Naive Bayes(参考wiki)
Using Bayes’ theorem, the conditional probability can be decomposed as
p(Ck∣x)=p(x)p(Ck)p(x∣Ck)
分母常数,分子是joint probability model
p(Ck,x1,…,xn)
p(Ck,x1,…,xn)=p(x1,…,xn,Ck)=p(x1∣x2,…,xn,Ck)p(x2,…,xn,Ck)=p(x1∣x2,…,xn,Ck)p(x2∣x3,…,xn,Ck)p(x3,…,xn,Ck)=…=p(x1∣x2,…,xn,Ck)p(x2∣x3,…,xn,Ck)…p(xn−1∣xn,Ck)p(Cn∣Ck)p(Ck)
Naive conditional independence assume that all features in x are mutually independent, conditional on the category
Ck:
p(xi∣xi+1,…,xn,Ck)=p(xi∣Ck)
Thus the joint model can be expressed as
p(Ck∣x1,…,xn)∝p(Ck,x1,…,xn)=p(Ck)p(x1∣Ck)p(x2∣Ck)p(x3∣Ck)⋯=p(Ck)i=1∏np(xi∣Ck)