代修订13.6

docs/chapter13/chapter13.md

@@ -69,7 +69,8 @@ $$\begin{aligned}
 \sum_{(x_i,y_i)\in D_l \wedge y_j=i} \cfrac{\partial ln(\alpha_i \cdot p(x_j| \mu_i,\Sigma_i))}{\partial\mu_i}\\
 &=\sum_{(x_i,y_i)\in D_l \wedge y_j=i}\cfrac{1}{p(x_j|\mu_i,\Sigma_i) }\cdot  \cfrac{\partial p(x_j|\mu_i,\Sigma_i)}{\partial\mu_i}\\
 &=\sum_{(x_i,y_i)\in D_l \wedge y_j=i}\cfrac{1}{p(x_j|\mu_i,\Sigma_i) }\cdot  p(x_j|\mu_i,\Sigma_i)  \cdot \Sigma_i^{-1}(x_j-\mu_i)\\
-&=\sum_{x_j \in D_u } \Sigma_i^{-1}(x_j-\mu_i)
+&=\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i}
+\boldsymbol{\Sigma}_{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}_{i}\right)
 \end{aligned}$$
 
 对于式(13.4)中的第 2 项$LL(D_u)$，求导结果与式(9.33)的推导过程一样
@@ -85,7 +86,7 @@ $$
 $$
 上式中， 可以作为常量提到求和号外面，而$\sum_{\left(x_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} 1=l_{i}$，即第 类样本的有标记 样本数目，因此
 $$
-\left(\sum_{x_{j} \in D_{u}} \gamma_{j i}+\sum_{\left(x_{j}, y_{j}\right) \in D_{l} \backslash y_{j}=i} 1\right) \mu_{i}=\sum_{x_{j} \in D_{u}} \gamma_{j i} \cdot x_{j}+\sum_{\left(x_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} x_{j}
+\left(\sum_{x_{j} \in D_{u}} \gamma_{j i}+\sum_{\left(x_{j}, y_{j}\right) \in D_{l} \backslash y_{j}=i}  \mu_{i} \right) =\sum_{x_{j} \in D_{u}} \gamma_{j i} \cdot x_{j}+\sum_{\left(x_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} x_{j}
 $$
 即得式(13.6)；
 ## 13.7