We proceed to find the marginal probability densitites of the Gaussian-binary RBM. We first marginalize over the binary hidden units to find \( p_{GB} (\boldsymbol{x}) \)
$$ \begin{align*} p_{GB} (\boldsymbol{x}) =& \sum_{\tilde{\boldsymbol{h}}}^{\tilde{\boldsymbol{H}}} p_{GB} (\boldsymbol{x}, \tilde{\boldsymbol{h}}) \nonumber \\ =& \frac{1}{Z_{GB}} \sum_{\tilde{\boldsymbol{h}}}^{\tilde{\boldsymbol{H}}} e^{-\vert\vert\frac{\boldsymbol{x} -\boldsymbol{a}}{2\boldsymbol{\sigma}}\vert\vert^2 + \boldsymbol{b}^T \tilde{\boldsymbol{h}} + (\frac{\boldsymbol{x}}{\boldsymbol{\sigma}^2})^T \boldsymbol{W}\tilde{\boldsymbol{h}}} \nonumber \\ =& \frac{1}{Z_{GB}} e^{-\vert\vert\frac{\boldsymbol{x} -\boldsymbol{a}}{2\boldsymbol{\sigma}}\vert\vert^2} \prod_j^N (1 + e^{b_j + (\frac{\boldsymbol{x}}{\boldsymbol{\sigma}^2})^T \boldsymbol{w}_{\ast j}} ) . \end{align*} $$