DDPM中建模的$q(\mathbf{x}_{t-1} \vert \mathbf{x}_t, \mathbf{x}_0)$满足正态分布，
$$
q(\mathbf{x}_{t-1} \vert \mathbf{x}_t, \mathbf{x}_0) = \mathcal{N}(\mathbf{x}_{t-1}; \tilde{\boldsymbol{\mu}}(\mathbf{x}_t, \mathbf{x}_0), \tilde{\beta}_t \mathbf{I}) \\
\tilde{\beta}_t = \frac{1 - \bar{\alpha}_{t-1}}{1 - \bar{\alpha}_t} \cdot \beta_t
$$
DDIM中建模的$q_\sigma(\mathbf{x}_{t-1} \vert \mathbf{x}_t, \mathbf{x}_0)$如下，第一个等式二三步用到了重参数技巧和多个独立高斯分布的等价形式，
$$
\begin{aligned}
\mathbf{x}_{t-1}
&= \sqrt{\bar{\alpha}_{t-1}}\mathbf{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1}}\boldsymbol{\epsilon}_{t-1} \\
&= \sqrt{\bar{\alpha}_{t-1}}\mathbf{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1} - \sigma_t^2} \boldsymbol{\epsilon}_t + \sigma_t\boldsymbol{\epsilon} \\
&= \sqrt{\bar{\alpha}_{t-1}}\mathbf{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1} - \sigma_t^2} \frac{\mathbf{x}_t - \sqrt{\bar{\alpha}_t}\mathbf{x}_0}{\sqrt{1 - \bar{\alpha}_t}} + \sigma_t\boldsymbol{\epsilon} \\
q_\sigma(\mathbf{x}_{t-1} \vert \mathbf{x}_t, \mathbf{x}_0)
&= \mathcal{N}(\mathbf{x}_{t-1}; \sqrt{\bar{\alpha}_{t-1}}\mathbf{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1} - \sigma_t^2} \frac{\mathbf{x}_t - \sqrt{\bar{\alpha}_t}\mathbf{x}_0}{\sqrt{1 - \bar{\alpha}_t}}, \sigma_t^2 \mathbf{I})
\end{aligned}
$$
DDIM中的的$\sigma_t$与DDPM中的$\tilde{\beta_t}$保持一致，并且添加了一个可学习参数控制方差，
$$
\sigma^2_t = \eta \cdot \tilde{\beta}_t
$$
当$\eta=0$时，采样过程是确定的；当$\eta=1$时，退化成DDPM的形式，以下给出推导，
$$
\begin{aligned}
\mu_{\mathbf{x}_{t-1}}
&= \sqrt{\bar{\alpha}_{t-1}}\mathbf{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1} - \sigma_t^2} \frac{\mathbf{x}_t - \sqrt{\bar{\alpha}_t}\mathbf{x}_0}{\sqrt{1 - \bar{\alpha}_t}} \\
&= \sqrt{\bar{\alpha}_{t-1}}\mathbf{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1} - \tilde{\beta}_t} \frac{\mathbf{x}_t - \sqrt{\bar{\alpha}_t}\mathbf{x}_0}{\sqrt{1 - \bar{\alpha}_t}} \\
&= \sqrt{\bar{\alpha}_{t-1}}\mathbf{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1} - \frac{(1 - \bar{\alpha}_{t-1}) \cdot (1 - \alpha_t)}{1 - \bar{\alpha}_t}} \frac{\mathbf{x}_t - \sqrt{\bar{\alpha}_t}\mathbf{x}_0}{\sqrt{1 - \bar{\alpha}_t}} \\
&= (\sqrt{\bar{\alpha}_{t-1}} + \sqrt{1 - \bar{\alpha}_{t-1} - \frac{(1 - \bar{\alpha}_{t-1}) \cdot (1 - \alpha_t)}{1 - \bar{\alpha}_t}} \cdot \frac{ - \sqrt{\bar{\alpha}_t}}{\sqrt{1 - \bar{\alpha}_t}}) \cdot \mathbf{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1} - \frac{(1 - \bar{\alpha}_{t-1}) \cdot (1 - \alpha_t)}{1 - \bar{\alpha}_t}} \cdot \frac{\mathbf{x}_t}{\sqrt{1 - \bar{\alpha}_t}} \\
&= (\sqrt{\bar{\alpha}_{t-1}} + \sqrt{(1 - \bar{\alpha}_{t-1}) \cdot (1 - \frac{1 - \alpha_t}{1 - \bar{\alpha}_t})} \cdot \frac{ - \sqrt{\bar{\alpha}_t}}{\sqrt{1 - \bar{\alpha}_t}}) \cdot \mathbf{x}_0 + \sqrt{(1 - \bar{\alpha}_{t-1}) \cdot (1 - \frac{1 - \alpha_t}{1 - \bar{\alpha}_t})} \cdot \frac{\mathbf{x}_t}{\sqrt{1 - \bar{\alpha}_t}} \\
&= (\sqrt{\bar{\alpha}_{t-1}} + \sqrt{(1 - \bar{\alpha}_{t-1}) \cdot \frac{\alpha_t - \bar\alpha_t}{1 - \bar{\alpha}_t}} \cdot \frac{ - \sqrt{\bar{\alpha}_t}}{\sqrt{1 - \bar{\alpha}_t}}) \cdot \mathbf{x}_0 + \sqrt{(1 - \bar{\alpha}_{t-1}) \frac{\alpha_t - \bar\alpha_t}{1 - \bar{\alpha}_t}} \cdot \frac{\mathbf{x}_t}{\sqrt{1 - \bar{\alpha}_t}} \\
&= (\sqrt{\bar{\alpha}_{t-1}} + \sqrt{1 - \bar{\alpha}_{t-1}} \cdot \frac{\sqrt{\alpha_t - \bar\alpha_t}}{\sqrt{{1 - \bar{\alpha}_t}}} \cdot \frac{ - \sqrt{\bar{\alpha}_t}}{\sqrt{1 - \bar{\alpha}_t}}) \cdot \mathbf{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1}} \cdot \frac{\sqrt{\alpha_t - \bar\alpha_t}}{\sqrt{{1 - \bar{\alpha}_t}}} \cdot \frac{\mathbf{x}_t}{\sqrt{1 - \bar{\alpha}_t}} \\
&= (\sqrt{\bar{\alpha}_{t-1}} - \sqrt{1 - \bar{\alpha}_{t-1}} \cdot \frac{\sqrt{1 - \bar\alpha_{t-1}} \cdot \sqrt{\alpha_t}}{\sqrt{{1 - \bar{\alpha}_t}}} \cdot \frac{\sqrt{\bar{\alpha}_t}}{\sqrt{1 - \bar{\alpha}_t}}) \cdot \mathbf{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1}} \cdot \frac{\sqrt{1 - \bar\alpha_{t-1}} \cdot \sqrt{\alpha_t}}{\sqrt{{1 - \bar{\alpha}_t}}} \cdot \frac{\mathbf{x}_t}{\sqrt{1 - \bar{\alpha}_t}} \\
&= (\sqrt{\bar{\alpha}_{t-1}} - \frac{\sqrt{\alpha_t} \cdot \sqrt{\bar\alpha_t} \cdot (1 - \bar\alpha_{t-1})}{1 - \bar\alpha_{t}}) \cdot \mathbf{x}_0 + \frac{\sqrt{\alpha_t} \cdot (1 - \bar\alpha_{t-1}) \cdot \mathbf{x}_t}{1 - \bar\alpha_{t}} \\
&= (\frac{\sqrt{\bar{\alpha}_{t-1}} - \bar\alpha_t \cdot \sqrt{\bar{\alpha}_{t-1}} - \sqrt{\alpha_t} \cdot \sqrt{\bar\alpha_t} \cdot + \sqrt{\alpha_t} \cdot \sqrt{\bar\alpha_t} \cdot \bar\alpha_{t-1}}{1 - \bar\alpha_{t}}) \cdot \mathbf{x}_0 + \frac{\sqrt{\alpha_t} \cdot (1 - \bar\alpha_{t-1}) \cdot \mathbf{x}_t}{1 - \bar\alpha_{t}} \\
&= (\frac{\sqrt{\bar{\alpha}_{t-1}} - \bar\alpha_t \cdot \sqrt{\bar{\alpha}_{t-1}} - \sqrt{\alpha_{t-1}} \cdot \bar\alpha_t + \sqrt{\bar\alpha_{t-1}} \cdot \bar\alpha_t}{1 - \bar\alpha_{t}}) \cdot \mathbf{x}_0 + \frac{\sqrt{\alpha_t} \cdot (1 - \bar\alpha_{t-1}) \cdot \mathbf{x}_t}{1 - \bar\alpha_{t}} \\
&= (\frac{\sqrt{\bar{\alpha}_{t-1}} - \sqrt{\alpha_{t-1}} \cdot \bar\alpha_t }{1 - \bar\alpha_{t}}) \cdot \mathbf{x}_0 + \frac{\sqrt{\alpha_t} \cdot (1 - \bar\alpha_{t-1}) \cdot \mathbf{x}_t}{1 - \bar\alpha_{t}} \\
&= \mu_{\mathbf{x}_{t-1}}^{DDPM}
\end{aligned}
$$
