【Practical】HMM中的概率推导

• 对HMM中出现的概率公式进行推导，关于HMM可以阅读《隐马尔可夫模型》.

符号约定.

• 状态转移概率矩阵 $A$ 中的元素：$$A[i,j]=P(z_t=s_j|z_{t-1}=s_i)$$
• 输出概率矩阵 $B$ 中的元素：$$B[j,k]=P(x_t=v_k|z_t=s_j)$$公式中有些地方会以 $B[j,\ast x_t]$ 表示上述概率。
• 状态空间：$$S=\{s_1,s_2,\cdots \}$$
• 输出空间：$$V=\{v_1,v_2,\cdots \}$$
• 初始状态概率：$${\pi }_i=P(z_1=s_i)$$
• 全体参数：$$\theta =(\mathcal{A},\mathcal{B},\pi )$$

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前向过程.

• 定义前向概率如下：$$\alpha (i,t)=P(x_1,x_2,\cdots ,x_t,z_t=s_i;\theta )$$
• 其边界条件：$$\begin{array}{rl}\alpha (i,1)& =P(x_1,z_1=s_i;\theta )\\ & =P(x_1|z_1=s_i;\theta )\cdot P(z_1=s_i)\\ & ={\pi }_i\cdot B[i,\ast x_1]\end{array}$$
• 考虑递推公式：$$\begin{array}{rl}\alpha (j,t+1)& =P(x_1,x_2,\cdots ,x_{t+1},z_{t+1}=s_j;\theta )\\ & =\sum _{i=1}^{|S|}P(x_1,x_2,\cdots ,x_{t+1},z_t=s_i,z_{t+1},z_{t+1}=s_j;\theta )\\ & =\sum _{i=1}^{|S|}P(x_{t+1}|x_1,x_2,\cdots ,x_t,z_t=s_i,z_{t+1}=s_j;\theta )\cdot P(z_{t+1}=s_j|z_t=s_i,x_1{x}_2,\cdots ,x_t;\theta )\\ & \cdot P(x_1,x_2,\cdots ,x_t,z_t=s_i;\theta )\\ & =\sum _{i=1}^{|S|}P(x_{t+1}|z_{t+1}=s_j;\theta )\cdot P(z_{t+1}=s_j|z_t=s_i;\theta )\cdot P(x_1,x_2,\cdots ,x_t,z_t=s_i;\theta )\\ & =\sum _{i=1}^{|S|}B[j,\ast x_{t+1}]\cdot A[i,j]\cdot \alpha (i,t)\end{array}$$$$\alpha (i,T)=P(x_1,x_2,\cdots ,x_T,z_T=s_i;\theta )$$
• 基于上述公式得到：$$\begin{array}{rl}P(x;\theta )& =P(x_1,x_2,\cdots ,x_T;\theta )\\ & =\sum _{i=1}^{|S|}P(x_1,x_2,\cdots ,x_T,z_T=s_i;\theta )\\ & =\sum _{i=1}^{|S|}\alpha (i,T)\end{array}$$

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后向过程.

• 定义后向概率如下：$$\beta (j,t)=P(x_{t+1},x_{t+2},\cdots ,x_T|z_t=s_j;\theta )$$
• 给定递推初始值：$$\beta (j,T)=1$$
• 考虑递推公式：$$\begin{array}{rl}\beta (i,t-1)& =P(x_t,x_{t+1},\cdots ,x_T|z_{t-1}=s_i;\theta )\\ & =\sum _{j=1}^{|S|}P(x_t,x_{t+1},\cdots ,x_T,z_t=s_j|z_{t-1}=s_i;\theta )\\ & =\sum _{j=1}^{|S|}P(x_t|x_{t+1},x_{t+2},\cdots ,x_T,z_t=s_j,z_{t-1}=s_i;\theta )\cdot P(x_{t+1},x_{t+2},\cdots ,x_T|z_t=s_j,z_{t-1}=s_i;\theta )\\ & \cdot P(z_t=s_j|z_{t-1}=s_i;\theta )\\ & =\sum _{j=1}^{|S|}P(x_t|z_t=s_j;\theta )\cdot P(x_{t+1},x_{t+2},\cdots ,x_T|z_t=s_i;\theta )\cdot P(z_t=s_j|z_{t-1}=s_i;\theta )\\ & =\sum _{j=1}^{|S|}B[j,\ast x_t]\cdot \beta (j,t)\cdot A[i,j]\end{array}$$$$\beta (i,1)=P(x_2,x_3,\cdots ,x_T|z_1=s_i;\theta )$$
• 基于上述公式得到：$$\begin{array}{rl}P(x;\theta )& =P(x_1,x_2,\cdots ,x_T;\theta )\\ & =\sum _{i=1}^{|S|}P(x_1,x_2,\cdots ,x_T,z_1=s_i;\theta )\\ & =\sum _{i=1}^{|S|}P(x_1|x_2,x_3,\cdots ,x_T,z_1=s_i;\theta )\\ & \cdot P(x_2,x_3,\cdots ,x_T|z_1=s_i;\theta )\cdot P(z_1=s_i;\theta )\\ & =\sum _{i=1}^{|S|}B[i,\ast x_1]\cdot \beta (i,1)\cdot {\pi }_i\end{array}$$

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前后向算法.

• 基于上面给出的前向概率与后向概率，我们得到：$$\begin{array}{rl}P(x;\theta )& =P(x_1,x_2,\cdots ,x_T;\theta )\\ & =\sum _{i=1}^{|S|}\sum _{j=1}^{|S|}P(x_1,x_2,\cdots ,x_T,z_t=s_i,z_{t+1}=s_j;\theta )\\ & =\sum _{i=1}^{|S|}\sum _{j=1}^{|S|}P(x_{t+2},x_{t+3},\cdots ,x_T|x_1,x_2,\cdots ,x_{t+1},z_t=s_i,z_{t+1}=s_j;\theta )\\ & \cdot P(x_{t+1}|x_1,x_2,\cdots ,x_t,z_t=s_i,z_{t+1}=s_j;\theta )\cdot P(z_{t+1}=s_j|z_t=s_i,x_1,x_2,\cdots ,x_t;\theta )\cdot P(x_1,x_2,\cdots ,x_t,z_t=s_i;\theta )\\ & =\sum _{i=1}^{|S|}\sum _{j=1}^{|S|}P(x_{t+2},x_{t+3},\cdots ,x_T|z_{t+1}=s_j;\theta )\cdot P(x_{t+1}|z_{t+1}=s_j;\theta )\cdot P(z_{t+1}=s_j|z_t=s_i;\theta )\\ & \cdot P(x_1,x_2,\cdots ,x_t,z_t=s_i;\theta )\\ & =\sum _{i=1}^{|S|}\sum _{j=1}^{|S|}\beta (j,t+1)\cdot B[j,\ast x_{t+1}]\cdot A[i,j]\cdot \alpha (i,t)\end{array}$$

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Viterbi Algorithm.

• 定义：$$\delta (i,t)=\underset{z_1,z_2,\cdots ,z_{t-1}}{\max }P(z_t=s_i,z_{t-1},\cdots ,z_1,x_t,x_{t-1},\cdots ,x_1;\theta )$$
• 给定边界条件：$$\delta (i,1)=P(z_1=s_i,x_1;\theta )=P(x_1|z_1=s_i;\theta )\cdot P(z_1=s_i;\theta )=B[1,\ast x_1]\cdot {\pi }_i$$
• 考虑如下递推式：$$\begin{array}{rl}\delta (j,t+1)& =\underset{z_1,z_2,\cdots ,z_t}{\max }P(z_{t+1}=s_j,z_t,\cdots ,z_1,x_{t+1},x_t,\cdots ,x_1;\theta )\\ & =\underset{z_1,z_2,\cdots ,z_t}{\max }P(x_{t+1}|z_{t+1}=s_j;\theta )\cdot P(z_{t+1}=s_j,z_t,\cdots ,z_1,x_t,x_{t-1},\cdots ,x_1;\theta )\\ & =P(x_{t+1}|z_{t+1}=s_j;\theta )\cdot \underset{z1,z_2,\cdots ,z_t}{\max }P(z_{t+1}=s_j,z_t,\cdots ,z_1,x_t,x_{t-1},\cdots ,x_1;\theta )\\ & =B[j,\ast x_{t+1}]\cdot \underset{i\in [1,|S|]}{\max }[P(z_{t+1}=s_j|z_t=s_i;\theta )\cdot \underset{z_1,z_2,\cdots ,z_{t-1}}{\max }P(z_t=s_i,z_{t-1},\cdots ,z_1,x_t,x_{t-1},\cdots ,x_1;\theta )]\\ & =B[j,\ast x_{t+1}]\cdot \underset{i\in [1,|S|]}{\max }[A[i,j]\cdot \delta (i,t)]\end{array}$$
• 当 $t=T$ 时我们得到：$$\delta (i,T)=\underset{z_1,z_2,\cdots ,z_{T-1}}{\max }P(z_T=s_i,z_{T-1},\cdots ,z_1,x_T,x_{T-1},\cdots ,x_1;\theta )$$因此对于给定的观测序列，概率最大的状态序列即为： $$\underset{i\in [1,|S|]}{\max }\delta (i,T)$$

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