q(xt−1∣xt)=q(xt,xt−1)q(xt)=q(xt∣xt−1)q(xt−1)q(xt)∵xt=αtxt−1+1−αtzt∼N(αtxt−1,(1−αt)I)xt=αt‾x0+1−αt‾zt∼N(αt‾x0,1−αt‾I)∴q(xt∣xt−1)∼N(αtxt−1,(1−αt)I)q(xt)∼N(αt‾x0,1−αt‾I)q(xt−1)∼N(αt−1‾x0,1−αt−1‾I)∵N(μ,σ)∝12πσexp(−(x−μ)22σ2)∴q(xt∣xt−1)q(xt−1)q(xt)∝exp{−12((xt−αtxt−1)21−αt+(xt−1−αt−1‾x0)21−αt−1‾−(xt−αt‾x0)21−αt‾)}∝exp{−12((αtβt+11−αt−1‾)xt−12−2(αtxtβt+αt−1‾x01−αt−1‾)xt−1+C‾)}∝exp{−12(Axt−12+Bxt−1+C)}∝exp{−12(A(xt−1+B2A)2+C)}q(x_{t-1}|x_t)=\frac{q(x_t,x_{t-1})}{q(x_t)}=\frac{q(x_t|x_{t-1})q(x_{t-1})}{q(x_t)} \\ \\ \begin{aligned} \because x_t&=\sqrt{\alpha_t}x_{t-1}+\sqrt{1-\alpha_t}z_t \sim N(\sqrt{\alpha_tx_{t-1}},(1-\alpha_t)I ) \\ x_t&=\sqrt{\overline{\alpha_t}}x_{0}+\sqrt{1-\overline{\alpha_t}}z_t \sim N(\sqrt{\overline{\alpha_t}}x_{0},\sqrt{1-\overline{\alpha_t}}I)\\ \quad \\ \therefore \quad& q(x_t|x_{t-1})\sim N(\sqrt{\alpha_tx_{t-1}},(1-\alpha_t)I )\\ \quad& q(x_t)\sim N(\sqrt{\overline{\alpha_t}}x_{0},\sqrt{1-\overline{\alpha_t}}I)\\ \quad& q(x_{t-1})\sim N(\sqrt{\overline{\alpha_{t-1}}}x_{0},\sqrt{1-\overline{\alpha_{t-1}}}I)\\ \quad \\ \because \quad & N(\mu,\sigma)\propto \frac{1}{\sqrt{2\pi}\sigma} exp(-\frac{(x-\mu)^2}{2\sigma^2}) \\ \therefore \quad& \frac{q(x_t|x_{t-1})q(x_{t-1})}{q(x_t)} \propto exp\lbrace -\frac{1}{2} (\frac{(x_t-\sqrt{\alpha_t}x_{t-1})^2}{1-\alpha_t} + \frac{(x_{t-1}-\sqrt{\overline{\alpha_{t-1}}}x_{0})^2}{1-\overline{\alpha_{t-1}}} - \frac{(x_t-\sqrt{\overline{\alpha_{t}}}x_{0})^2}{1-\overline{\alpha_{t}}})\rbrace \\ &\quad \quad \quad \quad \quad \quad \quad \quad \propto exp\lbrace -\frac{1}{2} ( (\frac{\alpha_t}{\beta_t}+\frac{1}{1-\overline{\alpha_{t-1}}})x_{t-1}^{2} - 2(\frac{\sqrt{\alpha_t}x_t}{\beta_t}+\frac{\sqrt{\overline{\alpha_{t-1}}}x_0}{1-\overline{\alpha_{t-1}}})x_{t-1}+\overline{C} ) \rbrace \\ &\quad \quad \quad \quad \quad \quad \quad \quad \propto exp \lbrace -\frac{1}{2}(Ax_{t-1}^{2}+Bx_{t-1}+C) \rbrace\\ &\quad \quad \quad \quad \quad \quad \quad \quad \propto exp \lbrace -\frac{1}{2}(A(x_{t-1}+\frac{B}{2A})^2+C) \rbrace \end{aligned} \\