(Diff1) \(p_\theta\left(\mathbf{x}_0\right)=\int p_\theta\left(\mathbf{x}_{0 : T}\right) d \mathbf{x}_{1: T}\).
(VAE1) \(p(\mathbf{x})=\int p(\mathbf{x}, \mathbf{z}) d \mathbf{z}\)
(Diff2) \(\log p_\theta\left(\mathrm{x}_0\right)=\log \int p_\theta\left(\mathrm{x}_{0 : T}\right) d \mathrm{x}_{1: T}\).
(VAE2) \(\log p(\mathbf{x})=\log \int p(\mathbf{x}, \mathbf{z}) d \mathbf{z}\).
(Diff3) \(\log p_\theta\left(\mathbf{x}_0\right)=\log \int q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right) \frac{p_\theta\left(\mathbf{x}_{0: T}\right)}{q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right)} d \mathbf{x}_{1: T}\).
(VAE3) \(\log p(\mathbf{x})=\log \int q(\mathbf{z} \mid \mathbf{x}) \frac{p(\mathbf{x}, \mathbf{z})}{q(\mathbf{z} \mid \mathbf{x})} d \mathbf{z}\).
( Jensen: \(\log \mathbb{E}[X] \geq \mathbb{E}[\log X]\) )
(Diff4) \(\log p_\theta\left(\mathrm{x}_0\right) =\log \int q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right) \frac{p_\theta\left(\mathbf{x}_{0: T}\right)}{q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right)} d \mathbf{x}_{1: T} \geq \int q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right) \log \frac{p_\theta\left(\mathbf{x}_{0: T}\right)}{q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right)} d \mathbf{x}_{1: T}\)
(VAE4) \(\log p(\mathbf{x})=\log \int q(\mathbf{z} \mid \mathbf{x}) \frac{p(\mathbf{x}, \mathbf{z})}{q(\mathbf{z} \mid \mathbf{x})} d \mathbf{z} \geq \int q(\mathbf{z} \mid \mathbf{x}) \log \frac{p(\mathbf{x}, \mathbf{z})}{q(\mathbf{z} \mid \mathbf{x})} d \mathbf{z}\).
(Diff5) \(\mathcal{L}(\mathbf{x})=\mathbb{E}_q\left[-\log \frac{p_\theta\left(\mathbf{x}_{0: T}\right)}{q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right)}\right]\).
(VAE5) \(\mathcal{L}(\mathbf{x})=\mathbb{E}_{q(\mathbf{z} \mid \mathbf{x})}[-\frac{\log p(\mathbf{x}, \mathbf{z})}{\log q(\mathbf{z} \mid \mathbf{x})}]\).