网上关于hmm 3个问题的求法基本已经烂大街了（等完事了我也烂一次大街嗯o(*￣▽￣*)o，但是对于3种求法的公式推导倒是基本没有，自己推了一天才推完3个公式，记下来以防以后忘了
- 前向公式$$\begin{aligned} \alpha_j(t)&=P(o_1...o_t,x_t=i|\mu)\\ \alpha_j(t+1)&=P(o_1...o_{t+1},x_{t+1}=j|\mu)\\ &=P(o_1...o_{t+1}|x_{t+1}=j,\mu)P(x_{t+1}=j|\mu)\\ &=P(o_1...o_t|x_{t+1}=j,\mu)P(o_{t+1}|x_{t+1}=j,\mu)P(x_{t+1}=j|\mu)\\ &=P(o_1...o_t,x_{t+1}=j|\mu)P(o_{t+1}|x_{t+1}=j,\mu)\\ &=\sum_{i=1...N}P(o_1...o_t,x_t=i,x_{t+1}=j|\mu)P(o_{t+1}|x_{t+1}=j,\mu)\\ &=\sum_{i=1...N}P(o_1...o_t,x_{t+1}=j|x_t=i,\mu)P(x_t=i|\mu)P(o_{t+1}|x_{t+1}=j,\mu)\\ &=\sum_{i=1...N}P(o_1...o_t,x_t=i,\mu)P(x_{t+1}=j|x_t=i,\mu)P(o_{t+1}|x_{t+1}=j,\mu)\\ &=\sum_{i=1...N}\alpha_i(t)a_{ij}b_{jo_{t+1}} \end{aligned}$$
后向公式
$$\begin{aligned} \beta_t(i)&=P(o_{t+1}...o_T|x_t=i,\mu)\\ \beta_{t+1}(j)&=P(o_{t+2}...o_T|x_{t+1}=j,\mu) \end{aligned}$$由此，可得：
$$\begin{aligned} \sum_{i=1...N}\beta_{t+1}(j)P(x_{t+1}=j|\mu)&=P(o_{t+2}...o_T|\mu)\\ \sum_{i=1...N}\beta_t(i)P(x_t=i|\mu)&=P(o_{t+1}...o_T|\mu)\\ &=P(o_{t+1}|\mu)P(o_{t+2}...o_T|\mu)\\ &=P(o_{t+1}|\mu)\sum_{j=1...N}\beta_{t+1}(j)P(x_{t+1}=j|\mu)\\ &=\sum_{j=1...N}\beta_{t+1}(j)P(x_{t+1}=j|\mu)b_{jo_{t+1}}\\ &=\sum_{j=1...N}\beta_{t+1}(j)\sum_{i=1...N}P(x_{t}=i|\mu)P(x_{t+1}=j|x_{t}=i,\mu)b_{jo_{t+1}}\\ &=\sum_{i=1...N}\sum_{j=1...N}\beta_{t+1}(j)P(x_{t}=i|\mu)P(x_{t+1}=j|x_{t}=i,\mu)b_{jo_{t+1}}\\ &=\sum_{i=1...N}P(x_{t}=i|\mu)\sum_{j=1...N}\beta_{t+1}(j)P(x_{t+1}=j|x_{t}=i,\mu)b_{jo_{t+1}} \end{aligned}$$易知，若上式恒成立，则：
$$\begin{aligned} \beta_t(i)&=\sum_{j=1...N}\beta_{t+1}(j)P(x_{t+1}=j|x_{t}=i,\mu)b_{jo_{t+1}}\\ &=\sum_{j=1...N}\beta_{t+1}(j)a_{ji}b_{jo_{t+1}} \end{aligned}$$前向后向公式
$$\begin{aligned} \alpha_t(i)\beta_t(i)&=P(o_1...o_t,x_t=i|\mu)P(o_{t+1}...o_T|x_t=i,\mu)\\ &=P(o_1...o_t|x_t=i,\mu)P(o_{t+1}...o_T|x_t=i,\mu)P(x_t=i|\mu)\\ &=P(o_1...o_t,o_{t+1}...o_T|x_t=i,\mu)P(x_t=i|\mu)\\ &=P(o_1...o_T,x_t=i|\mu)\\ \alpha_t(i)a_{ij}b_{jo_{t+1}}\beta_{t+1}(j)&=P(o_1...o_t,x_t=i|\mu)a_{ij}b_{jo_{t+1}}P(o_{t+2}...o_T|x_{t+1}=j,\mu)\\ &=P(o_1...o_t,x_t=i,\mu)P(x_{t+1}=j|x_t=i,\mu)P(o_{t+1}|x_{t+1}=j,\mu)P(o_{t+2}...o_T|x_{t+1}=j,\mu)\\ &=P(o_1...o_t,x_{t+1}=j|x_t=i,\mu)P(x_t=i|\mu)P(o_{t+1}|x_{t+1}=j,\mu)P(o_{t+2}...o_T|x_{t+1}=j,\mu)\\ &=P(o_1...o_t,x_t=i,x_{t+1}=j|\mu)P(o_{t+1}|x_{t+1}=j,\mu)P(o_{t+2}...o_T|x_{t+1}=j,\mu)\\ &=P(o_1...o_t,x_t=i,x_{t+1}=j|\mu)P(o_{t+1}...o_T|x_{t+1}=j,\mu)\\ &=P(x_t=i,x_{t+1}=j,O|\mu)\\ P(x_t=i,x_{t+1}=j|O,\mu)&=\frac{P(x_t=i,x_{t+1}=j,O|\mu)}{P(O|\mu)}\\ &=\frac{\alpha_t(i)a_{ij}b_{jo_{t+1}}\beta_{t+1}(j)}{\sum_{i=1}^N\sum_{j=1}^N\alpha_t(i)a_{ij}b_{jo_{t+1}}\beta_{t+1}(j)} \end{aligned}$$