(1) \(p_\theta\left(\mathbf{x}_0\right)=\int p_\theta\left(\mathbf{x}_{0 : T}\right) d \mathbf{x}_{1: T}\).
(2) \(\log p_\theta\left(\mathrm{x}_0\right)=\log \int p_\theta\left(\mathrm{x}_{0 : T}\right) d \mathrm{x}_{1: T}\).
(3) \(\log p_\theta\left(\mathbf{x}_0\right)=\log \int q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right) \frac{p_\theta\left(\mathbf{x}_{0: T}\right)}{q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right)} d \mathbf{x}_{1: T}\).
(4) \(\log p_\theta\left(\mathrm{x}_0\right) \geq \mathbb{E}_q\left[\log \frac{p_\theta\left(\mathbf{x}_{0: T}\right)}{q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right)}\right]\) = ELBO
(5) \(L=\mathbb{E}_q\left[-\log \frac{p_\theta\left(\mathbf{x}_{0: T}\right)}{q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right)}\right]\).
전개하기!
(5) \(L=\mathbb{E}_q\left[-\log \frac{p_\theta\left(\mathbf{x}_{0: T}\right)}{q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right)}\right]\).
- Forward: \(q\left(\mathrm{x}_{1: T} \mid \mathrm{x}_0\right)=\prod_{t=1}^T q\left(\mathrm{x}_t \mid \mathrm{x}_{t-1}\right)\)
- Reverse: \(p_\theta\left(\mathbf{x}_{0-T}\right)=p\left(\mathbf{x}_T\right) \prod_{t=1}^T p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)\).
(위에서 \(\mathbb{E}_q\)는 생략)
(6) \(\log \frac{p\left(\mathrm{x}_{\mathrm{T}}\right) \prod_{t=1}^T p_\theta\left(\mathrm{x}_{t-1} \mid \mathrm{x}_t\right)}{\prod_{t=1}^T q\left(\mathrm{x}_t \mid \mathrm{x}_{t-1}\right)}\).
(7) \(\log p\left(\mathbf{x}_T\right)+\sum_{t=1}^T \log p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)-\sum_{t=1}^T \log q\left(\mathbf{x}_t \mid \mathbf{x}_{t-1}\right)\).
(8) \(\log p\left(\mathbf{x}_T\right)+\sum_{t \geq 1} \log \frac{p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)}{q\left(\mathbf{x}_t \mid \mathbf{x}_{t-1}\right)}\)
그러면 아래 식이 이해가 갈 것!
\(\begin{aligned} L & =\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] \\ & =\mathbb{E}_q\left[-\log p\left(\mathbf{x}_T\right)-\sum_{t \geq 1} \log \frac{p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)}{q\left(\mathbf{x}_t \mid \mathbf{x}_{t-1}\right)}\right] \\ & =\mathbb{E}_q\left[-\log p\left(\mathbf{x}_T\right)-\sum_{t>1} \log \frac{p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)}{q\left(\mathbf{x}_t \mid \mathbf{x}_{t-1}\right)}-\log \frac{p_\theta\left(\mathbf{x}_0 \mid \mathbf{x}_1\right)}{q\left(\mathbf{x}_1 \mid \mathbf{x}_0\right)}\right] \\ & =\mathbb{E}_q\left[-\log p\left(\mathbf{x}_T\right)-\sum_{t>1} \log \frac{p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)}{q\left(\mathbf{x}_t \mid \mathbf{x}_{t-1}, \mathbf{x}_{0}\right)}-\log \frac{p_\theta\left(\mathbf{x}_0 \mid \mathbf{x}_1\right)}{q\left(\mathbf{x}_1 \mid \mathbf{x}_0\right)}\right] \\ & =\mathbb{E}_q\left[-\log p\left(\mathbf{x}_T\right)-\sum_{t>1} \log \frac{p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)}{\frac{q\left(x_t, x_{t-1}, x_0\right)}{q\left(x_{t-1}, x_0\right)} \cdot \frac{q\left(x_t, x_0\right)}{q\left(x_t, x_0\right)}}-\log \frac{p_\theta\left(\mathbf{x}_0 \mid \mathbf{x}_1\right)}{q\left(\mathbf{x}_1 \mid \mathbf{x}_0\right)}\right] \\& =\mathbb{E}_q\left[-\log p\left(\mathbf{x}_T\right)-\sum_{t>1} \log \frac{p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)}{q\left(x_{t-1} \mid x_t, x_0\right) \cdot \frac{q\left(x_t, x_0\right)}{q\left(x_{t-1}, x_0\right)}}-\log \frac{p_\theta\left(\mathbf{x}_0 \mid \mathbf{x}_1\right)}{q\left(\mathbf{x}_1 \mid \mathbf{x}_0\right)}\right] \\ & =\mathbb{E}_q\left[-\log p\left(\mathbf{x}_T\right)-\sum_{t>1} \log \frac{p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)}{q\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t, \mathbf{x}_0\right)} \cdot \frac{q\left(\mathbf{x}_{t-1} \mid \mathbf{x}_0\right)}{q\left(\mathbf{x}_t \mid \mathbf{x}_0\right)}-\log \frac{p_\theta\left(\mathbf{x}_0 \mid \mathbf{x}_1\right)}{q\left(\mathbf{x}_1 \mid \mathbf{x}_0\right)}\right] \\ & =\mathbb{E}_q\left[-\log p\left(\mathbf{x}_T\right)-\sum_{t>1} \log \frac{p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)}{q\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t, \mathbf{x}_0\right)} -\sum_{t>1} \log \frac{q\left(\mathbf{x}_{t-1} \mid \mathbf{x}_0\right)}{q\left(\mathbf{x}_t \mid \mathbf{x}_0\right)}-\log \frac{p_\theta\left(\mathbf{x}_0 \mid \mathbf{x}_1\right)}{q\left(\mathbf{x}_1 \mid \mathbf{x}_0\right)}\right] \\ & =\mathbb{E}_q\left[-\log p\left(\mathbf{x}_T\right)-\sum_{t>1} \log \frac{p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)}{q\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t, \mathbf{x}_0\right)} -\log \frac{q\left(x_1 \mid x_0\right)}{q\left(x_2 \mid x_0\right)}-\log \frac{q\left(x_2 \mid x_0\right)}{q\left(x_3 \mid x_0\right)}-\ldots -\log \frac{q\left(x_{T-1} \mid x_0\right)}{q\left(x_T \mid x_0\right)} -\log \frac{p_\theta\left(\mathbf{x}_0 \mid \mathbf{x}_1\right)}{q\left(\mathbf{x}_1 \mid \mathbf{x}_0\right)}\right] \\ & =\mathbb{E}_q\left[-\log \frac{p\left(\mathbf{x}_T\right)}{q\left(\mathbf{x}_T \mid \mathbf{x}_0\right)}-\sum_{t>1} \log \frac{p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)}{q\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t, \mathbf{x}_0\right)}-\log p_\theta\left(\mathbf{x}_0 \mid \mathbf{x}_1\right)\right]\\ & =\mathbb{E}_q\left[D_{\mathrm{KL}}\left(q\left(\mathbf{x}_T \mid \mathbf{x}_0\right) \| p\left(\mathbf{x}_T\right)\right)+\sum_{t>1} D_{\mathrm{KL}}\left(q\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t, \mathbf{x}_0\right) \| p_\theta\left(\mathbf{x}_{t-1} \mid \mathbf{x}_t\right)\right)-\log p_\theta\left(\mathbf{x}_0 \mid \mathbf{x}_1\right)\right] \end{aligned}\).