Joint Probability Density Function
$$
\begin{align*}
p_{GB} (\boldsymbol{x}, \boldsymbol{h},\boldsymbol{\Theta}) =& \frac{1}{Z_{GB}} e^{-\vert\vert\frac{\boldsymbol{x} -\boldsymbol{a}}{2\boldsymbol{\sigma}}\vert\vert^2 + \boldsymbol{b}^T \boldsymbol{h}
+ (\frac{\boldsymbol{x}}{\boldsymbol{\sigma}^2})^T \boldsymbol{W}\boldsymbol{h}} \nonumber \\
=& \frac{1}{Z_{GB}} e^{- \sum_i^M \frac{(x_i - a_i)^2}{2\sigma_i^2}
+ \sum_j^N b_j h_j
+\sum_{ij}^{M,N} \frac{x_i w_{ij} h_j}{\sigma_i^2}} \nonumber \\
=& \frac{1}{Z_{GB}} \prod_{ij}^{M,N} e^{-\frac{(x_i - a_i)^2}{2\sigma_i^2}
+ b_j h_j
+\frac{x_i w_{ij} h_j}{\sigma_i^2}}.
\end{align*}
$$