GMM
解决高斯分布的单峰性，引入多个高斯模型加权平均拟合数据
几何角度
$$p(x)=\sum _{k=1}^{K} \alpha _k N( \mu _k, \Sigma _k), \sum \alpha = 1$$
生成模型角度利用离散变量 z 来选择来自哪一个高斯分布
$$p(x)=\sum\limits_zp(x,z)=\sum\limits_{k=1}^Kp(x,z=k)=\sum\limits_{k=1}^Kp(z=k)p(x|z=k)$$
得到 $$p(x)=\sum\limits_{k=1}^Kp_k\mathcal{N}(x|\mu_k,\Sigma_k)$$
[[MLE]] 求解求解上面的 px
$$\theta_{MLE}=\mathop{argmax}\limits_{\theta}\log p(X)=\mathop{argmax}{\theta}\sum\limits{i=1}^N\log p(x_i)=\mathop{argmax}\theta\sum\limits{i=1}^N\log \sum\limits_{k=1}^Kp_k\mathcal{N}(x_i|\mu_k,\Sigma_k)
$$log 中有连加号存在，无法求出解析解。
[[EM]] 求解EM 目标求解：$\theta ^{(t+1)} = \arg \max _{\theta} \int _Z \log [p(X,Z|\theta)]p(Z|X, \theta ^{(t)})dZ = \arg \max _{\theta} E _{z|x,{\theta ^t}} [\log p(X,Z|\theta)]$
E-Step
- $$Q(\theta,\theta^t)=\sum\limits_z[\log\prod\limits_{i=1}^Np(x_i,z_i|\theta)]\prod \limits_{i=1}^Np(z_i|x_i,\theta^t)\
=\sum\limits_z[\sum\limits_{i=1}^N\log p(x_i,z_i|\theta)]\prod \limits_{i=1}^Np(z_i|x_i,\theta^t)
$$
+ 对第一个累加号展开，第一项为：
+ $$\sum\limits_z\log p(x_1,z_1|\theta)\prod\limits_{i=1}^Np(z_i|x_i,\theta^t)=\sum\limits_z\log p(x_1,z_1|\theta)p(z_1|x_1,\theta^t)\prod\limits_{i=2}^Np(z_i|x_i,\theta^t)\\
=\sum\limits_{z_1}\log p(x_1,z_1|\theta)
p(z_1|x_1,\theta^t)\sum\limits_{z_2,\cdots,z_K}\prod\limits_{i=2}^Np(z_i|x_i,\theta^t)\
=\sum\limits_{z_1}\log p(x_1,z_1|\theta)p(z_1|x_1,\theta^t)$$
+ 后面与 1 无关求和结果为1
+ $$Q(\theta,\theta^t)=\sum\limits_{i=1}^N\sum\limits_{z_i}\log p(x_i,z_i|\theta)p(z_i|x_i,\theta^t)$$
+ $$p(x,z|\theta)=p(z|\theta)p(x|z,\theta)=p_z\mathcal{N}(x|\mu_z,\Sigma_z)$$
+ $$p(z|x,\theta^t)=\frac{p(x,z|\theta^t)}{p(x|\theta^t)}=\frac{p_z^t\mathcal{N}(x|\mu_z^t,\Sigma_z^t)}{\sum\limits_kp_k^t\mathcal{N}(x|\mu_k^t,\Sigma_k^t)}$$
+ $$Q=\sum\limits_{i=1}^N\sum\limits_{z_i}\log p_{z_i}\mathcal{N(x_i|\mu_{z_i},\Sigma_{z_i})}\frac{p_{z_i}^t\mathcal{N}(x_i|\mu_{z_i}^t,\Sigma_{z_i}^t)}{\sum\limits_kp_k^t\mathcal{N}(x_i|\mu_k^t,\Sigma_k^t)}$$
M-Step
$$Q=\sum\limits_{k=1}^K\sum\limits_{i=1}^N[\log p_k + \log \mathcal{N}(x_i|\mu_k,\Sigma_k)]p(z_i=k|x_i,\theta^t)$$
$$p_k^{t+1}=\mathop{argmax}{p_k}\sum\limits{k=1}^K\sum\limits_{i=1}^N[\log p_k+\log \mathcal{N}(x_i|\mu_k,\Sigma_k)]p(z_i=k|x_i,\theta^t)\ s.t.\ \sum\limits_{k=1}^Kp_k=1$$
化简 $$p_k^{t+1}=\mathop{argmax}{p_k}\sum\limits{k=1}^K\sum\limits_{i=1}^N\log p_kp(z_i=k|x_i,\theta^t)\ s.t.\ \sum\limits_{k=1}^Kp_k=1$$
[[Lagrange Multiplier]] $$L(p_k,\lambda)=\sum\limits_{k=1}^K\sum\limits_{i=1}^N\log p_kp(z_i=k|x_i,\theta^t)-\lambda(1-\sum\limits_{k=1}^Kp_k)$$
[[Lagrange]] $$L(p_k,\lambda)=\sum\limits_{k=1}^K\sum\limits_{i=1}^N\log p_kp(z_i=k|x_i,\theta^t)-\lambda(1-\sum\limits_{k=1}^Kp_k)$$
$$\frac{\partial}{\partial p_k}L=\sum\limits_{i=1}^N\frac{1}{p_k}p(z_i=k|x_i,\theta^t)+\lambda=0\
\Rightarrow \sum\limits_k\sum\limits_{i=1}^N\frac{1}{p_k}p(z_i=k|x_i,\theta^t)+\lambda\sum\limits_kp_k=0\
\Rightarrow\lambda=-N$$
+ $$p_k^{t+1}=\frac{1}{N}\sum\limits_{i=1}^Np(z_i=k|x_i,\theta^t)$$
+ $$\mu_k,\Sigma_k$$ 无约束参数，直接求导
对比 MLE 和 EM 在求解 GMM 问题上的区别
$\theta_{MLE}=\mathop{argmax}\limits_{\theta}\log p(X)=\mathop{argmax}{\theta}\sum\limits{i=1}^N\log p(x_i)=\mathop{argmax}\theta\sum\limits{i=1}^N\log \sum\limits_{k=1}^Kp_k\mathcal{N}(x_i|\mu_k,\Sigma_k)
$$Q=\sum\limits_{k=1}^K\sum\limits_{i=1}^N[\log p_k + \log \mathcal{N}(x_i|\mu_k,\Sigma_k)]p(z_i=k|x_i,\theta^t)$
极大似然需要计算 $$P(X)$$，EM 计算 $$P(X,Z)$$。$$P(X) = \sum _Z P(X,Z)$$，每次单独考虑 $$P(X,Z)$$，避免在 log 中求和。
E 和 M 具体的含义还没有整理！