A=αtβt+11−αt−1‾B=−2(αtxtβt+αt−1‾x01−αt−1‾)∵μ=−B2Aσ2=1A∴σ2=1−αt−1‾1−αt‾βtμ=αt1−αt−1‾1−αt‾xt+αt−1‾βt1−αt‾x0∵xt=αtxt−1+1−αtztx0=1αt‾(xt−1−αt‾z~)∴μ=1αt(xt−βt1−αt‾z~)即：q(xt−1∣xt)∼N(1αt(xt−βt1−αt‾z~),1−αt−1‾1−αt‾βt)xt−1=1αt(xt−βt1−αt‾z~)+1−αt−1‾1−αt‾βtzz~=UNet(xt,t)z∼N(0,I) A=\frac{\alpha_t}{\beta_t}+\frac{1}{1-\overline{\alpha_{t-1}}} \quad B=-2(\frac{\sqrt{\alpha_t}x_t}{\beta_t}+\frac{\sqrt{\overline{\alpha_{t-1}}}x_0}{1-\overline{\alpha_{t-1}}}) \\ \quad \\ \because \mu=-\frac{B}{2A} \quad \sigma^2=\frac{1}{A}\\ \therefore \sigma^2=\frac{1-\overline{\alpha_{t-1}}}{1-\overline{\alpha_t}}\beta_t \quad \mu=\sqrt{\alpha_t}\frac{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\\ \quad \\ \begin{aligned} \because x_t&=\sqrt{\alpha_t}x_{t-1}+\sqrt{1-\alpha_t}z_t\\ x_0&=\frac{1}{\sqrt{\overline{\alpha_t}}}(x_t-\sqrt{1-\overline{\alpha_t}}\widetilde{z})\\ \therefore \mu&=\frac{1}{\sqrt{\alpha_t}}(x_t-\frac{\beta_t}{\sqrt{1-\overline{\alpha_t}}}\widetilde{z}) \end{aligned}\\ 即：q(x_{t-1}|x_t) \sim N(\frac{1}{\sqrt{\alpha_t}}(x_t-\frac{\beta_t}{\sqrt{1-\overline{\alpha_t}}}\widetilde{z}),\frac{1-\overline{\alpha_{t-1}}}{1-\overline{\alpha_t}}\beta_t)\\ x_{t-1}=\frac{1}{\sqrt{\alpha_t}}(x_t-\frac{\beta_t}{\sqrt{1-\overline{\alpha_t}}}\widetilde{z})+\sqrt{\frac{1-\overline{\alpha_{t-1}}}{1-\overline{\alpha_t}}\beta_t}z\\ \widetilde{z}=UNet(x_t,t) \qquad z \sim N(0,I)