Energy function of RBM The Hammersley-Clifford Theorem tells us that the distribution of a RBM must be Gibbs since it is  Markov Random Field, but how to prove that its energy function must be of the form:
$$E = -\sum_{i,j} w_{ij} \, v_i \, h_j -\sum_i \alpha_i \, v_i - \sum_i \beta_i \, h_i$$
 A: You only need to verify that the graphical model that represents a RBM fulfills the definition of a MRF as given in the document you refer to. See here for a picture.
Then it is warranted that you can write it as a factorized product of positive
functions defined on cliques that cover all the nodes and edges of G. Now, the theorem does not says that the decomposition is unique. It is mathematically convenient and meaningful to find a factorial decomposition through a energy function like that one. What you can do is very that this energy function gives you a valid factorial decomposition of the probability distribution in terms of positive functions which depend on the maximal cliques of the graph.
In this case the (maximal) cliques are rather trivial: just the pairs of nodes (again, take a look at the graph). Each pair has a corresponding term in the sum. When raising the exponential to $E$, the sum turns to a product of positive functions (the exponential is a positive function), where each function depends only on two variables corresponding to each of the cliques.
