# 概率计算问题 - 前向算法

## 定义

**前向概率**：给定马尔可夫模型$$\lambda$$，定义“到时刻t为止，部分观测序列为o1,o2,...,ot，且t时刻的状态为qi”的概率为前向概率，记作：

$$
\alpha\_t(i) = P(o\_1,o\_2,\cdots,o\_t,i\_t=q\_i|\lambda)
$$

## 原理

这是**状态DP**的思想。\
局部计算前向概率，利用路径结构将前向概率递推到全局（这一句没看懂）。\
每一次计算直接利用前一个时刻的计算结果，避免重复计算。

## 过程

**输入**：\
隐马尔可夫模型$$\lambda$$\
观测序列O\
**输出**：\
观测序列概率$$P(O|\lambda)$$ **过程**： 1. 初值：

$$
\alpha\_1(i) = \pi\_ib\_i(o\_1)
$$

1. 递推：  

   $$
   \alpha\_{t+1}(i) = \[\sum\_{j=1}^N\alpha\_t(j)a\_{ji}]b\_i(o\_{t+1})
   $$
2. 终止：  

   $$
   P(O|\lambda) = \sum\_{i=1}^N\alpha\_T(i)
   $$

**注意**$$\alpha$$**和**$$a$$**的不同**


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