p(c|x) = \frac{p(x|c)p(c)}{p(x)}
如果P(c₁|x) > P(c₂|x),那么属于类别c₁。 如果P(c₁|x) < P(c₂|x),那么属于类别c₂。
p(c1|x) = \frac{p(x|c1)p(c1)}{p(x)} p(c2|x) = \frac{p(x|c2)p(c2)}{p(x)} Therefore, comparing p(c1|x) and p(c2|x) are same as comparing \frac{p(x|c1)p(c1)}{p(x)} and \frac{p(x|c2)p(c2)}{p(x)} same as comparing p(x|c1)p(c1) and p(x|c2)p(c2)
p(x|c1)中,x是多个独立特征,即x=x_0,x_1...x_n, 则: p(x|c1)=p(x_0,x_1...x_n|c1) p(x|c1)=p(x_0|c1)p(x_1|c1)...p(x_n|c1)
为了解决下溢出问题,这是由于太多很小的数相乘造成的,所以程序会下溢出或者得到不正确的答案。 在代数中有ln(a*b) = ln(a)+ln(b),于是通过求对数可以避免下溢出或者浮点数舍入导致的错误。同时,采用自然对数进行处理不会有任何损失。 Therefore, comparing p(c1|x) and p(c2|x) same as comparing log(p(x_0|c1)) + log(p(x_1|c1)) + ... + log(p(x_n|c1) + log(p(c1))
and
log(p(x_0|c2)) + log(p(x_1|c2)) + ... + log(p(x_n|c2) + log(p(c2))