From 93bad89fa327eb01f03a38debc161e80fb88a26b Mon Sep 17 00:00:00 2001 From: Marc Lelarge Date: Fri, 15 Sep 2023 16:32:44 +0200 Subject: [PATCH] correct typo diffusion --- modules/18a-diffusion.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/modules/18a-diffusion.md b/modules/18a-diffusion.md index 20040d3..18b37c7 100644 --- a/modules/18a-diffusion.md +++ b/modules/18a-diffusion.md @@ -72,7 +72,7 @@ q(x_{t-1}|x_t,x_0) = \mathcal{N}(x_{t-1};\mu(x_t,x_0), \gamma_t I), \end{align*} with \begin{align*} -\mu(x_t,x_0) &= \frac{\sqrt{\alpha_t}(1-\overline{\alpha}_{t-1})}{1-\overline{\alpha}_{t}}x_t + \frac{\sqrt{\overline{\alpha}_{t-1}\beta_t}}{1-\overline{\alpha}_{t}}x_0\\ +\mu(x_t,x_0) &= \frac{\sqrt{\alpha_t}(1-\overline{\alpha}_{t-1})}{1-\overline{\alpha}_{t}}x_t + \frac{\beta_t\sqrt{\overline{\alpha}_{t-1}}}{1-\overline{\alpha}_{t}}x_0\\ \gamma_t &= \frac{1-\overline{\alpha}_{t-1}}{1-\overline{\alpha}_{t}}\beta_t \end{align*} but we know that $x_0 = 1/\sqrt{\overline{\alpha}_t}\left( x_t-\sqrt{1-\overline{\alpha}_t}\epsilon\right)$, hence we have