# explanation of RBM definition

I'm learning about RBM and I try to understand the notation used for it. We have the input vector $v=(v_i)$ and the output vector $h=(h_j)$, a weight matrix $W=(W_{ij})$ and finally two bias vectors - $a,b$.

Normally for an ANN we would say that - $$h=\sigma(v^TW+a)$$ where $\sigma$ is the network's activation function.

does this relation not hold for RBM? How does $b$ fit in here? Why is it common to talk about $h$ as a vector that stand for itself and not a deterministic result of the input? What part of this definition makes the network stochastic?

In an RBM, the joint distribution over all variables is: $$P(x,h) \propto \exp (b^Tv + a^Th + h^TWv)$$
In the case of binary RBM, if you condition on $x$, you'll end up with $$P(h|x) = \sigma(a+Wx)$$
Obviously, $b$ wouldn't show up in this equation because $x$ is fixed, and $b$ only influences the joint probability through $x$.