In a test experiment , there are also probability to wrong . For example , it is a True event , but after test the result is negative .
From above , we can get a table named as confusion matrix .
| real result > test result ^ |
True(cancer)(1%) | False(no cancer)(99%) |
|---|---|---|
| positive | 90%(1) | 5%(2) |
| nagative | 10%(3) | 95%(4) |
example
We make use of the above example , it says that , we have a sample where the cancer people occupy 1% , and the other who is no cancer is 99% . In %1 people , there are 90% test positive suffering from cancer , but 10% are not . (so it would be something wrong in this test , but we will repect the result of experments .) In the simarly way ,there are 5% people who isn’t suffering from cancer but the test result is show that they are . Here , how we calculate the real cancer probiblity in this test , still 1% , or maybe some compound of 90% with 5% ? From we can conclude :
- The test result can be wrong also , we can’t only believe in test result .
- However , We can make use of the piror knowledge about the test and combine with this test reuslt to get the best result .The piror knowledge in this example is we know 1% is suffer from cancer and the other is not .
We rewrite the table above .
| real result > test result ^ |
True(cancer)(1%) | False(no cancer)(99%) |
|---|---|---|
| positive (have cancer) | 90%(1) TF probability: 1%*90% = 0.9% |
5%(2) FP prob: 99%*5% = 4.95% |
| negative(not have cancer) | 10%(3) TN prob: 1%*10% = 0.1% |
95%(4) FN prob: 99%*95% = 94.05% |
(1) is called TP probability. The test result is positive and the real result is Ture .
(2) is called FP probability. The test result is negative but the real result is False .
The summary of (1) and (2) is the probability of all test positive , it is no relationship with True probability that is the prori knowledge .
And in the silmilar way with (3) , (4) .
Now we get the real cancer probability :
\[\begin{align*} Pr(real\quad cancer \quad after \quad test) & = \frac{Pr(test \quad positive \quad and \quad have \quad cancer\quad before\quad test) }{Pr(test\quad positive\quad in\quad test )} \\ & = \frac{0.9\%}{0.9\%+4.95\%} \\ & = 15.38 \% \end{align*}\]Insteresting , a positive mammogram only means somebody has a $15.38 \%$ chance really suffer from cancer , not $90\%$ , it is refute common sense . The reason is that there are also have some FP probability , a large people without cancer and misjudgement that they suffer from cancer .
formula
we can turn the process anove into a equation ,suppose event $A$ is someone suffering from cancer before test , $1\%$ , and $X$ is someone viewed as suffering from cancer after test .
The formula is :
End
Bayes told us that test experiment maybe wrong sometimes , we should make use of the prior belief to consider the experiment results .