Neyman-Pearson Criterion

I want to minimize the error of one class, but keep the error of the other class fixed.

$$\min P_1(e),\text{s.t.}{P}_2(e)=ϵ$$

To do this, we need to use the Langragian

$$L=P_1(e)+\lambda (P_2(e)-\epsilon )=(1-\lambda \epsilon )+{\int }_{R_1}[\lambda p(x|{\omega }_2)-p(x|{\omega }_1)]dx$$

where $R_1$ is the decision region of class ${\omega }_1$. Then we want to set $▽L=0$.

I have no idea what’s going on afterwards in solving for $L$. Maybe one day I can revisit this when it is clearer.