False Alarm in Classification

There are two types of error in Hypothesis Testing, the Type I error $\alpha$ and Type II error $\beta$. In classification term, the H0 is considered as the non-target class and the H1 is the target class. Therefore, the $\alpha$ is where we misclassify an observation as a target when it is not.

Therefore, the Type I error $\alpha$, here means false alarm (FA) or false positive.

\[P(FA)=\alpha\]

For example, The target class (H1) of the previous example is $\omega_1$ and the nontarget class (H0) is $\omega_2$ and the P(FA) at the decision boundary x=0.2 is

>> 0.4*normcdf(0.2,1,1)
ans =
    0.0847

Now for a reverse question, what if we want to set the P(FA) to 0.05, what would the decision boundary be?

Notice that
\[P(FA)=\int\limits_{-\infty}^C P(x|\omega_2)P(\omega_2)dx\]
We want to find the value x=C, such that
\[\int\limits_{-\infty}^C P(x|\omega_2)dx=\frac{P(FA)}{P(\omega_2)}\]
Thus,

>> norminv(0.05/0.4,1,1)

ans =

   -0.1503

The decision boundary is where x= -0.1503