EM算法推导小记

  EM算法（Expectation-Maximization），即期望-最大化算法。目标是估计某个概率分布函数中的参数$\theta$，使得在事件$x$发生的期望值$E_x(\log P(x|\theta ))$最大化。通常该问题可以通过最大似然估计的方法求解，但在许多情况下，数据的分布十分复杂，我们必须要引入一个隐变量$z$，假设$x$是由$z$产生的，而$z$的分布是可以自己假设的，从而减少了分布模型的复杂度。下面来简单的推导一下EM算法的过程，以帮助自己理解和记忆。
  首先，我们有对数似然函数$$L(\theta )=\log P(x|\theta )$$引入隐变量$z$，可得到$$\log P(x|\theta )=\log P(x,z|\theta )-\log P(z|x,\theta )$$证明过程如下，根据贝叶斯公式$P(A|B)=\frac{P(AB)}{P(B)}$可得$$P(x,z|\theta )=\frac{P(x,z,\theta )}{P(\theta )}\Rightarrow P(\theta )=\frac{P(x,z,\theta )}{P(x,z|\theta ))}$$$$P(z|x,\theta )=\frac{P(x,z,\theta )}{P(x,\theta )}\Rightarrow P(x,\theta )=\frac{P(x,z,\theta )}{P(z|x,\theta ))}$$$$P(x|\theta )=\frac{P(x,\theta )}{P(\theta )}=\frac{P(x,z|\theta )}{P(z|x,\theta )}$$两边取对数可得$$\log P(x|\theta )=\log \frac{P(x,z|\theta )}{P(z|x,\theta )}=\log P(x,z|\theta )-\log P(z|x,\theta )$$做一个恒等变换$$\log P(x|\theta )=\log P(x,z|\theta )-\log P(z)-[\log P(z|x,\theta )-\log P(z)]$$$$\log P(x|\theta )=\log \frac{P(x,z|\theta )}{P(z)}-\log \frac{P(z|x,\theta )}{P(z)}$$两边分别乘以$P(z)$并对$z$取积分可得$${\int }_{z}P(z)\log P(x|\theta )dz={\int }_{z}P(z)\log \frac{P(x,z|\theta )}{P(z)}dz-{\int }_{z}P(z)\log \frac{P(z|x,\theta )}{P(z)}dz$$由于$P(x|\theta )$是与$z$无关的函数，因此等号左边为$${\int }_{z}P(z)\log P(x|\theta )dz=\log P(x|\theta ){\int }_{z}P(z)dz=\log P(x|\theta )$$等号右边化简可得$$\begin{gathered} {\int }_{z}P(z)\log \frac{P(x,z|\theta )}{P(z)}dz-{\int }_{z}P(z)\log \frac{P(z|x,\theta )}{P(z)}dz \\ ={\int }_{z}P(z)\log \frac{P(x,z|\theta )}{P(z)}dz-[{\int }_{z}P(z)\log P(z|x,\theta )-P(z)\log P(z)dz] \\ ={\int }_{z}P(z)\log \frac{P(x,z|\theta )}{P(z)}dz+[{\int }_{z}P(z)\log P(z)-P(z)\log P(z|x,\theta )dz] \\ ={\int }_{z}P(z)\log \frac{P(x,z|\theta )}{P(z)}dz+{\int }_{z}P(z)\log \frac{P(z)}{P(z|x,\theta )}dz \\ ={\int }_{z}P(z)\log \frac{P(x,z|\theta )}{P(z)}dz+D_{KL}(P(z)\Vert P(z|x,\theta )) \end{gathered}$$其中$D_{KL}(P(z)\Vert P(z|x,\theta ))\ge 0$表示KL散度，因此可得$$\log P(x|\theta )\ge {\int }_{z}P(z)\log \frac{P(x,z|\theta )}{P(z)}dz$$我们称不等号的右侧式子为ELBO(evidence lower bound)，当且仅当$D_{KL}(P(z)\Vert P(z|x,\theta ))=0$时等号成立，即$$P(z)=P(z|x,\theta )$$$P(z|x,\theta )$表示$z$的后验概率，我们可以令第$i$次迭代时，$z$的后验概率为$$P(z)=P(z|x,{\theta }^{(i)})$$则$$\begin{gathered} ELBO={\int }_{z}P(z)\log \frac{P(x,z|\theta )}{P(z)}dz \\ ={\int }_{z}P(z|x,{\theta }^{(i)})\log \frac{P(x,z|\theta )}{P(z|x,{\theta }^{(i)})}dz \end{gathered}$$计算在参数${\theta }^{(i)}$的条件下，期望值$E_{z|x,{\theta }^{(i)}}[\log P(x,z|\theta )]=ELBO$的过程就是EM算法中的E步，核心在于计算$z$的后验概率$P(z|x,\theta )$。M步则是寻找更优的模型参数$\theta$以最大化对数似然函数$\log P(x|\theta )$，其等价于最大化ELBO，即$$\begin{gathered} {\theta }^{(i+1)}=\underset{\theta }{\mathrm{argmax}}\log P(x|\theta ) \\ =\underset{\theta }{\mathrm{argmax}}ELBO \\ =\underset{\theta }{\mathrm{argmax}}{\int }_{z}P(z|x,{\theta }^{(i)})\log \frac{P(x,z|\theta )}{P(z|x,{\theta }^{(i)})}dz \\ =\underset{\theta }{\mathrm{argmax}}{\int }_z[P(z|x,{\theta }^{(i)})\log P(x,z|\theta )-P(z|x,{\theta }^{(i)})\log P(z|x,{\theta }^{(i)})]dz \\ =\underset{\theta }{\mathrm{argmax}}{\int }_{z}P(z|x,{\theta }^{(i)})\log P(x,z|\theta )dz \end{gathered}$$上式最后一步推导，是因为$P(z|x,{\theta }^{(i)})\log P(z|x,{\theta }^{(i)})$是与$\theta$无关的常数项，因此可以可以直接忽略。EM算法就是，通过先固定参数$\theta$调整后验概率$P(z)$，然后再固定$P(z)$，调整$\theta$使得期望达到最大。重复迭代E和M步，直至收敛。