P(h|v) conditional probability proof of Restricted Boltzman Machine

Question

I have a question about the last step of the proof of conditional probability of the restricted boltzman machine in the deeplearningbook.org

Below are the derivation from the book: Shouldn't equation 20.15 be

$P(h|v) = \displaystyle\prod_{j=1}^{nh}\sigma((2h_j-1)\times(c_i+\textbf{W}_{:,j}^\intercal \textbf{v}))$ (my own derived formula)
instead of
$P(h|v) = \displaystyle\prod_{j=1}^{nh}\sigma((2\textbf{h}-1)\odot(\textbf{c}+\textbf{W}^\intercal \textbf{v}))_j$ (in the deeplearningbook.org)

What would that j even refer to in the original question in the textbook anyway?

Answer 1

$j$ indexes the hidden units in the RBM.

Assuming you meant $c_j$ instead of $c_i$ in your derived formula, it is equivalent to the provided formula.

$$ \begin{align} &\prod_j \sigma((2h-1) \odot (c + W^T v))_j \\ = &\prod_j \sigma((2h-1)_j \cdot (c + W^T v)_j) \\ = &\prod_j \sigma((2h_j-1) (c_j + (W^T)_j v)) \\ = &\prod_j \sigma((2h_j-1) (c_j + W_{:,j}^T v)) \\ \end{align} $$