How to inference the conditional probability about LDA?

Question

I'm studying the paper of Blei, "Latent Dirichlet Allocation" ( http://www.jmlr.org/papers/volume3/blei03a/blei03a.pdf ).

In his paper(page 1003), given equation is $p(\theta, z|w, \alpha, \beta)= \frac{p(\theta, z, w|\alpha, \beta)}{p(w|\alpha, \beta)}$

But I don't know how to derive the right term.

Could you give me a hint?

Answer 1

Let $a=(\theta,z)$, and $b=(\alpha,\beta)$. You want to show that $$p(a|w,b) = \frac{p(a,w|b)}{p(w|b)}.$$ Can you see that?

If you need more details, see LHS $= p(a,w,b)/p(w,b)$ and RHS $= p(a,w,b)/(p(b)p(w|b))$ Since $p(w,b)=p(b)p(w|b)$, you have the result.