推导2

$\gamma_{jk}$代表第j个观测值是否选择了第k个模型，1：是，0：否 $\hat \gamma_{jk} = E(\gamma_{jk}|y,\theta)$代表在y和$\theta$已知的情况下$\gamma_{jk}$的条件期望。

$$\begin{aligned} \hat \gamma_{jk} & = E(\gamma_{jk}|y,\theta) \\ & = P(\gamma_{jk}=1|y,\theta) && {1} \\ & = \frac{P(\gamma_{jk}=1, y_j | \theta)}{\sum_kP(\gamma_{jk}=1, y_j | \theta)} && {2} \\ & = \frac{P(y_j | \gamma_{jk}=1, \theta)P(\gamma_{jk}=1|\theta)}{\sum_kP(y_j | \gamma_{jk}=1, \theta)P(\gamma_{jk}=1|\theta)} && {3} \\ & = \frac{a_k\phi(y_j|\theta_k)}{\sum_k a_k\phi(y_j|\theta_k)} && {4} \end{aligned}$$

关于以上公式的说明： （1）：离散变量\gamma_{jk}的取值范围为0和1。根据离散型变量的期望公式，得：

$$E(\gamma_{jk}|y,\theta) = 1 * P(\gamma_{jk}=1|y,\theta) + 0 * P(\gamma_{jk}=0|y,\theta)$$

（2）：

$$\begin{aligned} P(\gamma_{jk}=1|y,\theta) & = P(\gamma_{jk}=1|y_j,\theta) & \text{只有}y_j\text{与}P(\gamma_{jk})\text{有关,其它y都可以忽略} \\ & = \frac{P(\gamma_{jk}=1|y_j,\theta)}{\sum_k P(\gamma_{jk}|y_j=1,\theta)} & \text{根据}\gamma_{jk}\text{的定义,}\sum_k P(\gamma_{jk}=1) = 1,\text{与}y_j\text{和}\theta\text{无关} \\ & = \frac{P(\gamma_{jk}=1|y_j,\theta)P(y_j|\theta)}{\sum_k P(\gamma_{jk}|y_j=1,\theta)P(y_j|\theta)} & \text{分子分母同乘以}P(y_j|\theta) \\ & = \frac{P(\gamma_{jk}=1, y_j|\theta)}{\sum_k P(\gamma_{jk}=1, y_j|\theta)} & \text{贝叶斯公式,}P(A|B)P(B) = P(A, B) \end{aligned}$$

（3）：贝叶斯公式，P(A, B) = P(B|A)P(A) （4）：$P(\gamma_{jk}=1|\theta)$代表选择模型k的概率，为ak $P(y_j | \gamma_{jk}=1, \theta)$代表通过第k的模型得到yj的概率，为$\phi(y_j|\theta_k)$