 Solution

Apply the minimum-integral decision region principle to $g(x)={\pi }_2{f}_2(x)-{\pi }_1{f}_1(x)$. The optimal Bayes classifier chooses class one when ${\pi }_1{f}_1(x)>{\pi }_2{f}_2(x)$, and class two for the reverse inequality. Ties can be assigned either way. An everywhere valid discriminant score isWhere both weighted densities are positive one may equivalently use $\log ({\pi }_1{f}_1(x)/({\pi }_2{f}_2(x)))$ and threshold zero. This chooses the largest posterior class probability, because both posterior probabilities have the same positive mixture-density denominator.