贝叶斯分类算法原理

1.贝叶斯定理：

\[P(X|c) = \prod\limits_{i = 1}^k {P(Xi|c)} \]

假设c正确的情况下样本X发生的概率

2.贝叶斯公式：

\[P(A|B) = \frac{{P(B|A)P(A)}}{{P(B)}} \]

3.极大后验假设

\[{c_{map}} = \mathop {\arg \max }\limits_{c \in C} P(c|X) = \mathop {\arg \max }\limits_{c \in C} \frac{{P(X|c)P(c)}}{{P(X)}} = \mathop {\arg \max }\limits_{c \in C} P(X|c)P(c) \]

最大可能性的假设（即c值）为极大后验假设

4.贝叶斯分类算法原理：
贝叶斯定理+极大后验假设

\[V(X) = \mathop {\arg \max }\limits_i P({c_i})P(X|{c_i}) \]

\[+ \]

\[P(X|{c_i}) = \prod\limits_{k = 1}^n {P({X_k}|{c_i})} \]

\[|| \]

\[V(X) = \mathop {\arg \max }\limits_i P({c_i})\prod\limits_{k = 1}^n {P({X_k}|{c_i})} \]

\[(贝叶斯分类公式) \]

总结：
贝叶斯分类的步骤：
(1)计算先验概率\(P({c_i})\)
(2)计算条件概率\({P({X_k}|{c_i})}\)
(3)代入贝叶斯分类公式，求出最大的\(P({c_{map}})\prod\limits_{k = 1}^n {P({X_k}|{c_{map}})} , then{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} V(X) = {c_{map}}\)

例子：

Day	Outlook	Temperature	Humidity	Wind	PlayTennis
D1	Sunny	Hot	High	Weak	No
D2	Sunny	Hot	High	Strong	No
D3	Overcast	Hot	High	Weak	Yes
D4	Rain	Mild	High	Weak	Yes
D5	Rain	Cool	Normal	Weak	Yes
D6	Rain	Cool	Normal	Strong	No
D7	Overcast	Cool	Normal	Strong	Yes
D8	Sunny	Mild	High	Weak	No
D9	Sunny	Cool	Normal	Weak	Yes
D10	Rain	Mild	Normal	Weak	Yes
D11	Sunny	Mild	Normal	Strong	Yes
D12	Overcast	Mild	High	Strong	Yes
D13	Overcast	Hot	Normal	Weak	Yes
D14	Rain	Mild	High	Strong	No

假设当前天气为：X={Sunny,Hot,High,Weak}
问：当前天气是否可以打网球？

解：
（1）计算先验概率P(Yse)=9/14   P(No)=5/14

（2）计算条件概率P(X|Yes)、P(X|No)
P(X|Yes) = P(Sunny|Yes)*P(Hot|Yes)*P(High|Yes)*P(Weak|Yes) = 8/729
P(X|No) = P(Sunny|No)*P(Hot|No)*P(High|No)*P(Weak|No) = 48/625

（3）带入贝叶斯公式
P(Yes)*P(X|Yes) = 9/14 * 8/729 ≈ 0.0071
P(No)*P(X|No) = 5/14 * 48/625 ≈ 0.027
0.0071 < 0.027,则假设playTennis=No为极大后验假设，则当前天气不可以打网球。