We next marginalize over the visible units. This is the first time we marginalize over continuous values. We rewrite the exponential factor dependent on \( \boldsymbol{x} \) as a Gaussian function before we integrate in the last step.
$$ \begin{align*} p_{GB} (\boldsymbol{h}) =& \int p_{GB} (\tilde{\boldsymbol{x}}, \boldsymbol{h}) d\tilde{\boldsymbol{x}} \nonumber \\ =& \frac{1}{Z_{GB}} \int e^{-\vert\vert\frac{\tilde{\boldsymbol{x}} -\boldsymbol{a}}{2\boldsymbol{\sigma}}\vert\vert^2 + \boldsymbol{b}^T \boldsymbol{h} + (\frac{\tilde{\boldsymbol{x}}}{\boldsymbol{\sigma}^2})^T \boldsymbol{W}\boldsymbol{h}} d\tilde{\boldsymbol{x}} \nonumber \\ =& \frac{1}{Z_{GB}} e^{\boldsymbol{b}^T \boldsymbol{h} } \int \prod_i^M e^{- \frac{(\tilde{x}_i - a_i)^2}{2\sigma_i^2} + \frac{\tilde{x}_i \boldsymbol{w}_{i\ast}^T \boldsymbol{h}}{\sigma_i^2} } d\tilde{\boldsymbol{x}} \nonumber \\ =& \frac{1}{Z_{GB}} e^{\boldsymbol{b}^T \boldsymbol{h} } \biggl( \int e^{- \frac{(\tilde{x}_1 - a_1)^2}{2\sigma_1^2} + \frac{\tilde{x}_1 \boldsymbol{w}_{1\ast}^T \boldsymbol{h}}{\sigma_1^2} } d\tilde{x}_1 \nonumber \\ & \times \int e^{- \frac{(\tilde{x}_2 - a_2)^2}{2\sigma_2^2} + \frac{\tilde{x}_2 \boldsymbol{w}_{2\ast}^T \boldsymbol{h}}{\sigma_2^2} } d\tilde{x}_2 \nonumber \\ & \times ... \nonumber \\ &\times \int e^{- \frac{(\tilde{x}_M - a_M)^2}{2\sigma_M^2} + \frac{\tilde{x}_M \boldsymbol{w}_{M\ast}^T \boldsymbol{h}}{\sigma_M^2} } d\tilde{x}_M \biggr) \nonumber \\ =& \frac{1}{Z_{GB}} e^{\boldsymbol{b}^T \boldsymbol{h}} \prod_i^M \int e^{- \frac{(\tilde{x}_i - a_i)^2 - 2\tilde{x}_i \boldsymbol{w}_{i\ast}^T \boldsymbol{h}}{2\sigma_i^2} } d\tilde{x}_i \nonumber \\ =& \frac{1}{Z_{GB}} e^{\boldsymbol{b}^T \boldsymbol{h}} \prod_i^M \int e^{- \frac{\tilde{x}_i^2 - 2\tilde{x}_i(a_i + \tilde{x}_i \boldsymbol{w}_{i\ast}^T \boldsymbol{h}) + a_i^2}{2\sigma_i^2} } d\tilde{x}_i \nonumber \\ =& \frac{1}{Z_{GB}} e^{\boldsymbol{b}^T \boldsymbol{h}} \prod_i^M \int e^{- \frac{\tilde{x}_i^2 - 2\tilde{x}_i(a_i + \boldsymbol{w}_{i\ast}^T \boldsymbol{h}) + (a_i + \boldsymbol{w}_{i\ast}^T \boldsymbol{h})^2 - (a_i + \boldsymbol{w}_{i\ast}^T \boldsymbol{h})^2 + a_i^2}{2\sigma_i^2} } d\tilde{x}_i \nonumber \\ =& \frac{1}{Z_{GB}} e^{\boldsymbol{b}^T \boldsymbol{h}} \prod_i^M \int e^{- \frac{(\tilde{x}_i - (a_i + \boldsymbol{w}_{i\ast}^T \boldsymbol{h}))^2 - a_i^2 -2a_i \boldsymbol{w}_{i\ast}^T \boldsymbol{h} - (\boldsymbol{w}_{i\ast}^T \boldsymbol{h})^2 + a_i^2}{2\sigma_i^2} } d\tilde{x}_i \nonumber \\ =& \frac{1}{Z_{GB}} e^{\boldsymbol{b}^T \boldsymbol{h}} \prod_i^M e^{\frac{2a_i \boldsymbol{w}_{i\ast}^T \boldsymbol{h} +(\boldsymbol{w}_{i\ast}^T \boldsymbol{h})^2 }{2\sigma_i^2}} \int e^{- \frac{(\tilde{x}_i - a_i - \boldsymbol{w}_{i\ast}^T \boldsymbol{h})^2}{2\sigma_i^2}} d\tilde{x}_i \nonumber \\ =& \frac{1}{Z_{GB}} e^{\boldsymbol{b}^T \boldsymbol{h}} \prod_i^M \sqrt{2\pi \sigma_i^2} e^{\frac{2a_i \boldsymbol{w}_{i\ast}^T \boldsymbol{h} +(\boldsymbol{w}_{i\ast}^T \boldsymbol{h})^2 }{2\sigma_i^2}} . \end{align*} $$