4-1 If P(y = 1) = 0.4 and P(y = 0) = 0.6, please calculate the probability of y = 1 given x = 1, 2, 3.

已知 \(P(y|x)=\frac{P(y\cap x)}{p(x)}\)

由題目中可以推得以下關係

\(P(x=1|y=1)=\frac{p(x=1 \cap y=1)}{p(y=1)}=0.1\), \(p(x=1 \cap y=1)=0.04\)

\(P(x=2|y=1)=\frac{p(x=2 \cap y=1)}{p(y=1)}=0.3\), \(p(x=2 \cap y=1)=0.12\)

\(P(x=3|y=1)=\frac{p(x=3 \cap y=1)}{p(y=1)}=0.6\), \(p(x=3 \cap y=1)=0.24\)

\(P(x=1)=0.4*0.1+0.6*0.5=0.34\),

\(P(x=2)=0.4*0.3+0.6*0.3=0.3\),

\(P(x=3)=0.4*0.6+0.6*0.2=0.36\)

有以上機率求

\(P(y=1|x=1)=\frac{p(y=1 \cap x=1)}{p(x=1)}=0.04/0.34=0.1176\)

\(P(y=1|x=2)=\frac{p(y=1 \cap x=2)}{p(x=2)}=0.12/0.3=0.4\)

\(P(y=1|x=3)=\frac{p(y=1 \cap x=3)}{p(x=3)}=0.24/0.36=0.6667\)

4-2 Please derived the predicted class of y given x using Bayes classifier.

Bayes classifier 可以寫成 \(P(y=c|x)=\frac{P(x|y=c)P(y=c)}{P(x)}\)

因此當 x = 1,2,3 時的條件機率如下

\(P(y=1|x=1)=\frac{p(y=1 \cap x=1)}{p(x=1)}=0.04/0.34=0.1176\)

\(P(y=1|x=2)=\frac{p(y=1 \cap x=2)}{p(x=2)}=0.12/0.3=0.4\)

\(P(y=1|x=3)=\frac{p(y=1 \cap x=3)}{p(x=3)}=0.24/0.36=0.6667\)

若機率 > 0.5 時分類至 y=1，那麼 x = 1,2 會被分到 y=0 這一類，x = 3 則會被分到 y = 1 這一類。