# 贝叶斯分类器

## 贝叶斯公式

主要根据贝叶斯公式：

$$p(b|a) = \frac { p(a|b)p(b) } { p(a) }$$

特征向量x和标签y：

$$p(y|x) = \frac { p(x|y)p(y) } { p(x) }$$

通常只需要判断哪一类概率最大，因此底数$$p(x)$$通常省略，变为寻找最大概率：

$$arg max\_{x} p(x|y)p(y)$$

## 贝叶斯计算

给定样本的特征向量，该样本属于类 $$c\_{i}$$的概率：

$$p(y=c\_{i}|x) = \frac { p(y=c\_{i})p(x|y=c\_{i})} { p(x) }$$

假设特征向量各个相互独立，

$$p(y=c\_{i}|x) = \frac { p(y=c\_{i})\prod\limits\_{j=1}^np(x\_j|y=c\_{i})} { Z }$$

## 概率计算-离散特征

$$p(x\_i=v|y=c)=\frac{N\_{x\_i=v,y=c}}{N\_{y=c}}$$


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