Possible typo in Eq. (8) of Section 3.3: missing square root on the z_t coefficient

[KaigeZheng]

According to the original DDPM paper (Ho et al., 2020), the reverse process is formulated as

$$ x_{t-1} = \mu_\theta(x_t,t) + \sigma_t z_t,\quad z_t \sim \mathcal{N}(0,I), $$

where $\sigma_t^2$ represents the variance of the reverse transition. Hence, the coefficient of
$z_t$ is given by the square root of the variance.

Based on this standard DDPM formulation, it seems that the noise term in Equation (8) of
Section 3.3 may be missing a square root. In particular, the coefficient in front of $z_t$
might be expected to take the form

$$ \sqrt{\frac{1-\bar{\alpha}_{t-1}}{1-\bar{\alpha}_t}\beta_t}\ z_t, $$

rather than a linear scaling without the square root.

We would appreciate it if the authors could kindly clarify whether this is a typographical
error or an intentional design choice.

[KaigeZheng]

While reading the paper, I felt that the reverse formula in Equation (8) looked a bit unusual, so I worked through the derivation by hand. I guess that there may be an error here (note: I have not carefully checked the code yet).