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I'm studying Topic Model, but I can't understand the following transformations.
$F$ is variational lower bound. $$\begin{eqnarray} F[q(z_{1:n}, \phi, \pi)] &=& \int \sum_{z_{1:n}} q(z_{1:n}) q(\phi) q(\pi) \log \frac{p(x_{1:n}, z_{1:n} | \phi, \pi) p(\phi | \eta) p(\pi|\alpha)}{q(z_{1:n}) q(\phi) q(\pi)}d\phi d\pi \\ &=& \int \sum_{z_{1:n}} q(z_{{1:n}}) q(\phi) q(\pi) \log \frac{p(x_{1:n}, z_{1:n} | \phi, \pi)}{q(z_{1:n})} d \phi d\pi \\ &+& \sum_{k=1}^{K} \int q(\phi_k) \log \frac{p(\phi_k | \eta)}{q(\phi_k)} d\phi_k + \int q(\pi) \log \frac{p(\pi | \alpha)}{q(\pi)} d\pi \end{eqnarray}$$

Please note that $z_{1:n} = \boldsymbol{z}_{1:n}$, $\phi = \boldsymbol{\phi}$, $\pi = \boldsymbol{\pi}$, and $x_{1:n} = \boldsymbol{x}_{1:n}$.

How can we derive the second from the first? Especially, I couldn't understand why $q_(z_{1:n})$, $q(\phi)$, and $q(\pi)$ aren't become factorials in log, since I thought $a \log b = \log b^a$.

user2978524
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  • Not sure what you mean they "aren't become factorials", of course you can write in both ways. The difference between the first and the second line is just we reorganize terms, e.g. if it doesn't contain $\pi$, then there is no need to integrate $\pi$ out. – TracyYXChen Feb 03 '20 at 20:56

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