b\in P^{-1}(a)}X^{(\ell)\mu}_{\beta b}\right)
^2\right]\propto \\&\exp\left[-\frac{1}{2}\sum_{b,c\in P^{-1}(a)}X^{(\ell)\mu}_{\beta b}\left(\frac{1}{\Delta_X^{(\ell)}}\delta_{bc}+\frac{1}{\Delta_X^{(\ell+1)}|P^{-1}(a)|^2}\right)X^{(\ell)\mu}_{\beta c}+\right.\\&\left.+\sum_{b\in P^{-1}(a)} X^{(\ell)\mu}_{\beta b}\left(\frac{\sigma(X^{(\ell-1)\mu}_{\beta b})}{\Delta_X^{(\ell)}}+\frac{X^{(\ell+1)\mu}_{\beta a}}{\Delta_X^{(\ell+1)}|P^{-1}(a)|}\right)\right]\propto \mathcal{N}(\{X^{(\ell)\mu}_{\beta,b}\}_{b\in P^{-1}(a)}|m^{(\ell)}_X)^\mu_\beta,\Sigma^{(\ell)}_X)
\end{align}
