The consistency term of the diffusion model is written as: $$\mathop{\mathbb{E_{q_\phi(x_{1:T}|x_0)}}} \left[\log\prod_{t=2}^T \frac{p(x_{t-1} | x_t)}{q_\phi(x_{t-1}|x_t, x_0)}\right]$$
$$= \sum_{t=2}^T \mathop{\mathbb{E_{q_\phi(x_t,x_{t-1}|x_0)}}} \log\frac{p(x_{t-1} | x_t)}{q_\phi(x_{t-1}|x_t, x_0)}$$
$$= -\sum_{t=2}^T \mathop{\mathbb{E_{q_\phi(x_t,x_{t-1}|x_0)}}} \mathop{\mathbb{D}_{KL}}(q_\theta(x_{t-1}|x_t, x_0) ||p(x_{t-1}|x_t)$$ I'm confused on why there is still a expectation $\mathop{\mathbb{E_{q_\phi(x_t,x_{t-1}|x_0)}}}$ in the last line.
Since $\mathop{\mathbb{D}_{KL}}(p||q) = \mathop{\mathbb{E}_{p(x)}}\log\frac{p(x)}{q(x)}$, shouldn't the last line be: $$= -\sum_{t=2}^T \mathop{\mathbb{D}_{KL}}(q_\theta(x_{t-1}|x_t, x_0) ||p(x_{t-1}|x_t)$$
