$p(x) = \int p(x,z) dz$

$\log p(x) = \log\int p(x,z) dz$

$\log\int p(x,z) dz =\log\int q(z|x)\frac{p(x,z)}{q(z|x)}dz$

Jensen Equality

$\log\int p(x) xdx >=\int\log p(x)xdx$

$\log\int q(z|x)\frac{p(x,z)}{q(z|x)}dz>=E_{q(z|x)}[\log \frac{p(x,z)}{q(z|x)}]$

$\log p(x)>=E_{q(z|x)}[\log \frac{p(x,z)}{q(z|x)}]=ELBO$

$ELBO =E_{q(z|x)}[\log \frac{p(x,z)}{q(z|x)}]=E_{q(z|x)}[\log { p(x,z)-\log q(z|x)}]$

$p(x,z)=p(x|z)p(z)$

$ELBO=E_{q(z|x)}[\log { p(x|z)+\log p(z)-\log q(z|x)}]$

$ELBO=E_{q(z|x)}[\log { p(x|z)+\log\frac{ p(z)}{q(z|x)}}]$

$ELBO=E_{q(z|x)}[\log { p(x|z)}]-D_{KL}(q(z|x)||p(z))$

Why Maximize ELBO?

$\log p(x) = \int q(z|x)\log p(x) dz=E_{q(z|x)}[\log p(x)]$