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1. DDPM Introduction q q q - 一个固定#xff08;或预定义#xff09;的正向扩散过程#xff0c;逐渐向图像添加高斯噪声#xff0c;直到最终得到纯噪声。 p θ p_θ p…看过来看过去唯有此up主非常牛 Video Explaination(Chinese)
2. DDPM Introduction q q q - 一个固定或预定义的正向扩散过程逐渐向图像添加高斯噪声直到最终得到纯噪声。 p θ p_θ pθ​ - 一个学习到的反向去噪扩散过程其中神经网络被训练以逐渐去噪图像从纯噪声开始直到最终得到实际图像。 正向和反向过程都由 t t t索引发生在有限时间步数 T T T内DDPM的作者使用 T T T1000。您从 t 0 t0 t0开始从数据分布中采样一个真实图像 x 0 x_0 x0​正向过程在每个时间步 t t t中从高斯分布中采样一些噪声然后将其添加到前一个时间步的图像上。在足够大的 T T T和每个时间步添加噪声的良好安排下通过逐渐的过程最终在 t T tT tT处得到所谓的各向同性高斯分布。
3. Forward Process q q q 这个过程是一个马尔可夫链 x t xt xt​ 只依赖于 x t − 1 x{t-1} xt−1​。 q ( x t ∣ x t − 1 ) q(x{t} | x{t-1}) q(xt​∣xt−1​) 在每个时间步 t t t 按照已知的方差计划 β t β_{t} βt​ 添加高斯噪声。 x 0 → q ( x 1 ∣ x 0 ) x 1 → q ( x 2 ∣ x 1 ) x 2 → ⋯ → x T − 1 → q ( x t ∣ x t − 1 ) x T x_0 \overset{q(x_1 | x_0)}{\rightarrow} x_1 \overset{q(x_2 | x_1)}{\rightarrow} x2 \rightarrow \dots \rightarrow x{T-1} \overset{q(x{t} | x{t-1})}{\rightarrow} x_T x0​→q(x1​∣x0​)x1​→q(x2​∣x1​)x2​→⋯→xT−1​→q(xt​∣xt−1​)xT​ x t 1 − β t × x t − 1 β t × ϵ t x_t \sqrt{1-βt}\times x{t-1} \sqrt{βt}\times ϵ{t} xt​1−βt​ ​×xt−1​βt​ ​×ϵt​ β t β_t βt​ is not constant at each time step t t t. In fact one defines a so-called “variance schedule”, which can be linear, quadratic, cosine, etc. 0 β 1 β 2 β 3 ⋯ β T 1 0 β_1 β_2 β_3 \dots βT 1 0β1​β2​β3​⋯βT​1 ϵ t ϵ{t} ϵt​ Gaussian noise, sampled from standard normal distribution. x t 1 − β t × x t − 1 β t × ϵ t x_t \sqrt{1-βt}\times x{t-1} \sqrt{βt} \times ϵ{t} xt​1−βt​ ​×xt−1​βt​ ​×ϵt​ Define a t 1 − β t a_t 1 - β_t at​1−βt​ x t a t × x t − 1 1 − a t × ϵ t xt \sqrt{a{t}}\times x_{t-1} \sqrt{1-at} \times ϵ{t} xt​at​ ​×xt−1​1−at​ ​×ϵt​ 2.1 Relationship between x t xt xt​ and x t − 2 x{t-2} xt−2​ x t − 1 a t − 1 × x t − 2 1 − a t − 1 × ϵ t − 1 x{t-1} \sqrt{a{t-1}}\times x{t-2} \sqrt{1-a{t-1}} \times ϵ_{t-1} xt−1​at−1​ ​×xt−2​1−at−1​ ​×ϵt−1​ ⇓ \Downarrow ⇓ x t a t ( a t − 1 × x t − 2 1 − a t − 1 ϵ t − 1 ) 1 − a t × ϵ t xt \sqrt{a{t}} (\sqrt{a{t-1}}\times x{t-2} \sqrt{1-a{t-1}} ϵ{t-1}) \sqrt{1-a_t} \times ϵ_t xt​at​ ​(at−1​ ​×xt−2​1−at−1​ ​ϵt−1​)1−at​ ​×ϵt​ ⇓ \Downarrow ⇓ x t a t a t − 1 × x t − 2 a t ( 1 − a t − 1 ) ϵ t − 1 1 − a t × ϵ t xt \sqrt{a{t}a{t-1}}\times x{t-2} \sqrt{a{t}(1-a{t-1})} ϵ_{t-1} \sqrt{1-a_t} \times ϵt xt​at​at−1​ ​×xt−2​at​(1−at−1​) ​ϵt−1​1−at​ ​×ϵt​ Because $N(\mu{1},\sigma{1}^{2}) N(\mu{2},\sigma{2}^{2}) N(\mu{1}\mu{2},\sigma{1}^{2} \sigma_{2}^{2})\( Proof x t a t a t − 1 × x t − 2 a t ( 1 − a t − 1 ) 1 − a t × ϵ x_t \sqrt{a_{t}a_{t-1}}\times x_{t-2} \sqrt{a_{t}(1-a_{t-1}) 1-a_t} \times ϵ xt​at​at−1​ ​×xt−2​at​(1−at−1​)1−at​ ​×ϵ ⇓ \Downarrow ⇓ x t a t a t − 1 × x t − 2 1 − a t a t − 1 × ϵ x_t \sqrt{a_{t}a_{t-1}}\times x_{t-2} \sqrt{1-a_{t}a_{t-1}} \times ϵ xt​at​at−1​ ​×xt−2​1−at​at−1​ ​×ϵ 2.2 Relationship between x t x_t xt​ and x t − 3 x_{t-3} xt−3​ x t − 2 a t − 2 × x t − 3 1 − a t − 2 × ϵ t − 2 x_{t-2} \sqrt{a_{t-2}}\times x_{t-3} \sqrt{1-a_{t-2}} \times ϵ_{t-2} xt−2​at−2​ ​×xt−3​1−at−2​ ​×ϵt−2​ ⇓ \Downarrow ⇓ x t a t a t − 1 ( a t − 2 × x t − 3 1 − a t − 2 ϵ t − 2 ) 1 − a t a t − 1 × ϵ x_t \sqrt{a_{t}a_{t-1}}(\sqrt{a_{t-2}}\times x_{t-3} \sqrt{1-a_{t-2}} ϵ_{t-2}) \sqrt{1-a_{t}a_{t-1}}\times ϵ xt​at​at−1​ ​(at−2​ ​×xt−3​1−at−2​ ​ϵt−2​)1−at​at−1​ ​×ϵ ⇓ \Downarrow ⇓ x t a t a t − 1 a t − 2 × x t − 3 a t a t − 1 ( 1 − a t − 2 ) ϵ t − 2 1 − a t a t − 1 × ϵ x_t \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} \sqrt{a_{t}a_{t-1}(1-a_{t-2})} ϵ_{t-2} \sqrt{1-a_{t}a_{t-1}}\times ϵ xt​at​at−1​at−2​ ​×xt−3​at​at−1​(1−at−2​) ​ϵt−2​1−at​at−1​ ​×ϵ ⇓ \Downarrow ⇓ x t a t a t − 1 a t − 2 × x t − 3 a t a t − 1 − a t a t − 1 a t − 2 ϵ t − 2 1 − a t a t − 1 × ϵ x_t \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} \sqrt{a_{t}a_{t-1}-a_{t}a_{t-1}a_{t-2}} ϵ_{t-2} \sqrt{1-a_{t}a_{t-1}}\times ϵ xt​at​at−1​at−2​ ​×xt−3​at​at−1​−at​at−1​at−2​ ​ϵt−2​1−at​at−1​ ​×ϵ ⇓ \Downarrow ⇓ x t a t a t − 1 a t − 2 × x t − 3 ( a t a t − 1 − a t a t − 1 a t − 2 ) 1 − a t a t − 1 × ϵ x_t \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} \sqrt{(a_{t}a_{t-1}-a_{t}a_{t-1}a_{t-2}) 1-a_{t}a_{t-1}} \times ϵ xt​at​at−1​at−2​ ​×xt−3​(at​at−1​−at​at−1​at−2​)1−at​at−1​ ​×ϵ ⇓ \Downarrow ⇓ x t a t a t − 1 a t − 2 × x t − 3 1 − a t a t − 1 a t − 2 × ϵ x_t \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} \sqrt{1-a_{t}a_{t-1}a_{t-2}} \times ϵ xt​at​at−1​at−2​ ​×xt−3​1−at​at−1​at−2​ ​×ϵ 2.3 Relationship between x t x_t xt​ and x 0 x_0 x0​ x t a t a t − 1 × x t − 2 1 − a t a t − 1 × ϵ x_t \sqrt{a_{t}a_{t-1}}\times x_{t-2} \sqrt{1-a_{t}a_{t-1}}\times ϵ xt​at​at−1​ ​×xt−2​1−at​at−1​ ​×ϵ x t a t a t − 1 a t − 2 × x t − 3 1 − a t a t − 1 a t − 2 × ϵ x_t \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} \sqrt{1-a_{t}a_{t-1}a_{t-2}}\times ϵ xt​at​at−1​at−2​ ​×xt−3​1−at​at−1​at−2​ ​×ϵ x t a t a t − 1 a t − 2 a t − 3 . . . a t − ( k − 2 ) a t − ( k − 1 ) × x t − k 1 − a t a t − 1 a t − 2 a t − 3 . . . a t − ( k − 2 ) a t − ( k − 1 ) × ϵ x_t \sqrt{a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{t-(k-2)}a_{t-(k-1)}}\times x_{t-k} \sqrt{1-a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{t-(k-2)}a_{t-(k-1)}}\times ϵ xt​at​at−1​at−2​at−3​...at−(k−2)​at−(k−1)​ ​×xt−k​1−at​at−1​at−2​at−3​...at−(k−2)​at−(k−1)​ ​×ϵ x t a t a t − 1 a t − 2 a t − 3 . . . a 2 a 1 × x 0 1 − a t a t − 1 a t − 2 a t − 3 . . . a 2 a 1 × ϵ x_t \sqrt{a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1}}\times x_{0} \sqrt{1-a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1}}\times ϵ xt​at​at−1​at−2​at−3​...a2​a1​ ​×x0​1−at​at−1​at−2​at−3​...a2​a1​ ​×ϵ a ˉ t : a t a t − 1 a t − 2 a t − 3 . . . a 2 a 1 \bar{a}_{t} : a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1} aˉt​:at​at−1​at−2​at−3​...a2​a1​ x t a ˉ t × x 0 1 − a ˉ t × ϵ , ϵ ∼ N ( 0 , I ) x_{t} \sqrt{\bar{a}_t}\times x_0 \sqrt{1-\bar{a}_t}\times ϵ , ϵ \sim N(0,I) xt​aˉt​ ​×x0​1−aˉt​ ​×ϵ,ϵ∼N(0,I) ⇓ \Downarrow ⇓ q ( x t ∣ x 0 ) 1 2 π 1 − a ˉ t e ( − 1 2 ( x t − a ˉ t x 0 ) 2 1 − a ˉ t ) q(x_{t}|x_{0}) \frac{1}{\sqrt{2\pi } \sqrt{1-\bar{a}_{t}}} e^{\left ( -\frac{1}{2}\frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}} \right ) } q(xt​∣x0​)2π ​1−aˉt​ ​1​e(−21​1−aˉt​(xt​−aˉt​ ​x0​)2​) 3.Reverse Process p p p Because P ( A ∣ B ) P ( B ∣ A ) P ( A ) P ( B ) P(A|B) \frac{ P(B|A)P(A) }{ P(B) } P(A∣B)P(B)P(B∣A)P(A)​ p ( x t − 1 ∣ x t , x 0 ) q ( x t ∣ x t − 1 , x 0 ) × q ( x t − 1 ∣ x 0 ) q ( x t ∣ x 0 ) p(x_{t-1}|x_{t},x_{0}) \frac{ q(x_{t}|x_{t-1},x_{0})\times q(x_{t-1}|x_0)}{q(x_{t}|x_0)} p(xt−1​∣xt​,x0​)q(xt​∣x0​)q(xt​∣xt−1​,x0​)×q(xt−1​∣x0​)​ \)\(x_{t} \sqrt{a_t}x_{t-1}\sqrt{1-a_t}\times ϵ\)\( ~ \)N(\sqrt{at}x{t-1}, 1-a{t})\( \)$x{t-1} \sqrt{\bar{a}_{t-1}}x0 \sqrt{1-\bar{a}{t-1}}\times ϵ$\( ~ \)N( \sqrt{\bar{a}_{t-1}}x0, 1-\bar{a}{t-1})\( \)\(x_{t} \sqrt{\bar{a}_{t}}x_0 \sqrt{1-\bar{a}_{t}}\times ϵ\)\( ~ \)N( \sqrt{\bar{a}_{t}}x0, 1-\bar{a}{t})$ q ( x t ∣ x t − 1 , x 0 ) 1 2 π 1 − a t e ( − 1 2 ( x t − a t x t − 1 ) 2 1 − a t ) q(x{t}|x{t-1},x{0}) \frac{1}{\sqrt{2\pi } \sqrt{1-a{t}}} e^{\left ( -\frac{1}{2}\frac{(x_{t}-\sqrt{at}x{t-1})^2}{1-a{t}} \right ) } q(xt​∣xt−1​,x0​)2π ​1−at​ ​1​e(−21​1−at​(xt​−at​ ​xt−1​)2​) q ( x t − 1 ∣ x 0 ) 1 2 π 1 − a ˉ t − 1 e ( − 1 2 ( x t − 1 − a ˉ t − 1 x 0 ) 2 1 − a ˉ t − 1 ) q(x{t-1}|x{0}) \frac{1}{\sqrt{2\pi } \sqrt{1-\bar{a}{t-1}}} e^{\left ( -\frac{1}{2}\frac{(x{t-1}-\sqrt{\bar{a}{t-1}}x0)^2}{1-\bar{a}{t-1}} \right ) } q(xt−1​∣x0​)2π ​1−aˉt−1​ ​1​e(−21​1−aˉt−1​(xt−1​−aˉt−1​ ​x0​)2​) q ( x t ∣ x 0 ) 1 2 π 1 − a ˉ t e ( − 1 2 ( x t − a ˉ t x 0 ) 2 1 − a ˉ t ) q(x{t}|x{0}) \frac{1}{\sqrt{2\pi } \sqrt{1-\bar{a}{t}}} e^{\left ( -\frac{1}{2}\frac{(x{t}-\sqrt{\bar{a}_{t}}x0)^2}{1-\bar{a}{t}} \right ) } q(xt​∣x0​)2π ​1−aˉt​ ​1​e(−21​1−aˉt​(xt​−aˉt​ ​x0​)2​) q ( x t ∣ x t − 1 , x 0 ) × q ( x t − 1 ∣ x 0 ) q ( x t ∣ x 0 ) [ 1 2 π 1 − a t e ( − 1 2 ( x t − a t x t − 1 ) 2 1 − a t ) ] ∗ [ 1 2 π 1 − a ˉ t − 1 e ( − 1 2 ( x t − 1 − a ˉ t − 1 x 0 ) 2 1 − a ˉ t − 1 ) ] ÷ [ 1 2 π 1 − a ˉ t e ( − 1 2 ( x t − a ˉ t x 0 ) 2 1 − a ˉ t ) ] \frac{ q(x{t}|x{t-1},x{0})\times q(x{t-1}|x0)}{q(x{t}|x0)} \left [ \frac{1}{\sqrt{2\pi} \sqrt{1-a{t}}} e^{\left ( -\frac{1}{2}\frac{(x_{t}-\sqrt{at}x{t-1})^2}{1-a{t}} \right ) } \right ] * \left [ \frac{1}{\sqrt{2\pi} \sqrt{1-\bar{a}{t-1}}} e^{\left ( -\frac{1}{2}\frac{(x{t-1}-\sqrt{\bar{a}{t-1}}x0)^2}{1-\bar{a}{t-1}} \right ) } \right ] \div \left [ \frac{1}{\sqrt{2\pi} \sqrt{1-\bar{a}{t}}} e^{\left ( -\frac{1}{2}\frac{(x{t}-\sqrt{\bar{a}_{t}}x0)^2}{1-\bar{a}{t}} \right ) } \right ] q(xt​∣x0​)q(xt​∣xt−1​,x0​)×q(xt−1​∣x0​)​[2π ​1−at​ ​1​e(−21​1−at​(xt​−at​ ​xt−1​)2​)]∗[2π ​1−aˉt−1​ ​1​e(−21​1−aˉt−1​(xt−1​−aˉt−1​ ​x0​)2​)]÷[2π ​1−aˉt​ ​1​e(−21​1−aˉt​(xt​−aˉt​ ​x0​)2​)] ⇓ \Downarrow ⇓ 2 π 1 − a ˉ t 2 π 1 − a t 2 π 1 − a ˉ t − 1 e [ − 1 2 ( ( x t − a t x t − 1 ) 2 1 − a t ( x t − 1 − a ˉ t − 1 x 0 ) 2 1 − a ˉ t − 1 − ( x t − a ˉ t x 0 ) 2 1 − a ˉ t ) ] \frac{\sqrt{2\pi} \sqrt{1-\bar{a}{t}}}{\sqrt{2\pi} \sqrt{1-a{t}} \sqrt{2\pi} \sqrt{1-\bar{a}{t-1}} } e^{\left [ -\frac{1}{2} \left ( \frac{(x{t}-\sqrt{at}x{t-1})^2}{1-a{t}} \frac{(x{t-1}-\sqrt{\bar{a}_{t-1}}x0)^2}{1-\bar{a}{t-1}} - \frac{(x{t}-\sqrt{\bar{a}{t}}x0)^2}{1-\bar{a}{t}} \right ) \right ] } 2π ​1−at​ ​2π ​1−aˉt−1​ ​2π ​1−aˉt​ ​​e[−21​(1−at​(xt​−at​ ​xt−1​)2​1−aˉt−1​(xt−1​−aˉt−1​ ​x0​)2​−1−aˉt​(xt​−aˉt​ ​x0​)2​)] ⇓ \Downarrow ⇓ 1 2 π ( 1 − a t 1 − a ˉ t − 1 1 − a ˉ t ) e x p [ − 1 2 ( ( x t − a t x t − 1 ) 2 1 − a t ( x t − 1 − a ˉ t − 1 x 0 ) 2 1 − a ˉ t − 1 − ( x t − a ˉ t x 0 ) 2 1 − a ˉ t ) ] \frac{1}{\sqrt{2\pi} \left ( \frac{ \sqrt{1-at} \sqrt{1-\bar{a}{t-1}} } {\sqrt{1-\bar{a}{t}}} \right ) } exp{\left [ -\frac{1}{2} \left ( \frac{(x{t}-\sqrt{at}x{t-1})^2}{1-at} \frac{(x{t-1}-\sqrt{\bar{a}_{t-1}}x0)^2}{1-\bar{a}{t-1}} - \frac{(x{t}-\sqrt{\bar{a}{t}}x0)^2}{1-\bar{a}{t}} \right ) \right ] } 2π ​(1−aˉt​ ​1−at​ ​1−aˉt−1​ ​​)1​exp[−21​(1−at​(xt​−at​ ​xt−1​)2​1−aˉt−1​(xt−1​−aˉt−1​ ​x0​)2​−1−aˉt​(xt​−aˉt​ ​x0​)2​)] ⇓ \Downarrow ⇓ 1 2 π ( 1 − a t 1 − a ˉ t − 1 1 − a ˉ t ) e x p [ − 1 2 ( x t 2 − 2 a t x t x t − 1 a t x t − 1 2 1 − a t x t − 1 2 − 2 a ˉ t − 1 x 0 x t − 1 a ˉ t − 1 x 0 2 1 − a ˉ t − 1 − ( x t − a ˉ t x 0 ) 2 1 − a ˉ t ) ] \frac{1}{\sqrt{2\pi} \left ( \frac{ \sqrt{1-at} \sqrt{1-\bar{a}{t-1}} } {\sqrt{1-\bar{a}{t}}} \right ) } exp \left[ -\frac{1}{2} \left ( \frac{ x{t}^2-2\sqrt{at}x{t}x_{t-1}{at}x{t-1}^2 }{1-at} \frac{ x{t-1}^2-2\sqrt{\bar{a}_{t-1}}x0x{t-1}\bar{a}_{t-1}x0^2 }{1-\bar{a}{t-1}} - \frac{(x{t}-\sqrt{\bar{a}{t}}x0)^2}{1-\bar{a}{t}} \right) \right] 2π ​(1−aˉt​ ​1−at​ ​1−aˉt−1​ ​​)1​exp[−21​(1−at​xt2​−2at​ ​xt​xt−1​at​xt−12​​1−aˉt−1​xt−12​−2aˉt−1​ ​x0​xt−1​aˉt−1​x02​​−1−aˉt​(xt​−aˉt​ ​x0​)2​)] ⇓ \Downarrow ⇓ 1 2 π ( 1 − a t 1 − a ˉ t − 1 1 − a ˉ t ) e x p [ − 1 2 ( x t − 1 − ( a t ( 1 − a ˉ t − 1 ) 1 − a ˉ t x t a ˉ t − 1 ( 1 − a t ) 1 − a ˉ t x 0 ) ) 2 ( 1 − a t 1 − a ˉ t − 1 1 − a ˉ t ) 2 ] \frac{1}{\sqrt{2\pi} \left ( {\color{Red} \frac{ \sqrt{1-at} \sqrt{1-\bar{a}{t-1}} } {\sqrt{1-\bar{a}{t}}}} \right ) } exp \left[ -\frac{1}{2} \frac{ \left( x{t-1} - \left( {\color{Purple} \frac{\sqrt{at}(1-\bar{a}{t-1})}{1-\bar{a}_t}xt \frac{\sqrt{\bar{a}{t-1}}(1-a_t)}{1-\bar{a}_t}x_0} \right) \right) ^2 } { \left( {\color{Red} \frac{ \sqrt{1-at} \sqrt{1-\bar{a}{t-1}} } {\sqrt{1-\bar{a}{t}}}} \right)^2 } \right] 2π ​(1−aˉt​ ​1−at​ ​1−aˉt−1​ ​​)1​exp ​−21​(1−aˉt​ ​1−at​ ​1−aˉt−1​ ​​)2(xt−1​−(1−aˉt​at​ ​(1−aˉt−1​)​xt​1−aˉt​aˉt−1​ ​(1−at​)​x0​))2​ ​ ⇓ \Downarrow ⇓ p ( x t − 1 ∣ x t ) ∼ N ( a t ( 1 − a ˉ t − 1 ) 1 − a ˉ t x t a ˉ t − 1 ( 1 − a t ) 1 − a ˉ t x 0 , ( 1 − a t 1 − a ˉ t − 1 1 − a ˉ t ) 2 ) p(x{t-1}|x_{t}) \sim N\left( {\color{Purple} \frac{\sqrt{at}(1-\bar{a}{t-1})}{1-\bar{a}_t}xt \frac{\sqrt{\bar{a}{t-1}}(1-a_t)}{1-\bar{a}_t}x_0} , \left( {\color{Red} \frac{ \sqrt{1-at} \sqrt{1-\bar{a}{t-1}} } {\sqrt{1-\bar{a}{t}}}} \right)^2 \right) p(xt−1​∣xt​)∼N(1−aˉt​at​ ​(1−aˉt−1​)​xt​1−aˉt​aˉt−1​ ​(1−at​)​x0​,(1−aˉt​ ​1−at​ ​1−aˉt−1​ ​​)2) Because x t a ˉ t × x 0 1 − a ˉ t × ϵ x{t} \sqrt{\bar{a}_t}\times x_0 \sqrt{1-\bar{a}_t}\times ϵ xt​aˉt​ ​×x0​1−aˉt​ ​×ϵ, x 0 x t − 1 − a ˉ t × ϵ a ˉ t x_0 \frac{x_t - \sqrt{1-\bar{a}_t}\times ϵ}{\sqrt{\bar{a}_t}} x0​aˉt​ ​xt​−1−aˉt​ ​×ϵ​. Substitute x 0 x0 x0​ with this formula. p ( x t − 1 ∣ x t ) ∼ N ( a t ( 1 − a ˉ t − 1 ) 1 − a ˉ t x t a ˉ t − 1 ( 1 − a t ) 1 − a ˉ t × x t − 1 − a ˉ t × ϵ a ˉ t , β t ( 1 − a ˉ t − 1 ) 1 − a ˉ t ) p(x{t-1}|x_{t}) \sim N\left( {\color{Purple} \frac{\sqrt{at}(1-\bar{a}{t-1})}{1-\bar{a}_t}xt \frac{\sqrt{\bar{a}{t-1}}(1-a_t)}{1-\bar{a}_t}\times \frac{x_t - \sqrt{1-\bar{a}_t}\times ϵ}{\sqrt{\bar{a}t}} } , {\color{Red} \frac{ \beta{t} (1-\bar{a}{t-1}) } { 1-\bar{a}{t}}} \right) p(xt−1​∣xt​)∼N(1−aˉt​at​ ​(1−aˉt−1​)​xt​1−aˉt​aˉt−1​ ​(1−at​)​×aˉt​ ​xt​−1−aˉt​ ​×ϵ​,1−aˉt​βt​(1−aˉt−1​)​)