机器学习第五次作业
6.1
由第一次检测：
P(⊕∣cancer)P(cancer)=0.0078P(⊕∣cancer)P(cancer)=0.0078P(⊕∣cancer)P(cancer)=0.0078
P(⊕∣¬cancer)P(¬cancer)=0.0298P(⊕∣\neg cancer)P(\neg cancer)=0.0298P(⊕∣¬cancer)P(¬cancer)=0.0298、
将第一次检测结果作为新的先验概率，归一化得到：
P(cancer∣⊕)=0.0078/0.0376=0.2074P(cancer∣⊕)=0.0078/0.0376=0.2074P(cancer∣⊕)=0.0078/0.0376=0.2074
P(¬cancer∣⊕)=0.0298/0.0376=0.7926P(\neg cancer∣⊕)=0.0298/0.0376=0.7926P(¬cancer∣⊕)=0.0298/0.0376=0.7926
第二次检测的后验概率：
P(cancer∣⊕,⊕)=P(⊕∣cancer)∗P(cancer∣⊕)=.098∗0.2074=0.2033P(cancer∣⊕,⊕)=P(⊕∣cancer)*P(cancer∣⊕)=.098*0.2074=0.2033P(cancer∣⊕,⊕)=P(⊕∣cancer)∗P(cancer∣⊕)=.098∗0.2074=0.2033
P(¬cancer∣⊕,⊕)=P(⊕∣¬cancer)∗P(¬cancer∣⊕)=0.03∗0.7926=0.0238P(\neg cancer∣⊕,⊕)=P(⊕∣\neg cancer)*P(\neg cancer∣⊕)=0.03*0.7926=0.0238P(¬cancer∣⊕,⊕)=P(⊕∣¬cancer)∗P(¬cancer∣⊕)=0.03∗0.7926=0.0238
归一化得：
P(cancer∣⊕,⊕)=0.2033/0.2271=0.8952P(cancer∣⊕,⊕)=0.2033/0.2271=0.8952P(cancer∣⊕,⊕)=0.2033/0.2271=0.8952
P(¬cancer∣⊕,⊕)=0.0238/0.2271=0.1048P(\neg cancer∣⊕,⊕)=0.0238/0.2271=0.1048P(¬cancer∣⊕,⊕)=0.0238/0.2271=0.1048
6.2
根据贝叶斯公式：
P(cancer∣⊕)=P(⊕∣cancer)∗P(cancer)P(⊕)=P(⊕∣cancer)∗P(cancer)P(⊕∣cancer)P(cancer)+P(⊕∣¬cancer)P(¬cancer)=1P(⊕∣cancer)P(cancer)+P(⊕∣¬cancer)P(¬cancer)P(⊕∣cancer)∗P(cancer)=11+P(⊕∣¬cancer)P(¬cancer)P(⊕∣cancer)∗P(cancer)P(cancer∣⊕)=\frac{P(⊕∣cancer)*P(cancer)}{P(⊕)}\\=\frac{P(⊕∣cancer)*P(cancer)}{P(⊕∣cancer)P(cancer)+P(⊕∣\neg cancer)P(\neg cancer)}\\=\frac{1}{\frac{P(⊕∣cancer)P(cancer)+P(⊕∣\neg cancer)P(\neg cancer)}{P(⊕∣cancer)*P(cancer)}}\\=\frac{1}{1+\frac{P(⊕∣\neg cancer)P(\neg cancer)}{P(⊕∣cancer)*P(cancer)}}P(cancer∣⊕)=P(⊕)P(⊕∣cancer)∗P(cancer)=P(⊕∣cancer)P(cancer)+P(⊕∣¬cancer)P(¬cancer)P(⊕∣cancer)∗P(cancer)=P(⊕∣cancer)∗P(cancer)P(⊕∣cancer)P(cancer)+P(⊕∣¬cancer)P(¬cancer)1=1+P(⊕∣cancer)∗P(cancer)P(⊕∣¬cancer)P(¬cancer)1
而将P(⊕∣cancer)P(cancer)P(⊕∣cancer)P(cancer)P(⊕∣cancer)P(cancer)和P(⊕∣¬cancer)P(¬cancer)P(⊕∣\neg cancer)P(\neg cancer)P(⊕∣¬cancer)P(¬cancer) 归一化，得到的正是P(⊕∣cancer)P(cancer)P(⊕)\frac{P(⊕∣cancer)P(cancer)}{P(⊕)}P(⊕)P(⊕∣cancer)P(cancer)和P(⊕∣¬cancer)P(¬cancer)P(⊕)\frac{P(⊕∣\neg cancer)P(\neg cancer)}{P(⊕)}P(⊕)P(⊕∣¬cancer)P(¬cancer)，而前者正是P(cancer∣⊕)P(cancer∣⊕)P(cancer∣⊕)，因此这样是正确的