#epsilon-parameterization

Question 1: Single choice

In DDPM, $x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1 - \bar\alpha_t}\,\epsilon$ with $\epsilon \sim \mathcal{N}(0, I)$ , and the forward-process posterior $q(x_{t-1} \mid x_t, x_0)$ has the mean $\tilde\mu = \frac{\sqrt{\alpha_t}(1 - \bar\alpha_{t-1})}{1 - \bar\alpha_t}x_t + \frac{\sqrt{\bar\alpha_{t-1}}\,\beta_t}{1 - \bar\alpha_t}x_0$ . Written in terms of $x_t$ and the noise $\epsilon$ instead of $x_0$ , what is $\tilde\mu$ ?

$\frac{1}{\sqrt{\alpha_t}}\Big(x_t - \frac{\beta_t}{\sqrt{1 - \bar\alpha_t}}\,\epsilon\Big)$ 1
$\frac{1}{\sqrt{\bar\alpha_t}}\Big(x_t - \sqrt{1 - \bar\alpha_t}\,\epsilon\Big)$ 2
$\frac{1}{\sqrt{\alpha_t}}\Big(x_t - \sqrt{\beta_t}\,\epsilon\Big)$ 3
$\sqrt{\alpha_t}\,x_t - \frac{\beta_t}{\sqrt{1 - \bar\alpha_t}}\,\epsilon$ 4

#epsilon-parameterization

Timed test