理解条件概率的链式法则

• 条件概率:
  \(\text{x}=x\) 事件发生时 \(\text{y}=y\) 事件发生的概率:
  \[ P(\text{y}=y|\text{x}=x)=\frac{P(\text{x}=x,\text{y}=y)}{P(\text{x}=x)} \]

• 条件概率的链式法则
  也称为条件概率的乘法法则

\[ \begin{aligned} P(a,b,c) &= P(a|b,c)P(b,c) \\ &= P(a|b,c)P(b|c)P(c) \end{aligned} \]

• 推广到一般情况有:
  \[ \begin{aligned} P(\text{x}^{(1)},\text{x}^{(2)},\cdots,\text{x}^{(n)}) &= P(\text{x}^{(n)}|\text{x}^{(n-1)},\cdots,\text{x}^{(1)})P(\text{x}^{(1)},\cdots,\text{x}^{(n-1)})\\ &=P(\text{x}^{(n)}|\text{x}^{(n-1)},\cdots,\text{x}^{(1)})P(\text{x}^{(n-1)}|\text{x}^{(n-2)},\cdots,\text{x}^{(1)})P(\text{x}^{(1)},\cdots,\text{x}^{(n-2)})\\ &=P(\text{x}^{(n)}|\text{x}^{(n-1)},\cdots,\text{x}^{(1)})P(\text{x}^{(n-1)}|\text{x}^{(n-2)},\cdots,\text{x}^{(1)})\cdots P(\text{x}^{(2)}|\text{x}^{(1)})P(\text{x}^{(1)})\\ &=P(\text{x}^{(1)})\prod_2^nP(\text{x}^{(i))}|\text{x}^{(1)}\cdots\text{x}^{(i-1)}) \end{aligned} \]

通俗点讲，条件概率的链式法则可以如下理解：
以 \(P(\text{x}^{(1)},\text{x}^{(2)},\cdots,\text{x}^{(n)})\) 为例，可以看作 \(P(\text{x}^{(1)})\) 发生后，\(P(\text{x}^{(2)}|\text{x}^{(1)})P(\text{x}^{(1)})\) 是\(\text{x}^{(1)},\text{x}^{(2)}\) 同时发生的概率，\(P(\text{x}^{(3)}|\text{x}^{(1)},\text{x}^{(2)})P(\text{x}^{(2)}|\text{x}^{(1)})P(\text{x}^{(1)})\) 是 \(\text{x}^{(1)},\text{x}^{(2)},\text{x}^{(3)}\) 同时发生的概率，依次类推下去，便可以得到条件概率的链式法则公式。

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参考资料：Deep Learning 3.5

转载于:https://www.cnblogs.com/yuyin/articles/10958971.html