Cost-sensitive Bayes classifier ($c(2\mid 1){\pi }_1{f}_1(x)\ge c(1\mid 2){\pi }_2{f}_2(x)$)

With zero cost for correct classifications, assign class one when the displayed inequality holds and class two otherwise. It compares conditional expected losses; ties may be resolved arbitrarily. For positive costs and densities, the log likelihood ratio threshold is $\log (c(1\mid 2){\pi }_2/(c(2\mid 1){\pi }_1))$.