回顾贝叶斯定理
首先,我们先来复习一下贝叶斯定理:
在这个简简单单的式子中,蕴含了我们要掌握的很多重要内容:
贝叶斯定理当中的四个概率表达式,他们都非常重要,在这一届的内容中将反复出现,我们来一一解析一下:
P(A):先验分布。反映的是在观则到数据之前我。们对待估计的参数的了解和认识
P(A|B):后验分布后验分布就是我们通过贝叶斯定理得到的最终的分析结果,反映的是在给定观测数据的基础上,我们对于参数的新的认知。说的更直白一点,就是最开始没有观填 测数据的时候,我们依据以往的经验赋予了参数一个先验分布,然后来了实际的观测数据之后,我们就对先验进行了更新,得到了这次分析过程的后验分布:可能性函数或调整函数,也就是在已知的情况下,对P(A)进行调整的因子。
举个简单的例子
有两个桶,1号桶里有40个球,其中30个白球,10个黑球;2号桶里也有40个球,其中20个白球,20个黑球。
条件概率:从1号桶中抽取白球的概率P(B|A)=30/(30+10)=75%;从2号桶中抽取白球的概率P(B|C)=20/(20+20)=50%,这些都是条件概率。
全概率:抽取一个球,为白球的概率,这个概率叫全概率,是1号桶中抽取白球和2号桶中抽取白球两个事件的概率之和,即 P(B)=P(B|A)P(A)+P(B|C)P(C)=75% 50% + 50% 50%= 62.5%。
逆概率/贝叶斯定理:抓取了一个球是白球,那么这个白球来自1号桶的概率P(A|B)是多少,这就是典型的贝叶斯定理解决的问题。
如何使用贝叶斯定理?
- 先求P(A),即抓取一个球,球来自A桶的概率,也就是在没有约束条件(约束条件为抽取的是白球)下事件A发生的概率,这个很好计算为50%。
- 再求P(B|A)/P(B),也就是在已知条件下抓取的是白球的情况下,对P(A)进行调整的因子。根据上文计算的P(B|A)= 75%,P(B)= 62.5%,得出调整因子为75%/62.5%=1.2。
- 最后求P(A|B),P(A|B)= P(A) (P(B|A)/P(B)) = 50% 1.2 = 60% 。
也就是说,抽取一个球,在信息不完整的情况下,这个球来自1号桶的概率为50%;在我们知道这个球是白球的条件下,那么这个球来自1号桶的可能性提高了20%(调整因子为1.2),则最终抽取的是白球且来自1号桶的概率将提升到60%。
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