1、来源
论文连接1：http://ganguli-gang.stanford.edu/pdf/DeepUnsupDiffusion.pdf
论文连接2(带appendix)：https://arxiv.org/pdf/1503.03585v7
代码链接：https://github.com/Sohl-Dickstein/Diffusion-Probabilistic-Models
代码的环境配置(基于theano)参考：https://blog.csdn.net/u010948546/article/details/146217516?spm=1001.2014.3001.5501
2、论文推理过程
扩散模型的流程如下图所示，可以看出 q ( x 0 , 1 , 2 ⋯ , T − 1 , T ) q(x^{0,1,2\cdots ,T-1, T}) q(x0,1,2⋯,T−1,T)为正向加噪音过程， p ( x 0 , 1 , 2 ⋯ , T − 1 , T ) p(x^{0,1,2\cdots ,T-1, T}) p(x0,1,2⋯,T−1,T)为逆向去噪音过程，具体过程参考https://blog.csdn.net/u010948546/article/details/144902864?spm=1001.2014.3001.5501。可以看出，逆向去噪的末端得到的图上还散布一些噪点。
2.1、名词解释
q ( x 0 ) q(x^0) q(x0)： x 0 x^0 x0 表示数据集的图像分布，例如在使用MNIST数据集时， x 0 x^0 x0就表示MNIST数据集中的图像，而 q ( x 0 ) q(x^0) q(x0)就表示数据集MNIST中数据集的分布情况。
p ( x T ) p(x^T) p(xT)： x T x^T xT表示 x 0 x^0 x0的加噪结果， x T x^T xT是逆向去噪的起点，因此 p ( x T ) p(x^T) p(xT)是去噪起点的分布情况。与 π ( x T ) \pi(x^T) π(xT)相同。
值得注意的是 p ( x t ) p(x^t) p(xt)与 q ( x t ) q(x^t) q(xt)是相同的。
2.2、推理过程
正向加噪过程满足马尔可夫性质，因此有公式1。
q ( x 0 , 1 , 2 ⋯ , T − 1 , T ) = q ( x 0 ) ⋅ ∏ t = 1 T q ( x t ∣ x t − 1 ) = q ( x 0 ) ⋅ q ( x 1 ∣ x 0 ) ⋅ q ( x 2 ∣ x 1 ) ... q ( x T ∣ x T − 1 ) . q ( x 1 , 2 ⋯ T ∣ x 0 ) = q ( x 1 ∣ x 0 ) ⋅ q ( x 2 ∣ x 1 ) ... q ( x T ∣ x T − 1 ) ) . \begin{equation} \begin{split} q(x^{0,1,2\cdots,T-1,T})&=q(x^0)\cdot \prod_{t=1}^{T}{q(x^t|x^{t-1})}=q(x^0)\cdot q(x^1|x^0)\cdot q(x^2|x^1)\dots q(x^T|x^{T-1}). \\ q(x^{1,2 \cdots T}|x^0)&=q(x^1|x^0)\cdot q(x^2|x^1)\dots q(x^T|x^{T-1})). \end{split} \end{equation} q(x0,1,2⋯,T−1,T)q(x1,2⋯T∣x0)=q(x0)⋅t=1∏Tq(xt∣xt−1)=q(x0)⋅q(x1∣x0)⋅q(x2∣x1)...q(xT∣xT−1).=q(x1∣x0)⋅q(x2∣x1)...q(xT∣xT−1)).
逆向去噪过程如公式2。
p θ ( x 0 , 1 , 2 ⋯ , T − 1 , T ) = p θ ( x T ) ⋅ ∏ t = 1 T p θ ( x t − 1 ∣ x t ) = p θ ( x T ) ⋅ p θ ( x T − 1 ∣ x T ) ⋅ p θ ( x T − 2 ∣ x T − 1 ) ... p θ ( x 0 ∣ x 1 ) . \begin{equation} p_{\theta}(x^{0,1,2\cdots,T-1,T})=p_{\theta}(x^T)\cdot \prod_{t=1}^{T}{p_{\theta}(x^{t-1}|x^{t})}=p_{\theta}(x^T)\cdot p_{\theta}(x^{T-1}|x^T)\cdot p_{\theta}(x^{T-2}|x^{T-1})\dots p_{\theta}(x^{0}|x^{1}). \end{equation} pθ(x0,1,2⋯,T−1,T)=pθ(xT)⋅t=1∏Tpθ(xt−1∣xt)=pθ(xT)⋅pθ(xT−1∣xT)⋅pθ(xT−2∣xT−1)...pθ(x0∣x1).
公式2中的参数 θ \theta θ就是深度学习模型中需要学习的参数。为了方便，省略公式2中的 θ \theta θ，因此公式2被重写为公式3。
p ( x 0 , 1 , 2 ⋯ , T − 1 , T ) = p ( x T ) ⋅ ∏ t = 1 T p ( x t − 1 ∣ x t ) = p ( x T ) ⋅ p ( x T − 1 ∣ x T ) ⋅ p ( x T − 2 ∣ x T − 1 ) ... p ( x 0 ∣ x 1 ) . \begin{equation} p(x^{0,1,2\cdots,T-1,T})=p(x^T)\cdot \prod_{t=1}^{T}{p(x^{t-1}|x^{t})}=p(x^T)\cdot p(x^{T-1}|x^T)\cdot p(x^{T-2}|x^{T-1})\dots p(x^{0}|x^{1}). \end{equation} p(x0,1,2⋯,T−1,T)=p(xT)⋅t=1∏Tp(xt−1∣xt)=p(xT)⋅p(xT−1∣xT)⋅p(xT−2∣xT−1)...p(x0∣x1).
逆向去噪的目标是使得其终点与正向加噪的起点相同。也就是使得 p ( x 0 ) p(x^0) p(x0)最大，即使得 逆向去噪过程为 x 0 x^0 x0的概率最大。
p ( x 0 ) = ∫ p ( x 0 , x 1 ) d x 1 ( 联合分布概率公式 ) = ∫ p ( x 1 ) ⋅ p ( x 0 ∣ x 1 ) d x 1 ( 贝叶斯概率公式 ) = ∫ ∫ p ( x 1 , x 2 ) d x 2 ⋅ p ( x 0 ∣ x 1 ) d x 1 ( 积分套积分 ) = ∫ ∫ p ( x 2 ) ⋅ p ( x 1 ∣ x 2 ) ⋅ p ( x 0 ∣ x 1 ) d x 1 d x 2 ( 改写为二重积分 ) = ∫ ∫ p ( x 2 ) ⋅ p ( x 1 ∣ x 2 ) ⋅ p ( x 0 ∣ x 1 ) d x 1 d x 2 = ⋮ = ∫ ∫ ⋯ ∫ p ( x T ) ⋅ p ( x T − 1 ∣ x T ) ⋅ p ( x T − 2 ∣ x − 1 ) ⋯ p ( x 0 ∣ x 1 ) ⋅ d x 1 d x 2 ⋯ d x T = ∫ p ( x 0 , 1 , 2 ⋯ T ) d x 1 , 2 ⋯ T ( T − 1 重积分 ) = ∫ d x 1 , 2 ⋯ T ⋅ p ( x 0 , 1 , 2 ⋯ T ) ⋅ q ( x 1 , 2 ⋯ T ∣ x 0 ) q ( x 1 , 2 ⋯ T ∣ x 0 ) = ∫ d x 1 , 2 ⋯ T ⋅ q ( x 1 , 2 ⋯ T ∣ x 0 ) ⋅ p ( x 0 , 1 , 2 ⋯ T ) q ( x 1 , 2 ⋯ T ∣ x 0 ) = ∫ d x 1 , 2 ⋯ T ⋅ q ( x 1 , 2 ⋯ T ∣ x 0 ) ⋅ p ( x T ) ⋅ p ( x T − 1 ∣ x T ) ⋅ p ( x T − 2 ∣ x T − 1 ) ... p ( x 0 ∣ x 1 ) q ( x 1 ∣ x 0 ) ⋅ q ( x 2 ∣ x 1 ) ... q ( x T ∣ x T − 1 ) = ∫ d x 1 , 2 ⋯ T ⋅ q ( x 1 , 2 ⋯ T ∣ x 0 ) ⋅ p ( x T ) ⋅ p ( x T − 1 ∣ x T ) ⋅ p ( x T − 2 ∣ x T − 1 ) ... p ( x 0 ∣ x 1 ) q ( x 1 ∣ x 0 ) ⋅ q ( x 2 ∣ x 1 ) ... q ( x T ∣ x T − 1 ) = ∫ d x 1 , 2 ⋯ T ⋅ q ( x 1 , 2 ⋯ T ∣ x 0 ) ⋅ p ( x T ) ⋅ ∏ t = 1 T p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) = E x 1 , 2 , ⋯ T ∼ q ( x 1 , 2 ⋯ T ∣ x 0 ) p ( x T ) ⋅ ∏ t = 1 T p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ( 改写为期望的形式 ) \begin{equation} \begin{split} p(x^0)&=\int p(x^0,x^1)dx^{1} (联合分布概率公式)\\ &=\int p(x^1)\cdot p(x^0|x^1)dx^1 (贝叶斯概率公式) \\ &=\int \int p(x1,x2)dx^2 \cdot p(x^0|x^1)dx^1 (积分套积分)\\ &=\int \int p(x^2)\cdot p(x^1|x^2) \cdot p(x^0|x^1)dx^1 dx^2(改写为二重积分)\\ &= \int \int p(x^2) \cdot p(x^1|x^2) \cdot p(x^0|x^1) dx^1 dx^2 \\ &= \vdots \\ &= \int \int \cdots \int p(x^T)\cdot p(x^{T-1}|x^{T})\cdot p(x^{T-2}|x^{-1})\cdots p(x^0|x^1) \cdot dx^1 dx^2 \cdots dx^T \\ &= \int p(x^{0,1,2 \cdots T})dx^{1,2\cdots T} (T-1重积分) \\ &= \int dx^{1,2\cdots T} \cdot p(x^{0,1,2 \cdots T}) \cdot \frac{q(x^{1,2 \cdots T}| x^0)}{q(x^{1,2 \cdots T}|x^0)} \\ &= \int dx^{1,2\cdots T} \cdot q(x^{1,2 \cdots T}| x^0) \cdot \frac{ p(x^{0,1,2 \cdots T}) }{q(x^{1,2 \cdots T}|x^0)} \\ &= \int dx^{1,2\cdots T} \cdot q(x^{1,2 \cdots T}| x^0) \cdot \frac{ p(x^T)\cdot p(x^{T-1}|x^T)\cdot p(x^{T-2}|x^{T-1})\dots p(x^{0}|x^{1})}{q(x^1|x^0)\cdot q(x^2|x^1)\dots q(x^T|x^{T-1})} \\ &= \int dx^{1,2\cdots T} \cdot q(x^{1,2 \cdots T}| x^0) \cdot p(x^T)\cdot \frac{ p(x^{T-1}|x^T)\cdot p(x^{T-2}|x^{T-1})\dots p(x^{0}|x^{1})}{q(x^1|x^0)\cdot q(x^2|x^1)\dots q(x^T|x^{T-1})} \\ &= \int dx^{1,2\cdots T} \cdot q(x^{1,2 \cdots T}| x^0) \cdot p(x^T)\cdot \prod_{t=1}^{T} \frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})} \\ &= E_{x^{1,2, \cdots T} \sim q(x^{1,2 \cdots T} | x^0)} p(x^T)\cdot \prod_{t=1}^{T} \frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})} (改写为期望的形式)\\ \end{split} \end{equation} p(x0)=∫p(x0,x1)dx1(联合分布概率公式)=∫p(x1)⋅p(x0∣x1)dx1(贝叶斯概率公式)=∫∫p(x1,x2)dx2⋅p(x0∣x1)dx1(积分套积分)=∫∫p(x2)⋅p(x1∣x2)⋅p(x0∣x1)dx1dx2(改写为二重积分)=∫∫p(x2)⋅p(x1∣x2)⋅p(x0∣x1)dx1dx2=⋮=∫∫⋯∫p(xT)⋅p(xT−1∣xT)⋅p(xT−2∣x−1)⋯p(x0∣x1)⋅dx1dx2⋯dxT=∫p(x0,1,2⋯T)dx1,2⋯T(T−1重积分)=∫dx1,2⋯T⋅p(x0,1,2⋯T)⋅q(x1,2⋯T∣x0)q(x1,2⋯T∣x0)=∫dx1,2⋯T⋅q(x1,2⋯T∣x0)⋅q(x1,2⋯T∣x0)p(x0,1,2⋯T)=∫dx1,2⋯T⋅q(x1,2⋯T∣x0)⋅q(x1∣x0)⋅q(x2∣x1)...q(xT∣xT−1)p(xT)⋅p(xT−1∣xT)⋅p(xT−2∣xT−1)...p(x0∣x1)=∫dx1,2⋯T⋅q(x1,2⋯T∣x0)⋅p(xT)⋅q(x1∣x0)⋅q(x2∣x1)...q(xT∣xT−1)p(xT−1∣xT)⋅p(xT−2∣xT−1)...p(x0∣x1)=∫dx1,2⋯T⋅q(x1,2⋯T∣x0)⋅p(xT)⋅t=1∏Tq(xt∣xt−1)p(xt−1∣xt)=Ex1,2,⋯T∼q(x1,2⋯T∣x0)p(xT)⋅t=1∏Tq(xt∣xt−1)p(xt−1∣xt)(改写为期望的形式)
因此公式3中的参数 θ \theta θ应满足
θ = a r g max θ p ( x 0 ) . \begin{equation} \theta= arg \underset {\theta}{\text{max}} p(x^0). \end{equation} θ=argθmaxp(x0).
公式4是对数据集中的一张图片进行求解，然而数据集中通常是有成千上万张图像的。假设数据集中有 N N N张图像，因此有公式6，其目的是求得一组参数 θ \theta θ，使得 L L L取得最大值。值得注意的是 q ( x 0 ) q(x^0) q(x0)表示数据集中每张图片被采样出来的概率。
L = ∑ n = 0 N q ( x 0 ) ⋅ l o g ( p ( x 0 ) ) = ∫ d x 0 ⋅ q ( x 0 ) ⋅ l o g ( p ( x 0 ) ) = ∫ d x 0 ⋅ q ( x 0 ) ⋅ l o g [ E x 1 , 2 , ⋯ T ∼ q ( x 1 , 2 ⋯ T ∣ x 0 ) p ( x T ) ⋅ ∏ t = 1 T p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ] ≥ ∫ d x 0 ⋅ q ( x 0 ) ⋅ E x 1 , 2 , ⋯ T ∼ q ( x 1 , 2 ⋯ T ∣ x 0 ) l o g [ p ( x T ) ⋅ ∏ t = 1 T p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ] = ∫ d x 0 ⋅ q ( x 0 ) ∫ q ( x 1 , 2 ⋯ T ∣ x 0 ) ⋅ l o g [ p ( x T ) ⋅ ∏ t = 1 T p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ] ⋅ d x 1 , 2 ⋯ T = ∫ d x 0 , 1 , 2 ⋯ T q ( x 0 ) ⋅ q ( x 1 , 2 ⋯ T ∣ x 0 ) ⋅ l o g [ p ( x T ) ⋅ ∏ t = 1 T p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ] = ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x T ) ⋅ ∏ t = 1 T p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ] = ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ ∏ t = 1 T p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ] + ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x T ) ] = K \begin{equation} \begin{split} L&=\sum_{n=0}^{N} q(x^0)\cdot log(p(x^0)) \\ &=\int dx^0\cdot q(x^0)\cdot log(p(x^0)) \\ &=\int dx^0\cdot q(x^0)\cdot log [ E_{x^{1,2, \cdots T} \sim q(x^{1,2 \cdots T} | x^0)} p(x^T)\cdot \prod_{t=1}^{T} \frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})}] \\ & \geq \int dx^0\cdot q(x^0)\cdot E_{x^{1,2, \cdots T} \sim q(x^{1,2 \cdots T} | x^0)} log [p(x^T)\cdot \prod_{t=1}^{T} \frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})}]\\ &= \int dx^0\cdot q(x^0) \int q(x^{1,2 \cdots T}| x^0) \cdot log [p(x^T)\cdot \prod_{t=1}^{T} \frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})}] \cdot dx^{1,2\cdots T}\\ &= \int dx^{0,1,2\cdots T} q(x^0) \cdot q(x^{1,2 \cdots T}| x^0) \cdot log [p(x^T)\cdot \prod_{t=1}^{T} \frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})}] \\ &= \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log [p(x^T)\cdot \prod_{t=1}^{T} \frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})}] \\ &= \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log [\prod_{t=1}^{T} \frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})}] + \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log [p(x^T)] \\ &= K \\ \end{split} \end{equation} L=n=0∑Nq(x0)⋅log(p(x0))=∫dx0⋅q(x0)⋅log(p(x0))=∫dx0⋅q(x0)⋅log[Ex1,2,⋯T∼q(x1,2⋯T∣x0)p(xT)⋅t=1∏Tq(xt∣xt−1)p(xt−1∣xt)]≥∫dx0⋅q(x0)⋅Ex1,2,⋯T∼q(x1,2⋯T∣x0)log[p(xT)⋅t=1∏Tq(xt∣xt−1)p(xt−1∣xt)]=∫dx0⋅q(x0)∫q(x1,2⋯T∣x0)⋅log[p(xT)⋅t=1∏Tq(xt∣xt−1)p(xt−1∣xt)]⋅dx1,2⋯T=∫dx0,1,2⋯Tq(x0)⋅q(x1,2⋯T∣x0)⋅log[p(xT)⋅t=1∏Tq(xt∣xt−1)p(xt−1∣xt)]=∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[p(xT)⋅t=1∏Tq(xt∣xt−1)p(xt−1∣xt)]=∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[t=1∏Tq(xt∣xt−1)p(xt−1∣xt)]+∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[p(xT)]=K
因此有公式
K = ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ ∏ t = 1 T p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ] ⏟ K 1 + ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x T ) ] ⏟ K 2 = K 1 + K 2 \begin{equation} \begin{split} K &= \underbrace{\int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log [\prod_{t=1}^{T} \frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})}]}{K1} + \underbrace{\int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log [p(x^T)]}{K_2} \\ &=K_1 + K_2 \end{split} \end{equation} K=K1 ∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[t=1∏Tq(xt∣xt−1)p(xt−1∣xt)]+K2 ∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[p(xT)]=K1+K2
首先考虑 K K K中的第二项 K 2 K_2 K2。
K 2 = ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x T ) ] = ∫ q ( x 0 ) ⋅ q ( x 1 ∣ x 0 ) ⋅ q ( x 2 ∣ x 1 ) ⋯ q ( x T ∣ x T − 1 ) ⋅ l o g [ p ( x T ) ] ⋅ d x 0 d x 1 ⋯ d x T = ∫ ( ∫ q ( x 1 , x 0 ) ⋅ d x 0 ) ⋅ q ( x 2 ∣ x 1 ) ⋯ q ( x T ∣ x T − 1 ) ⋅ l o g [ p ( x T ) ] ⋅ d x 1 ⋯ d x T = ∫ q ( x 1 ) ⋅ q ( x 2 ∣ x 1 ) ⋯ q ( x T ∣ x T − 1 ) ⋅ l o g [ p ( x T ) ] ⋅ d x 1 ⋯ d x T = ∫ q ( x T ) ⋅ l o g [ p ( x T ) ] ⋅ d x T = ∫ p ( x T ) ⋅ l o g [ p ( x T ) ] ⋅ d x T = − H p ( x T ) \begin{equation} \begin{split} K_2 &= \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log [p(x^T)] \\ &= \int q(x^0)\cdot q(x^1|x^0) \cdot q(x^2|x^1) \cdots q(x^{T}|x^{T-1})\cdot log [p(x^T)] \cdot dx^0 dx^1\cdots dx^{T} \\ &= \int \bigg( \int q(x^1, x^0) \cdot dx^0 \bigg) \cdot q(x^2|x^1) \cdots q(x^{T}|x^{T-1})\cdot log [p(x^T)] \cdot dx^1\cdots dx^{T} \\ &= \int q(x^1) \cdot q(x^2|x^1) \cdots q(x^{T}|x^{T-1})\cdot log [p(x^T)] \cdot dx^1\cdots dx^{T} \\ &= \int q(x^T) \cdot log [p(x^T)] \cdot dx^{T} \\ &= \int p(x^T) \cdot log [p(x^T)] \cdot dx^{T} \\ &=-H_p(x^T) \end{split} \end{equation} K2=∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[p(xT)]=∫q(x0)⋅q(x1∣x0)⋅q(x2∣x1)⋯q(xT∣xT−1)⋅log[p(xT)]⋅dx0dx1⋯dxT=∫(∫q(x1,x0)⋅dx0)⋅q(x2∣x1)⋯q(xT∣xT−1)⋅log[p(xT)]⋅dx1⋯dxT=∫q(x1)⋅q(x2∣x1)⋯q(xT∣xT−1)⋅log[p(xT)]⋅dx1⋯dxT=∫q(xT)⋅log[p(xT)]⋅dxT=∫p(xT)⋅log[p(xT)]⋅dxT=−Hp(xT)
p ( x T ) p(x^T) p(xT)是一个均值为0，方差为1的高斯分布。参考【正态分布系列】正态分布的熵，可以计算出 K 2 K_2 K2如下所示。
K 2 = − H p ( x T ) = − ( 1 2 l o g [ 2 π σ 2 ] + 1 2 ) = − ( 1 2 l o g [ 2 π ] + 1 2 ) \begin{equation} \begin{split} K_2 &=-H_p(x^T) \\ &=-\bigg( \frac{1}{2} log[2 \pi \sigma^2] + \frac{1}{2} \bigg)\\ &=-\bigg( \frac{1}{2} log[2 \pi ] + \frac{1}{2} \bigg) \end{split} \end{equation} K2=−Hp(xT)=−(21log[2πσ2]+21)=−(21log[2π]+21)
在代码中的计算过程如下图红框所示。
接下来考虑 K 1 K_1 K1。值得注意的是，论文中说明，为了避免边界效应，因此强迫 p ( x 0 ∣ x 1 ) = q ( x 1 ∣ x 0 ) p(x^{0}|x^1)=q(x^1|x^{0}) p(x0∣x1)=q(x1∣x0)。
K 1 = ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ ∏ t = 1 T p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ] = ∑ t = 1 T ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ] = ∑ t = 2 T ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ] + ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x 0 ∣ x 1 ) q ( x 1 ∣ x 0 ) ] = ∑ t = 2 T ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x t − 1 ∣ x t ) q ( x t ∣ x t − 1 ) ] = ∑ t = 2 T ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t ) ⋅ q ( x t − 1 ) q ( x t ) ] = ∑ t = 2 T ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) ⋅ q ( x t − 1 ∣ x 0 ) q ( x t ∣ x 0 ) ] = ∑ t = 2 T ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) ] + ∑ t = 2 T ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ q ( x t − 1 ∣ x 0 ) q ( x t ∣ x 0 ) ] = ∑ t = 2 T ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) ] + ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ ∏ t = 2 T q ( x t − 1 ∣ x 0 ) q ( x t ∣ x 0 ) ] = ∑ t = 2 T ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) ] + ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ q ( x 1 ∣ x 0 ) q ( x T ∣ x 0 ) ] = ∑ t = 2 T ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) ] + ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ q ( x 1 ∣ x 0 ) ] − ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ q ( x T ∣ x 0 ) ] = ∑ t = 2 T ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ p ( x t − 1 ∣ x t ) q ( x t − 1 ∣ x t , x 0 ) ] + ∫ d x 0 d x 1 ⋯ d x T ⋅ q ( x 0 ) ⋅ q ( x 1 ∣ x 0 ) ⋅ q ( x 2 ∣ x 1 ) ⋯ q ( x T ∣ x T − 1 ) ⋅ l o g [ q ( x 1 ∣ x 0 ) ] − ∫ d x 0 , 1 , 2 ⋯ T ⋅ q ( x 0 , 1 , 2 ⋯ T ) ⋅ l o g [ q ( x T ∣ x 0 ) ] \begin{equation} \begin{split} K_1 &=\int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\prod_{t=1}^{T} \frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})}\bigg] \\ &= \sum_{t=1}^{T} \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})}\bigg] \\ &= \sum_{t=2}^{T} \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})}\bigg] + \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{ p(x^{0}|x^1)}{q(x^1|x^{0})}\bigg] \\ &= \sum_{t=2}^{T} \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{ p(x^{t-1}|x^t)}{q(x^t|x^{t-1})}\bigg] \\ &= \sum_{t=2}^{T} \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{ p(x^{t-1}|x^t)}{q(x^{t-1}|x^{t})}\cdot \frac{q(x^{t-1})}{q(x^t)}\bigg] \\ &= \sum_{t=2}^{T} \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{ p(x^{t-1}|x^t)}{q(x^{t-1}|x^{t}, x^0)}\cdot \frac{q(x^{t-1}|x^0)}{q(x^t|x^0)}\bigg] \\ &= \sum_{t=2}^{T} \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{ p(x^{t-1}|x^t)}{q(x^{t-1}|x^{t}, x^0)}\bigg] + \sum_{t=2}^{T} \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{q(x^{t-1}|x^0)}{q(x^t|x^0)}\bigg]\\ &= \sum_{t=2}^{T} \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{ p(x^{t-1}|x^t)}{q(x^{t-1}|x^{t}, x^0)}\bigg] + \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\prod_{t=2}^{T} \frac{q(x^{t-1}|x^0)}{q(x^t|x^0)}\bigg]\\ &= \sum_{t=2}^{T} \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{ p(x^{t-1}|x^t)}{q(x^{t-1}|x^{t}, x^0)}\bigg] + \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{q(x^{1}|x^0)}{q(x^T|x^0)}\bigg]\\ &= \sum_{t=2}^{T} \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{ p(x^{t-1}|x^t)}{q(x^{t-1}|x^{t}, x^0)}\bigg] + \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[q(x^{1}|x^0)\bigg] - \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[q(x^T|x^0)\bigg]\\ &= \sum_{t=2}^{T} \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[\frac{ p(x^{t-1}|x^t)}{q(x^{t-1}|x^{t}, x^0)}\bigg] + \int dx^{0}dx^{1} \cdots dx^{T} \cdot q(x^{0}) \cdot q(x^1|x^0) \cdot q(x^2|x^1) \cdots q(x^T|x^{T-1}) \cdot log \bigg[q(x^{1}|x^0)\bigg] - \int dx^{0,1,2\cdots T} \cdot q(x^{0,1,2 \cdots T}) \cdot log \bigg[q(x^T|x^0)\bigg]\\ \end{split} \end{equation} K1=∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[t=1∏Tq(xt∣xt−1)p(xt−1∣xt)]=t=1∑T∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xt∣xt−1)p(xt−1∣xt)]=t=2∑T∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xt∣xt−1)p(xt−1∣xt)]+∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(x1∣x0)p(x0∣x1)]=t=2∑T∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xt∣xt−1)p(xt−1∣xt)]=t=2∑T∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xt−1∣xt)p(xt−1∣xt)⋅q(xt)q(xt−1)]=t=2∑T∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xt−1∣xt,x0)p(xt−1∣xt)⋅q(xt∣x0)q(xt−1∣x0)]=t=2∑T∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xt−1∣xt,x0)p(xt−1∣xt)]+t=2∑T∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xt∣x0)q(xt−1∣x0)]=t=2∑T∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xt−1∣xt,x0)p(xt−1∣xt)]+∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[t=2∏Tq(xt∣x0)q(xt−1∣x0)]=t=2∑T∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xt−1∣xt,x0)p(xt−1∣xt)]+∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xT∣x0)q(x1∣x0)]=t=2∑T∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xt−1∣xt,x0)p(xt−1∣xt)]+∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(x1∣x0)]−∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xT∣x0)]=t=2∑T∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xt−1∣xt,x0)p(xt−1∣xt)]+∫dx0dx1⋯dxT⋅q(x0)⋅q(x1∣x0)⋅q(x2∣x1)⋯q(xT∣xT−1)⋅log[q(x1∣x0)]−∫dx0,1,2⋯T⋅q(x0,1,2⋯T)⋅log[q(xT∣x0)]