I'm trying to understand the following proof from Deep Unsupervised Learning Using Nonequilibrium Thermodynamics.
This particular version of the proof is found in: Denoising Diffusion Probabilistic Models.
I'm confused by the move from (19) to (20) and the move from (20) to (21) though.
For (19) to (20) I tried applying Bayes Rule to $q(x_{t-1}|x_t,x_0)$ but that didn't help.
I also see:
$$ \begin{align*} \frac{p_\theta(x_{t-1}|x_t)}{q(x_t|x_{t-1})}\\ &= \frac{p_\theta(x_{t-1}|x_t)}{q(x_t|x_{t-1})} \cdot \frac{q(x_t|x_{t-1}) \cdot q(x_{t-1}|x_0)}{q(x_t|x_0)}\\ \end{align*} $$
but I can't simplify any further to (20).
I know it's not going to help you much but.
from 19 to 20.
Like the first paper said, $q(x_t|x_{t-1})$ is Markov process, it can be $q(x_t|x_{t-1}, x_0)$
Then Bayes Rule comes, becomes (20), then you can substitute 2 to T in t in mid term, then there is some elimination, you can derive (21).
I still don't get $D_{KL}$ part (22). I am trying to figure this out.