I am reading a paper where we define, $$F(\theta, \gamma, \phi) = \mathbb{E}_{q(\mathbf{a},\mathbf{z}|\mathbf{x},\mathbf{u})}\left[\log\left(\frac{p_{\theta}(\mathbf{x}|\mathbf{a}) p_{\gamma}(\mathbf{a}|\mathbf{z}) p_{\gamma}(\mathbf{z}|\mathbf{u})}{q(\mathbf{a}, \mathbf{z}|\mathbf{x}, \mathbf{u})}\right)\right].$$ Then we structure $q$ such that, $$q(\mathbf{a},\mathbf{z}|\mathbf{x},\mathbf{u}) = q_{\phi}(\mathbf{a}|\mathbf{x})p_{\gamma}(\mathbf{z}|\mathbf{a},\mathbf{u}).$$ The the authors claim that, $$F(\theta, \gamma,\phi) = \mathbb{E}_{q_{\phi}(\mathbf{a}|\mathbf{x})}\left[\log\left(\frac{p_{\theta}(\mathbf{x}|\mathbf{a})}{q_{\phi}(\mathbf{a}|\mathbf{x})}\right) + \mathbb{E}_{p_{\gamma}(\mathbf{z}|\mathbf{a}, \mathbf{u})}\left[\log\left(\frac{p_{\gamma}(\mathbf{a}|\mathbf{z})p_{\gamma}(\mathbf{z}|\mathbf{u})}{p_{\gamma}(\mathbf{z}|\mathbf{a},\mathbf{u})}\right)\right]\right].$$
I was trying to prove this by substituting the expression for $q$ in the first equation. This gives: $$ \begin{align} F(\theta,\gamma,\phi) &= \int \int \left[\log\left(\frac{p_{\theta}(\mathbf{x}|\mathbf{a}) p_{\gamma}(\mathbf{a}|\mathbf{z}) p_{\gamma}(\mathbf{z}|\mathbf{u})}{q(\mathbf{a}, \mathbf{z}|\mathbf{x}, \mathbf{u})}\right)\right] q(\mathbf{a},\mathbf{z}|\mathbf{x},\mathbf{u})\\ &= \int \int \left[\log\left(\frac{p_{\theta}(\mathbf{x}|\mathbf{a})}{q_{\phi}(\mathbf{a}|\mathbf{x})}\right) p_{\gamma}(\mathbf{z}|\mathbf{a},\mathbf{u}) + \left[\log\left(\frac{p_{\gamma}(\mathbf{a}|\mathbf{z})p_{\gamma}(\mathbf{z}|\mathbf{u})}{p_{\gamma}(\mathbf{z}|\mathbf{a},\mathbf{u})}\right)\right]p_{\gamma}(\mathbf{z}|\mathbf{a},\mathbf{u})\right]q_{\phi}(\mathbf{a}|\mathbf{x})\\ &=\int \log\left(\frac{p_{\theta}(\mathbf{x}|\mathbf{a})}{q_{\phi}(\mathbf{a}|\mathbf{x})}\right) q_{\phi}(\mathbf{a}|\mathbf{x}) + \int \mathbb{E}_{p_{\gamma}(\mathbf{z}|\mathbf{a},\mathbf{u})}\left[\log\left(\frac{p_{\gamma}(\mathbf{a}|\mathbf{z})p_{\gamma}(\mathbf{z}|\mathbf{u})}{p_{\gamma}(\mathbf{z}|\mathbf{a},\mathbf{u})}\right)\right] q_{\phi}(\mathbf{a}|\mathbf{x})\\ &= \mathbb{E}_{q_{\phi}(\mathbf{a}|\mathbf{x})} \left[\log\left(\frac{p_{\theta}(\mathbf{x}|\mathbf{a})}{q_{\phi}(\mathbf{a}|\mathbf{x})}\right)\right] + \mathbb{E}_{p_{\gamma}(\mathbf{z}|\mathbf{a}, \mathbf{u})}\left[\log\left(\frac{p_{\gamma}(\mathbf{a}|\mathbf{z})p_{\gamma}(\mathbf{z}|\mathbf{u})}{p_{\gamma}(\mathbf{z}|\mathbf{a},\mathbf{u})}\right)\right]\end{align}. $$
I don't understand why one would choose to nest the expectation within another expectation. Perhaps someone can explain this.
By the definition of $q$ and the linearity of expectations,$$F(\theta,\,\gamma,\,\phi)=\color{blue}{\Bbb E_{q_\phi(a|x)p_\gamma(z|a,\,u)}\ln\frac{p_\theta(x|a)}{q_\phi(a|x)}}+\color{red}{\Bbb E_{q_\phi(a|x)p_\gamma(z|a,\,u)}\ln\frac{p_\gamma(a|z)p_\gamma(z|u)}{p_\gamma(z|a,\,u)}}.$$The authors claim$$F(\theta,\,\gamma,\,\phi)=\color{blue}{\Bbb E_{q_\phi(a|x)}\ln\frac{p_\theta(x|a)}{q_\phi(a|x)}}+\color{red}{\Bbb E_{q_\phi(a|x)}\Bbb E_{p_\gamma(z|a,\,u)}\ln\frac{p_\gamma(a|z)p_\gamma(z|u)}{p_\gamma(z|a,\,u)}}.$$The blue (red) term can be rewritten as shown because its logarithm's argument has no implicit dependence on $z$ ($x$).