Proof taken from http://image.diku.dk/igel/paper/AItRBM-proof.pdf (page 24)
I understand everything up to and including:
(1)
$$p(\textbf{v}) = \frac{1}{Z}e^{\sum_{j=1}^mb_jv_j} \prod_{i=1}^n\sum e^{h_i(c_i + \sum_{h_i}w_{ij}v_j)}$$
My question is how do we go from (1) to (2):
(2)
$$p(\textbf{v}) = \frac{1}{Z}\prod_{j=1}^m e^{b_jv_j}\prod_{i=1}^n(1+e^{c_i+\sum_{j=1}^m})$$.
(I see where $\frac{1}{Z}\prod_{j=1}^m e^{b_jv_j}$ comes from but not the rest of the formula.)
Also, how easy would it be to extend this to multi-class? (Currently, ($\textbf{v,h}$) $\in \{ 0,1\}^{m+n}$).
Algebraic fact: For every collection $(t_i)_{i\in I}$, $$ \sum_h\prod_{i\in I}\mathrm e^{h_it_i}=\prod_{i\in I}\sum_{\ell=0}^1\mathrm e^{\ell t_i}=\prod_{i\in I}(1+\mathrm e^{t_i}), $$ where the sum on the LHS runs over every $h=(h_i)_{i\in I}$ in $\{0,1\}^I$.