Downsampling and Stretching

Lots of math here. You can reprove this by hand from Jensen’s inequality or look at your notes

Basically, start with log(p(x)). —Then express it using a latent variable model decomposition —choose an arbitrary q(z) as your ‘weighting’ int( log( q(z) * p(x,z)/q(z) ) dz) —apply Jensen’s inequality when you get to p(x,z) split it up into p(z|x) and p(x) that will let you take out p(x), and also give you a KL term

Maximizing p(x) with respect to the parameters for q(z) and p(z|x) involves expectation maximization, this means —can improve ELBO’s lower bound by changing p(z|x) slash p(x|z)’s parametrization OR —can make ELBO bound more tight by changing q(z)’s parametrization Last Reviewed: 1/19/25