Representing the wave function
The wavefunction should be a probability amplitude depending on \( \boldsymbol{x} \). The RBM model is given by the joint distribution of \( \boldsymbol{x} \) and \( \boldsymbol{h} \)
$$
\begin{align}
F_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}.
\tag{24}
\end{align}
$$
To find the marginal distribution of \( \boldsymbol{x} \) we set:
$$
\begin{align}
F_{rbm}(\mathbf{x}) &= \sum_\mathbf{h} F_{rbm}(\mathbf{x}, \mathbf{h})
\tag{25}\\
&= \frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}.
\tag{26}
\end{align}
$$
Now this is what we use to represent the wave function, calling it a neural-network quantum state (NQS)
$$
\begin{align}
\Psi (\mathbf{X}) &= F_{rbm}(\mathbf{x})
\tag{27}\\
&= \frac{1}{Z}\sum_{\boldsymbol{h}} e^{-E(\mathbf{x}, \mathbf{h})}
\tag{28}\\
&= \frac{1}{Z} \sum_{\{h_j\}} e^{-\sum_i^M \frac{(x_i - a_i)^2}{2\sigma^2} + \sum_j^N b_j h_j + \sum_{i,j}^{M,N} \frac{x_i w_{ij} h_j}{\sigma^2}}
\tag{29}\\
&= \frac{1}{Z} e^{-\sum_i^M \frac{(x_i - a_i)^2}{2\sigma^2}} \prod_j^N (1 + e^{b_j + \sum_i^M \frac{x_i w_{ij}}{\sigma^2}}).
\tag{30}\\
\tag{31}
\end{align}
$$