I was following some tutorials on variational inference and ELBO loss functions and I think I understand it quite well, but I'm struggling with math. If I understand it correctly we went from this equation: $D_{KL}(q(z|x)\|p(z|x))=\displaystyle\int_{z}^{}q(z|x)\left( \log\frac{q(z|x)}{p(x,z)} + \log p(x)\right) dz$
to:
$D_{KL}(q(z|x)\|p(z|x))=\displaystyle\int_{z}^{}q(z|x)\left( \log\frac{q(z|x)}{p(x,z)}\right)dz + \log p(x)\displaystyle\int_{z}^{}q(z|x)dz$
by multiplying $q(z|x)$ by both terms in parentheses, which gave us sum under integral and then we rewrote it as sum of two integrals. Next we can extract $\log p(x)$infront of the second integral because we treat it as a constant when integrating over $z$. Am I wrong about it or this is exactly what we are doing. Thanks for the answers!
