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13.1

$$ p(\boldsymbol{x})=\sum_{i=1}^{N} \alpha_{i} \cdot p\left(\boldsymbol{x} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right) $$

[解析]： 高斯混合分布的定义式。

13.2

$$ \begin{aligned} f(\boldsymbol{x}) &=\underset{j \in \mathcal{Y}}{\arg \max } p(y=j | \boldsymbol{x}) \\ &=\underset{j \in \mathcal{Y}}{\arg \max } \sum_{i=1}^{N} p(y=j, \Theta=i | \boldsymbol{x}) \\ &=\underset{j \in \mathcal{Y}}{\arg \max } \sum_{i=1}^{N} p(y=j | \Theta=i, \boldsymbol{x}) \cdot p(\Theta=i | \boldsymbol{x}) \end{aligned} $$

[解析]：从公式第 1 行到第 2 行是对概率进行边缘化(marginalization)；通过引入$\Theta$并对其求和 $\sum_{i=1}^N$以抵消引入的影响。从公式第 2 行到第 3 行推导如下 $$ \begin{aligned}p(y=j, \Theta=i | \boldsymbol{x}) &=\frac{p(y=j, \Theta=i, \boldsymbol{x})}{p(\boldsymbol{x})} \&=\frac{p(y=j, \Theta=i, \boldsymbol{x})}{p(\Theta=i, \boldsymbol{x})} \cdot \frac{p(\Theta=i, \boldsymbol{x})}{p(\boldsymbol{x})} \&=p(y=j | \Theta=i, \boldsymbol{x}) \cdot p(\Theta=i | \boldsymbol{x})\end{aligned} $$

13.3

$$ p(\Theta=i | \boldsymbol{x})=\frac{\alpha_{i} \cdot p\left(\boldsymbol{x} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)}{\sum_{i=1}^{N} \alpha_{i} \cdot p\left(\boldsymbol{x} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)} $$

[解析]：根据 13.1 $$ p(\boldsymbol{x})=\sum_{i=1}^{N} \alpha_{i} \cdot p\left(\boldsymbol{x} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right) $$ 因此 $$ \begin{aligned}p(\Theta=i | \boldsymbol{x})&=\frac{p(\Theta=i , \boldsymbol{x})}{P(x)}\&=\frac{\alpha_{i} \cdot p\left(\boldsymbol{x} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)}{\sum_{i=1}^{N} \alpha_{i} \cdot p\left(\boldsymbol{x} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)}\end{aligned} $$

13.4

$$ \begin{aligned} L L\left(D_{l} \cup D_{u}\right)=& \sum_{\left(x_{j}, y_{j}\right) \in D_{l}} \ln \left(\sum_{i=1}^{N} \alpha_{i} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right) \cdot p\left(y_{j} | \Theta=i, \boldsymbol{x}_{j}\right)\right) \\ &+\sum_{x_{j} \in D_{u}} \ln \left(\sum_{i=1}^{N} \alpha_{i} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)\right) \end{aligned} $$

[解析]：第二项很好解释，当不知道类别信息的时候，样本$x_j$的概率可以用式 13.1 表示，所有无类别信息的样本$D_u$的似然是所有样本的乘积，因为$\ln$函数是单调的，所以也可以将$\ln$函数作用于这个乘积消除因为连乘产生的数值计算问题。第一项引入了样本的标签信息，由 $$ p(y=j | \Theta=i, \boldsymbol{x})=\left\{\begin{array}{ll}1, & i=j \0, & i \neq j\end{array}\right. $$ 可知，这项限定了样本$x_j$只可能来自于$y_j$所对应的高斯分布。

13.5

$$ \gamma_{j i}=\frac{\alpha_{i} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)}{\sum_{i=1}^{N} \alpha_{i} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)} $$

[解析]：参见式 13.3，这项可以理解成样本$x_j$属于类别标签$i$(或者说由第$i$个高斯分布生成)的后验概率。其中$\alpha_i,\boldsymbol{\mu}{i}\boldsymbol{\Sigma}i$可以通过有标记样本预先计算出来。即： $$ \begin{array}{l}\alpha{i}=\frac{l{i}}{\left|D_{l}\right|}, \text { where }\left|D_{l}\right|=\sum_{i=1}^{N} l_{i} \\boldsymbol{\mu}{i}=\frac{1}{l{i}} \sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \boldsymbol{x}_{j} \\boldsymbol{\Sigma}{i}=\frac{1}{l{i}} \sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}\end{array} $$

13.6

$$ \boldsymbol{\mu}{i}=\frac{1}{\sum_{\boldsymbol{x}_{j} \in D{u}} \gamma_{j i}+l_{i}}\left(\sum_{\boldsymbol{x}_{j} \in D_{u}} \gamma_{j i} \boldsymbol{x}_{j}+\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \boldsymbol{x}_{j}\right) $$

[推导]：这项可以由$$\cfrac{\partial LL(D_l \cup D_u) }{\partial \mu_i}=0$$而得，将式 13.4 的两项分别记为： $$ \begin{aligned}LL(D_l)&=\sum_{(\boldsymbol{x_j},y_j \in D_l)}\ln\left(\sum_{s=1}^{N}\alpha_s \cdot p(\boldsymbol{x_j}\vert \boldsymbol{\mu}s,\boldsymbol{\Sigma}s) \cdot p(y_i|\Theta = s,\boldsymbol{x_j})\right)\&=\sum{(\boldsymbol{x_j},y_j \in D_l)}\ln\left(\sum{s=1}^{N}\alpha_{y_j} \cdot p(\boldsymbol{x_j} \vert \boldsymbol{\mu}{y_j},\boldsymbol{\Sigma}{y_j})\right)\LL(D_u)&=\sum_{\boldsymbol{x_j} \in D_u} \ln\left(\alpha_s \cdot p(\boldsymbol{x_j} | \boldsymbol{\mu}s,\boldsymbol{\Sigma}s)\right)\end{aligned} $$ 首先，$LL(D_l)$对$\boldsymbol{\mu_i}$求偏导，$LL(D_l)$求和号中只有$y_j=i$ 的项能留下来，即 $$ \begin{aligned}\frac{\partial L L\left(D{l}\right)}{\partial \boldsymbol{\mu}{i}} &=\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \frac{\partial \ln \left(\alpha_{i} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)\right)}{\partial \boldsymbol{\mu}{i}} \&=\sum{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \frac{1}{p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)} \cdot \frac{\partial p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)}{\partial \boldsymbol{\mu}{i}} \&=\sum{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \frac{1}{p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right) \cdot \boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right) \&=\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\end{aligned} $$ $LL(D_u)$对$\boldsymbol{\mu_i}$求导，参考 9.33 的推导： $$ \begin{aligned} \frac{\partial L L\left(D_{u}\right)}{\partial \boldsymbol{\mu}{i}} &=\sum_{\boldsymbol{x}_{j} \in D{u}} \frac{\alpha_{i}}{\sum_{s=1}^{N} \alpha_{s} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{s}, \boldsymbol{\Sigma}{s}\right)} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right) \cdot \boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right) \\ &=\sum_{\boldsymbol{x}_{j} \in D_{u}} \gamma_{j i} \cdot \boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right) \end{aligned} $$

综上， $$ \begin{aligned}\frac{\partial L L\left(D_{l} \cup D_{u}\right)}{\partial \boldsymbol{\mu}{i}} &=\sum{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)+\sum_{\boldsymbol{x}_{j} \in D_{u}} \gamma_{j i} \cdot \boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right) \&=\boldsymbol{\Sigma}{i}^{-1}\left(\sum{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)+\sum_{\boldsymbol{x}_{j} \in D{u}} \gamma_{j i} \cdot\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\right) \&=\boldsymbol{\Sigma}{i}^{-1}\left(\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \boldsymbol{x}_{j}+\sum_{\boldsymbol{x}_{j} \in D_{u}} \gamma_{j i} \cdot \boldsymbol{x}_{j}-\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \boldsymbol{\mu}{i}-\sum_{\boldsymbol{x}_{j} \in D{u}} \gamma_{j i} \cdot \boldsymbol{\mu}{i}\right)\end{aligned} $$ 令$\frac{\partial L L\left(D{l} \cup D_{u}\right)}{\partial \boldsymbol{\mu}{i}}=0$，两边同时左乘$\Sigma_i$并移项： $$ \sum_{\boldsymbol{x}_{j} \in D{u}} \gamma_{j i} \cdot \boldsymbol{\mu}{i}+\sum{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \boldsymbol{\mu}{i}=\sum_{\boldsymbol{x}_{j} \in D{u}} \gamma_{j i} \cdot \boldsymbol{x}_{j}+\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \boldsymbol{x}_{j} $$ 上式中，$\boldsymbol{\mu_i}$ 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$i$类样本的有标记 样本数目，因此

$$ \left(\sum_{x_{j} \in D_{u}} \gamma_{j i}+\sum_{\left(x_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} 1\right) \boldsymbol{\mu}{i}=\sum{x_{j} \in D_{u}} \gamma_{j i} \cdot \boldsymbol{x}_{j}+\sum_{\left(x_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \boldsymbol{x}_{j} $$

即得式 13.6。

13.7

$$ \begin{aligned}\boldsymbol{\Sigma}{i}=& \frac{1}{\sum_{\boldsymbol{x}_{j} \in D{u}} \gamma_{j i}+l_{i}}\left(\sum_{\boldsymbol{x}_{j} \in D_{u}} \gamma_{j i} \cdot\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}\right.\&\left.+\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}\right)\end{aligned} $$

[推导]：首先$LL(D_l)$对$\boldsymbol{\Sigma_i}$求偏导 ，类似于 13.6 $$ \begin{aligned} \frac{\partial L L\left(D_{l}\right)}{\partial \boldsymbol{\Sigma}{i}} &=\sum{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \frac{\partial \ln \left(\alpha_{i} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)\right)}{\partial \boldsymbol{\Sigma}{i}} \\ &=\sum{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \frac{1}{p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)} \cdot \frac{\partial p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)}{\partial \boldsymbol{\Sigma}{i}} \\ &=\sum{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \frac{1}{p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right) \cdot\left(\boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}-\boldsymbol{I}\right) \cdot \frac{1}{2} \boldsymbol{\Sigma}{i}^{-1}\\ &=\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i}\left(\boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}-\boldsymbol{I}\right) \cdot \frac{1}{2} \boldsymbol{\Sigma}{i}^{-1} \end{aligned} $$ 然后$LL(D_u)$ 对$\boldsymbol{\Sigma_i}$求偏导，类似于 9.35 $$ \frac{\partial L L\left(D_{u}\right)}{\partial \boldsymbol{\Sigma}{i}}=\sum_{\boldsymbol{x}_{j} \in D{u}} \gamma_{j i} \cdot\left(\boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}-\boldsymbol{I}\right) \cdot \frac{1}{2} \boldsymbol{\Sigma}{i}^{-1} $$ 综合可得： $$ \begin{aligned} \frac{\partial L L\left(D_{l} \cup D_{u}\right)}{\partial \boldsymbol{\Sigma}{i}}=& \sum_{\boldsymbol{x}_{j} \in D{u}} \gamma_{j i} \cdot\left(\boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}-\boldsymbol{I}\right) \cdot \frac{1}{2} \boldsymbol{\Sigma}{i}^{-1} \\ &+\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i}\left(\boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}-\boldsymbol{I}\right) \cdot \frac{1}{2} \boldsymbol{\Sigma}{i}^{-1} \=&\left(\sum_{\boldsymbol{x}_{j} \in D_{u}} \gamma_{j i} \cdot\left(\boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}-\boldsymbol{I}\right)\right.\\ &\left.+\sum{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i}\left(\boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}-\boldsymbol{I}\right)\right) \cdot \frac{1}{2} \boldsymbol{\Sigma}{i}^{-1} \end{aligned} $$ 令$\frac{\partial L L\left(D_{l} \cup D_{u}\right)}{\partial \boldsymbol{\Sigma}{i}}=0$，两边同时右乘$2\Sigma_i$并移项： $$ \begin{aligned} \sum_{\boldsymbol{x}_{j} \in D{u}} \gamma_{j i} \cdot \boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}+& \sum{\left(\boldsymbol{x}_{j}, y_{j} \in D_{l} \wedge y_{j}=i\right.} \boldsymbol{\Sigma}{i}^{-1}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top} \=& \sum_{\boldsymbol{x}_{j} \in D{u}} \gamma_{j i} \cdot \boldsymbol{I}+\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \boldsymbol{I} \\ &=\left(\sum_{\boldsymbol{x}_{j} \in D_{u}} \gamma_{j i}+l_{i}\right) \boldsymbol{I} \end{aligned} $$ 两边同时左乘以$\Sigma_i$： $$ \sum_{\boldsymbol{x}_{j} \in D_{u}} \gamma_{j i} \cdot\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}+\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i}\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)\left(\boldsymbol{x}_{j}-\boldsymbol{\mu}{i}\right)^{\top}=\left(\sum_{\boldsymbol{x}_{j} \in D_{u}} \gamma_{j i}+l_{i}\right) \boldsymbol{\Sigma}_{i} $$ 即得式 13.7。

13.8

$$ \alpha_{i}=\frac{1}{m}\left(\sum_{\boldsymbol{x}_{j} \in D_{u}} \gamma_{j i}+l_{i}\right) $$

[推导]：类似于式 9.36，写出$LL(D_l \cup D_u)$的拉格朗日形式 $$ \begin{aligned}\mathcal{L}\left(D_{l} \cup D_{u}, \lambda\right) &=L L\left(D_{l} \cup D_{u}\right)+\lambda\left(\sum_{s=1}^{N} \alpha_{s}-1\right) \&=L L\left(D_{l}\right)+L L\left(D_{u}\right)+\lambda\left(\sum_{s=1}^{N} \alpha_{s}-1\right)\end{aligned} $$

类似于式 9.37，对$\alpha_i$求偏导。对于$LL(D_u)$，求导结果与式 9.37 的推导过程一样 $$ \frac{\partial L L\left(D_{u}\right)}{\partial \alpha_{i}}=\sum_{\boldsymbol{x}_{j} \in D_{u}} \frac{1}{\sum_{s=1}^{N} \alpha_{s} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{s}, \boldsymbol{\Sigma}{s}\right)} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right) $$

对于$LL(D_l)$，类似于 13.6 和 13.7 的推导过程 $$ \begin{aligned}\frac{\partial L L\left(D_{l}\right)}{\partial \alpha_{i}} &=\sum_{\left(x_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \frac{\partial \ln \left(\alpha_{i} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)\right)}{\partial \alpha_{i}} \&=\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \frac{1}{\alpha_{i} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)} \cdot \frac{\partial\left(\alpha_{i} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)\right)}{\partial \alpha_{i}} \&=\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \frac{1}{\alpha_{i} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right) \&=\sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} \frac{1}{\alpha_{i}}=\frac{1}{\alpha_{i}} \cdot \sum_{\left(\boldsymbol{x}_{j}, y_{j}\right) \in D_{l} \wedge y_{j}=i} 1=\frac{l_{i}}{\alpha_{i}}\end{aligned} $$

上式推导过程中，重点注意变量是$\alpha_i$ ，$p(x_j|\mu_i,\Sigma_i)$是常量；最后一行$\alpha_i$相对于求和变量为常量，因此作为公因子提到求和号外面； $l_i$ 为第$i$类样本的有标记样本数目。

综合两项结果： $$ \frac{\partial \mathcal{L}\left(D_{l} \cup D_{u}, \lambda\right)}{\partial \alpha_{i}}=\frac{l_{i}}{\alpha_{i}}+\sum_{\boldsymbol{x}_{j} \in D_{u}} \frac{p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)}{\sum_{s=1}^{N} \alpha_{s} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{s}, \boldsymbol{\Sigma}{s}\right)}+\lambda $$

令$\cfrac{\partial LL(D_l \cup D_u) }{\partial \alpha_i}=0$ 并且两边同乘以$\alpha_i$，得 $$ \alpha_{i} \cdot \frac{l_{i}}{\alpha_{i}}+\sum_{\boldsymbol{x}_{j} \in D_{u}} \frac{\alpha_{i} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{i}, \boldsymbol{\Sigma}{i}\right)}{\sum_{s=1}^{N} \alpha_{s} \cdot p\left(\boldsymbol{x}_{j} | \boldsymbol{\mu}{s}, \boldsymbol{\Sigma}{s}\right)}+\lambda \cdot \alpha_{i}=0 $$

结合式 9.30 发现，求和号内即为后验概率$\gamma_{ji}$,即 $$ l_i+\sum_{x_i \in D_u} \gamma_{ji}+\lambda \alpha_i = 0 $$ 对所有混合成分求和，得 $$ \sum_{i=1}^N l_i+\sum_{i=1}^N \sum_{x_i \in D_u} \gamma_{ji}+\sum_{i=1}^N \lambda \alpha_i = 0 $$

这里$\sum_{i=1}^N \alpha_i =1$ ，因此$\sum_{i=1}^N \lambda \alpha_i=\lambda\sum_{i=1}^N \alpha_i=\lambda$，根据 9.30 中$\gamma_{ji}$表达式可知 $$ \sum_{i=1}^N \gamma_{ji} = \sum_{i =1}^{N} \cfrac{\alpha_i \cdot p(x_j|\mu_i,\Sigma_i)}{\Sigma_{s=1}^N \alpha_s \cdot p(x_j| \mu_s, \Sigma_s)}= \cfrac{\sum_{i =1}^{N}\alpha_i \cdot p(x_j|\mu_i,\Sigma_i)}{\sum_{s=1}^N \alpha_s \cdot p(x_j| \mu_s, \Sigma_s)}=1 $$

再结合加法满足交换律，所以 $$ \sum_{i=1}^N \sum_{x_i \in D_u} \gamma_{ji}=\sum_{x_i \in D_u} \sum_{i=1}^N \gamma_{ji} =\sum_{x_i \in D_u} 1=u $$

以上分析过程中，$\sum_{x_j\in D_u}$ 形式与$\sum_{j=1}^u$等价，其中u为未标记样本集的样本个数； $\sum_{i=1}^Nl_i=l$其中$l$为有标记样本集的样本个数；将这些结果代入 $$ \sum_{i=1}^N l_i+\sum_{i=1}^N \sum_{x_i \in D_u} \gamma_{ji}+\sum_{i=1}^N \lambda \alpha_i = 0 $$

解出$l+u+\lambda = 0$ 且$l+u =m$ 其中$m$为样本总个数，移项即得$\lambda = -m$ 最后代入整理解得 $$ l_i + \sum_{x_j \in{D_u}} \gamma_{ji}-\lambda \alpha_i = 0 $$ 整理即得式 13.8。

13.9

$$ \min {\boldsymbol{w}, \boldsymbol{b}, \boldsymbol{y}, \boldsymbol{\xi}} \frac{1}{2}|\boldsymbol{w}|{2}^{2}+C_{l} \sum_{i=1}^{l} \xi_{i}+C_{u} \sum_{i=l+1}^{m} \xi_{i}\\ \begin{aligned} \text { s.t. } &y_{i}\left(\boldsymbol{w}^{\mathrm{T}} \boldsymbol{x}_{i}+b\right) \geqslant 1-\xi_{i}, \quad i=1,2, \ldots, l\\ &\hat{y}{i}\left(\boldsymbol{w}^{\mathrm{T}} \boldsymbol{x}_{i}+b\right) \geqslant 1-\xi{i}, \quad i=l+1, l+2, \ldots, m\\ &\xi_{i} \geqslant 0, \quad i=1,2, \dots, m \end{aligned} $$

[解析]：这个公式和公式 6.35 基本一致，除了引入了无标记样本的松弛变量$\xi_i, i=l+1,\cdots m$和对应的权重系数$C_u$和无标记样本的标记指派$\hat{y}_i$。

13.12

$$ \begin{aligned} E(f) &=\frac{1}{2} \sum_{i=1}^{m} \sum_{j=1}^{m}(\mathbf{W}){i j}\left(f\left(\boldsymbol{x}_{i}\right)-f\left(\boldsymbol{x}_{j}\right)\right)^{2} \\ &=\frac{1}{2}\left(\sum_{i=1}^{m} d{i} f^{2}\left(\boldsymbol{x}_{i}\right)+\sum_{j=1}^{m} d_{j} f^{2}\left(\boldsymbol{x}_{j}\right)-2 \sum_{i=1}^{m} \sum_{j=1}^{m}(\mathbf{W}){i j} f\left(\boldsymbol{x}_{i}\right) f\left(\boldsymbol{x}_{j}\right)\right) \\ &=\sum_{i=1}^{m} d{i} f^{2}\left(\boldsymbol{x}_{i}\right)-\sum_{i=1}^{m} \sum_{j=1}^{m}(\mathbf{W})_{i j} f\left(\boldsymbol{x}_{i}\right) f\left(\boldsymbol{x}_{j}\right) \\ &=\boldsymbol{f}^{\mathrm{T}}(\mathbf{D}-\mathbf{W}) \boldsymbol{f} \end{aligned} $$

[解析]：首先解释下这个能量函数的定义。原则上，我们希望能量函数$E(f)$越小越好，对于节点$i,j$，如果它们不相邻，则$(\mathbf{W}){i j}=0$，如果它们相邻，则最小化能量函数要求$f(x_i)$和$f(x_j)$尽量相似，和逻辑相符。下面进行公式的推导，首先由二项展开可得： $$ \begin{aligned} E(f) &=\frac{1}{2} \sum_{i=1}^{m} \sum{j=1}^{m}(\mathbf{W}){i j}\left(f\left(\boldsymbol{x}_{i}\right)-f\left(\boldsymbol{x}_{j}\right)\right)^{2} \\ &=\frac{1}{2} \sum_{i=1}^{m} \sum{j=1}^{m}(\mathbf{W}){i j}\left(f^{2}\left(\boldsymbol{x}_{i}\right)-2 f\left(\boldsymbol{x}_{i}\right) f\left(\boldsymbol{x}_{j}\right)+f^{2}\left(\boldsymbol{x}_{j}\right)\right) \\ &=\frac{1}{2}\left( \sum_{i=1}^{m} \sum{j=1}^{m}(\mathbf{W}){i j} f^{2}\left(\boldsymbol{x}_{i}\right)+ \sum_{i=1}^{m} \sum{j=1}^{m}(\mathbf{W}){i j} f^{2}\left(\boldsymbol{x}_{j}\right)-2\sum_{i=1}^{m} \sum{j=1}^{m}(\mathbf{W}){i j} f\left(\boldsymbol{x}_{i}\right) f\left(\boldsymbol{x}_{j}\right)\right) \end{aligned} $$ 由于$\mathbf{W}$是一个对称矩阵，可以通过变量替换得到 $$ \begin{aligned} \sum_{i=1}^{m} \sum{j=1}^{m}(\mathbf{W}){i j} f^{2}\left(\boldsymbol{x}_{j}\right)&=\sum{j=1}^{m} \sum_{i=1}^{m}(\mathbf{W}){j i} f^{2}\left(\boldsymbol{x}_{i}\right)\\ &=\sum_{i=1}^{m} \sum{j=1}^{m}(\mathbf{W}){i j} f^{2}\left(\boldsymbol{x}_{i}\right)\\ &= \sum_{i=1}^{m} \sum{j=1}^{m}(\mathbf{W}){i j} f^{2}\left(\boldsymbol{x}_{j}\right) \end{aligned} $$ 因此$E(f)$可化简为 $$ \begin{aligned} E(f) &= \sum_{i=1}^{m} \sum{j=1}^{m}(\mathbf{W}){i j} f^{2}\left(\boldsymbol{x}_{i}\right)-\sum_{i=1}^{m} \sum{j=1}^{m}(\mathbf{W}){i j} f\left(\boldsymbol{x}_{i}\right) f\left(\boldsymbol{x}_{j}\right) \end{aligned} $$ 根据定义 $d_i=\sum{j=1}^{l+u}\left(\mathbf{W}\right){ij}$，且$m=l+u$则 $$ \begin{aligned} E(f)&=\sum_{i=1}^{m} d{i} f^{2}\left(\boldsymbol{x}_{i}\right)-\sum_{i=1}^{m} \sum_{j=1}^{m}(\mathbf{W})_{i j} f\left(\boldsymbol{x}_{i}\right) f\left(\boldsymbol{x}_{j}\right)\\ &=\boldsymbol{f}^{\mathrm{T}}\mathbf{D}\boldsymbol{f}-\boldsymbol{f}^{\mathrm{T}}\mathbf{W}\boldsymbol{f}\\ &=\boldsymbol{f}^{\mathrm{T}}(\mathbf{D}-\mathbf{W}) \boldsymbol{f} \end{aligned} $$

13.13

$$ \begin{aligned} E(f) &=\left(\boldsymbol{f}{l}^{\mathrm{T}} \boldsymbol{f}{u}^{\mathrm{T}}\right)\left(\left[\begin{array}{ll} \mathbf{D}{l l} & \mathbf{0}{l u} \\ \mathbf{0}{u l} & \mathbf{D}{u u} \end{array}\right]-\left[\begin{array}{ll} \mathbf{W}{l l} & \mathbf{W}{l u} \\ \mathbf{W}{u l} & \mathbf{W}{u u} \end{array}\right]\right)\left[\begin{array}{l} \boldsymbol{f}{l} \\ \boldsymbol{f}{u} \end{array}\right] \\ &=\boldsymbol{f}{l}^{\mathrm{T}}\left(\mathbf{D}{l l}-\mathbf{W}{l l}\right) \boldsymbol{f}{l}-2 \boldsymbol{f}{u}^{\mathrm{T}} \mathbf{W}{u l} \boldsymbol{f}{l}+\boldsymbol{f}{u}^{\mathrm{T}}\left(\mathbf{D}{u u}-\mathbf{W}{u u}\right) \boldsymbol{f}_{u} \end{aligned} $$

[解析]：这里第一项西瓜书中的符号有歧义，应该表示成$\left[\begin{array}{ll} \boldsymbol{f}{l}^{\mathrm{T}} & \boldsymbol{f}{u}^{\mathrm{T}} \end{array}\right]$即一个$\mathbb{R}^{1\times(l+u)}$的行向量，根据矩阵乘法的定义，有： $$ \begin{aligned} E(f) &=\left[\begin{array}{ll} \boldsymbol{f}{l}^{\mathrm{T}} & \boldsymbol{f}{u}^{\mathrm{T}} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{D}{l l}-\boldsymbol{W}{l l} & -\boldsymbol{W}{l u} \\ -\boldsymbol{W}{u l} & \boldsymbol{D}{u u}-\boldsymbol{W}{u u} \end{array}\right]\left[\begin{array}{l} \boldsymbol{f}{l} \\ \boldsymbol{f}{u} \end{array}\right] \\ &=\left[\begin{array}{ll}\boldsymbol{f}{l}^{\mathrm{T}}\left(\boldsymbol{D}{l l}-\boldsymbol{W}{l l}\right)-\boldsymbol{f}{u}^{\mathrm{T}} \boldsymbol{W}{u l} & -\boldsymbol{f}{l}^{\mathrm{T}} \boldsymbol{W}{l u}+\boldsymbol{f}{u}^{\mathrm{T}}\left(\boldsymbol{D}{u u}-\boldsymbol{W}{u u}\right)\end{array}\right]\left[\begin{array}{l} \boldsymbol{f}{l} \\ \boldsymbol{f}{u} \end{array}\right] \\ &=\left(\boldsymbol{f}{l}^{\mathrm{T}}\left(\boldsymbol{D}{l l}-\boldsymbol{W}{l l}\right)-\boldsymbol{f}{u}^{\mathrm{T}} \boldsymbol{W}{u l}\right) \boldsymbol{f}{l}+\left(-\boldsymbol{f}{l}^{\mathrm{T}} \boldsymbol{W}{l u}+\boldsymbol{f}{u}^{\mathrm{T}}\left(\boldsymbol{D}{u u}-\boldsymbol{W}{u u}\right)\right) \boldsymbol{f}{u} \\ &=\boldsymbol{f}{l}^{\mathrm{T}}\left(\boldsymbol{D}{l l}-\boldsymbol{W}{l l}\right) \boldsymbol{f}{l}-\boldsymbol{f}{u}^{\mathrm{T}} \boldsymbol{W}{u l} \boldsymbol{f}{l}-\boldsymbol{f}{l}^{\mathrm{T}} \boldsymbol{W}{l u} \boldsymbol{f}{u}+\boldsymbol{f}{u}^{\mathrm{T}}\left(\boldsymbol{D}{u u}-\boldsymbol{W}{u u}\right) \boldsymbol{f}{u} \\ &=\boldsymbol{f}{l}^{\mathrm{T}}\left(\boldsymbol{D}{l l}-\boldsymbol{W}{l l}\right) \boldsymbol{f}{l}-2 \boldsymbol{f}{u}^{\mathrm{T}} \boldsymbol{W}{u l} \boldsymbol{f}{l}+\boldsymbol{f}{u}^{\mathrm{T}}\left(\boldsymbol{D}{u u}-\boldsymbol{W}{u u}\right) \boldsymbol{f}{u} \end{aligned} $$ 其中最后一步，$\boldsymbol{f}{l}^{\mathrm{T}} \boldsymbol{W}{l u} \boldsymbol{f}{u}=\left(\boldsymbol{f}{l}^{\mathrm{T}} \boldsymbol{W}{l u} \boldsymbol{f}{u}\right)^{\mathrm{T}}=f{u}^{\mathrm{T}} \boldsymbol{W}{u l} \boldsymbol{f}{l}$，因为这个式子的结果是一个标量。

13.14

$$ \begin{aligned} E(f) &=\left(\boldsymbol{f}{l}^{\mathrm{T}} \boldsymbol{f}{u}^{\mathrm{T}}\right)\left(\left[\begin{array}{ll} \mathbf{D}{l l} & \mathbf{0}{l u} \\ \mathbf{0}{u l} & \mathbf{D}{u u} \end{array}\right]-\left[\begin{array}{ll} \mathbf{W}{l l} & \mathbf{W}{l u} \\ \mathbf{W}{u l} & \mathbf{W}{u u} \end{array}\right]\right)\left[\begin{array}{l} \boldsymbol{f}{l} \\ \boldsymbol{f}{u} \end{array}\right] \\ &=\boldsymbol{f}{l}^{\mathrm{T}}\left(\mathbf{D}{l l}-\mathbf{W}{l l}\right) \boldsymbol{f}{l}-2 \boldsymbol{f}{u}^{\mathrm{T}} \mathbf{W}{u l} \boldsymbol{f}{l}+\boldsymbol{f}{u}^{\mathrm{T}}\left(\mathbf{D}{u u}-\mathbf{W}{u u}\right) \boldsymbol{f}_{u} \end{aligned} $$

[解析]：参考 13.13

13.15

$$ \boldsymbol{f}{u}=\left(\mathbf{D}{u u}-\mathbf{W}{u u}\right)^{-1} \mathbf{W}{u l} \boldsymbol{f}_{l} $$

[解析]：由 13.13，有 $$ \begin{aligned} \frac{\partial E(f)}{\partial \boldsymbol{f}{u}} &=\frac{\partial \boldsymbol{f}{l}^{\mathrm{T}}\left(\boldsymbol{D}{l l}-\boldsymbol{W}{l l}\right) \boldsymbol{f}{l}-2 \boldsymbol{f}{u}^{\mathrm{T}} \boldsymbol{W}{u l} \boldsymbol{f}{l}+\boldsymbol{f}{u}^{\mathrm{T}}\left(\boldsymbol{D}{u u}-\boldsymbol{W}{u u}\right) \boldsymbol{f}{u}}{\partial \boldsymbol{f}{u}} \\ &=-2 \boldsymbol{W}{u l} \boldsymbol{f}{l}+2\left(\boldsymbol{D}{u u}-\boldsymbol{W}{u u}\right) \boldsymbol{f}{u} \end{aligned} $$ 令结果等于 0 即得 13.15。

13.16

$$ \begin{aligned} \mathbf{P} &=\mathbf{D}^{-1} \mathbf{W}=\left[\begin{array}{cc} \mathbf{D}{l l}^{-1} & \mathbf{0}{l u} \\ \mathbf{0}{u l} & \mathbf{D}{u u}^{-1} \end{array}\right]\left[\begin{array}{ll} \mathbf{W}{l l} & \mathbf{W}{l u} \\ \mathbf{W}{u l} & \mathbf{W}{u u} \end{array}\right] \\ &=\left[\begin{array}{ll} \mathbf{D}{l l}^{-1} \mathbf{W}{l l} & \mathbf{D}{l l}^{-1} \mathbf{W}{l u} \\ \mathbf{D}{u u}^{-1} \mathbf{W}{u l} & \mathbf{D}{u u}^{-1} \mathbf{W}{u u} \end{array}\right] \end{aligned} $$

[解析]：根据矩阵乘法的定义计算可得该式，其中需要注意的是，对角矩阵$\mathbf{D}$的拟等于其各个对角元素的倒数。

13.17

$$ \begin{aligned} \boldsymbol{f}{u} &=\left(\mathbf{D}{u u}\left(\mathbf{I}-\mathbf{D}{u u}^{-1} \mathbf{W}{u u}\right)\right)^{-1} \mathbf{W}{u l} \boldsymbol{f}{l} \\ &=\left(\mathbf{I}-\mathbf{D}{u u}^{-1} \mathbf{W}{u u}\right)^{-1} \mathbf{D}{u u}^{-1} \mathbf{W}{u l} \boldsymbol{f}{l} \\ &=\left(\mathbf{I}-\mathbf{P}{u u}\right)^{-1} \mathbf{P}{u l} \boldsymbol{f}{l} \end{aligned} $$

[解析]：第一项到第二项是根据矩阵乘法逆的定义：$(\mathbf{A}\mathbf{B})^{-1}=\mathbf{B}^{-1}\mathbf{A}^{-1}$，在这个式子中​ $$ \begin{aligned} \mathbf{P}{u u}&=\mathbf{D}{u u}^{-1} \mathbf{W}{u u}\\ \mathbf{P}{ul}&=\mathbf{D}{u u}^{-1} \mathbf{W}{u l} \end{aligned} $$ 均可以根据$\mathbf{W}_{ij}$计算得到，因此可以通过标记$\mathbf{f}_l$计算未标记数据的标签$\mathbf{f}_u$。

13.20

$$ \mathbf{F}^{*}=\lim _{t \rightarrow \infty} \mathbf{F}(t)=(1-\alpha)(\mathbf{I}-\alpha \mathbf{S})^{-1} \mathbf{Y} $$

[解析]：由 13.19 $$ \mathbf{F}(t+1)=\alpha \mathbf{S} \mathbf{F}(t)+(1-\alpha) \mathbf{Y} $$ 当 t取不同的值时，有： $$ \begin{aligned} t=0: \mathbf{F}(1) &=\alpha \mathbf{S F}(0)+(1-\alpha) \mathbf{Y}\\ &=\alpha \mathbf{S} \mathbf{Y}+(1-\alpha) \mathbf{Y} \\ t=1: \mathbf{F}(2) &=\alpha \mathbf{S F}(1)+(1-\alpha) \mathbf{Y}=\alpha \mathbf{S}(\alpha \mathbf{S} \mathbf{Y}+(1-\alpha) \mathbf{Y})+(1-\alpha) \mathbf{Y} \\ &=(\alpha \mathbf{S})^{2} \mathbf{Y}+(1-\alpha)\left(\sum_{i=0}^{1}(\alpha \mathbf{S})^{i}\right) \mathbf{Y} \\ t=2:\mathbf{F}(3)&=\alpha\mathbf{S}\mathbf{F}(2)+(1-\alpha)\mathbf{Y}\&=\alpha \mathbf{S}\left((\alpha \mathbf{S})^{2} \mathbf{Y}+(1-\alpha)\left(\sum_{i=0}^{1}(\alpha \mathbf{S})^{i}\right) \mathbf{Y}\right)+(1-\alpha) \mathbf{Y} \\ &=(\alpha \mathbf{S})^{3} \mathbf{Y}+(1-\alpha)\left(\sum_{i=0}^{2}(\alpha \mathbf{S})^{i}\right) \mathbf{Y}\\ \end{aligned} $$ 可以观察到规律 $$ \mathbf{F}(t)=(\alpha \mathbf{S})^{t} \mathbf{Y}+(1-\alpha)\left(\sum_{i=0}^{t-1}(\alpha \mathbf{S})^{i}\right) \mathbf{Y} $$ 则 $$ \mathbf{F}^{*}=\lim _{t \rightarrow \infty}\mathbf{F}(t)=\lim _{t \rightarrow \infty}(\alpha \mathbf{S})^{t} \mathbf{Y}+\lim {t \rightarrow \infty}(1-\alpha)\left(\sum{i=0}^{t-1}(\alpha \mathbf{S})^{i}\right) \mathbf{Y} $$ 其中第一项由于$\mathbf{S}=\mathbf{D}^{-\frac{1}{2}} \mathbf{W} \mathbf{D}^{-\frac{1}{2}}$的特征值介于[-1, 1]之间(这里省略详细推导，可以参见 https://en.wikipedia.org/wiki/Laplacian_matrix 其中对称拉普拉斯矩阵的特征值介于 0 和 2 之间)，而$\alpha\in(0,1)$，所以$\lim _{t \rightarrow \infty}(\alpha \mathbf{S})^{t}=0$，第二项由等比数列公式 $$ \lim {t \rightarrow \infty} \sum{i=0}^{t-1}(\alpha \mathbf{S})^{i}=\frac{\mathbf{I}-\lim _{t \rightarrow \infty}(\alpha \mathbf{S})^{t}}{\mathbf{I}-\alpha \mathbf{S}}=\frac{\mathbf{I}}{\mathbf{I}-\alpha \mathbf{S}}=(\mathbf{I}-\alpha \mathbf{S})^{-1} $$ 综合可得式 13.20。