How to prove the Bayes formula below? $$
p(W,H,β|V) = \frac{p(V|W,H)  p(W|β)  p(H|β)  p(β)}{p(V)}
$$
I have tried a lot but I cannot prove this formula. The paper is:  


*

*Ioannis Psorakis, I, Roberts, S, Ebden, M, & Sheldon, B. (2011). Overlapping community detection using Bayesian non-negative matrix factorization.  PHYSICAL REVIEW E 83, 066114. (pdf)


which gives a graphical model structure for the variables in question as:

 A: According to the paper and the graphical model (Fig. 1) which specifies dependencies/independencies between the variables, authors didn't use the Bayes formula. They used the definition of the conditional probability.
The Eq. (2) stems from Eq. (1) by using the conditional probability definition. The Eq. (1) is the joint distribution over all model variables.
If you look at the graphical model, the dependencies are specified by it. Just use the basic result of dependencies to read the graph and write the joint probability.
A: The formula can be proved under the assumptions that $V$ and $\beta$ are independent and that $W$ and $H$ are conditionally independent given $\beta$, which assumptions might well be deducible from the graphical model in the paper (to which I don't have access). 
\begin{align}
p(W,H,\beta\mid V) &= \frac{p(W,H,\beta,V)}{p(V)}
& {\scriptstyle{\text{definition of conditional probability}}}\\
&= \frac{p(V\mid W,H,\beta)\cdot p(W,H,\beta)}{p(V)}
& {\scriptstyle{\text{conditional probability def. used backwards}}}\\
&= \frac{p(V\mid W,H)\cdot p(W,H,\beta)}{p(V)}
& {\scriptstyle{\text{assuming that}~V~\text{is independent of}}~\beta}\\
&= \frac{p(V\mid W,H)\cdot p(W,H\mid\beta)\cdot p(\beta)}{p(V)}
& {\scriptstyle{\text{conditional probability def. used backwards again}}}\\
&= \frac{p(V\mid W,H)\cdot p(W\mid\beta)\cdot p(H\mid \beta)\cdot p(\beta)}{p(V)}
& {\scriptstyle{\text{assuming}~W~\text{and}~H~\text{conditionally independent given}~\beta}}\\
\end{align}
