Unequal costs

Topic: Statistical classifiers and discriminant analysis · more questions on this topic
‹ A Bayes rule in one dimension LDA with unequal priors ›

A test has posterior probability \tau_1(x)=0.3 of disease. Missing a case costs c_{12}=5 and a false alarm costs c_{21}=1. Which class does the unequal-cost Bayes rule choose? At which value of \tau_1 does the decision switch?

Worked answer (click to show)

Compare posterior expected losses. The more probable class need not be the cheaper decision when the costs differ. Predicting class 1, with \tau_1=0.3 and \tau_2=0.7, has expected cost

\tau_2 c_{21}=0.7\times 1=0.7.

Predicting class 2 has expected cost

\tau_1 c_{12}=0.3\times 5=1.5.

Class 1 is chosen, even though it is the less probable class. The decision switches when \tau_2 c_{21}=\tau_1 c_{12}, that is 1-\tau_1=5\tau_1, so \tau_1=1/6\approx 0.167. Class 1 is predicted whenever \tau_1>1/6.