极大似然函数期望$E(LL)$如下式：
$$
\begin{aligned}
E(LL)&=\sum_{(x_j,y_j)\in D_l}\ln p(x_j,y_j)+\sum_{x_j\in D_u}\sum_i\gamma_{ji}\ln p(x_j,y_j=i)\
&=\sum_{(x_j,y_j)\in D_l}\ln[\alpha_{y_j}\Bbb{N}(x_j|\mu_{y_j},\Sigma_{y_j})]
+\sum_{x_j\in D_u}\sum_i\gamma_{ji}\ln[\alpha_i\Bbb{N}(x_j|\mu_i,\Sigma_i)]\
\end{aligned}
$$
多维的高斯分布的似然函数对均值矩阵和协方差矩阵求导为：
$$
\begin{aligned}
&\Bbb{N}(x_j|\mu_i,\Sigma_i)=\frac{1}{(2\pi)^{n/2}|\Sigma_i|^{1/2}}exp[-\frac{1}{2}(x_j-\mu_i)^T\Sigma_i^{-1}(x_j-\mu_i)]\
&\nabla_{\mu_i}\ln\Bbb{N}(x_j|\mu_i,\Sigma_i)=\Sigma_i^{-1}(x_j-\mu_i)\
&\nabla_{\Sigma_i^{-1}}\ln\Bbb{N}(x_j|\mu_i,\Sigma_i)=\frac{1}{2}[\Sigma_i-(x_j-\mu_i)(x_j-\mu_i)^T]
\end{aligned}
$$
最大化似然函数，并对均值矩阵和协方差矩阵求导，令其为0即可得$mu_i$和$\Sigma_i$:
$$
\begin{aligned}
&\bf\nabla_{\mu_i}E(LL)\
&=\sum_{(x_j,y_j)\in D_l}I(y_j=i)\Sigma_{i}^{-1}(x_j-\mu_i)+\sum_{x_j\in D_u}\gamma_{ji}\Sigma_{i}^{-1}(x_j-\mu_i)]\
&=0\
&\Rightarrow \mu_i=\frac{1}{l_i+\sum_{D_u}\gamma_{ji}}[\sum_{(x_j,y_j)\in D_l}I(y_j=i)x_j+\sum_{x_j\in D_u}\gamma_{ji},x_j]
\end{aligned}
$$
$$
\begin{aligned}
&\bf\nabla_{\Sigma_i^{-1}}E(LL)\\
&=\sum_{(x_j,y_j)\in D_l}I(y_j=i)\frac{1}{2}[\Sigma_i-(x_j-\mu_i)(x_j-\mu_i)^T]\\
&\quad+\sum_{x_j\in D_u}\gamma_{ji}\frac{1}{2}[\Sigma_i-(x_j-\mu_i)(x_j-\mu_i)^T\\
&=0\\
&\Rightarrow \Sigma_i=\frac{1}{l_i+\sum_{D_u}\gamma_{ji}}[\sum_{(x_j,y_j)\in D_l}I(y_j=i)(x_j-\mu_i)(x_j-\mu_i)^T\\
&\qquad\qquad+\sum_{x_j\in D_u}\gamma_{ji},(x_j-\mu_i)(x_j-\mu_i)^T]
\end{aligned}
$$
${\alpha_i}$可采用拉格朗日方法求解约束问题。