- VAE:
- Hierarchical VAE:
- DDPM(Hierarchical VAE with deterministic forward process):
Notice that $\phi$ has been removed from $q_\phi()$.
Also see that:
- The ELBO for a Variational Autoencoder (VAE) is given by:
$$
\text{ELBO (VAE)} = \mathbb{E}_{q_\phi(z|x)} \left[ \log \frac{p(x|z)}{q_\phi(z|x)} \right]
$$
- For Denoising Diffusion Models (DDPM), the ELBO is expressed as:$$
\text{ELBO (DM)} = \mathbb{E}_{q(x_{1:T}|x_0)} \left[ \log \frac{p(x_{0:T})}{q(x_{1:T}|x_0)} \right]
$$