Formulas for threshold computations

Given a continuous variable S, which density functions on two subgroups omega + and omega - of a population omega are known (with for instance a higher mean value on omega + than on omega -), we first define two strategies for classification in these groups; the first one (MWC) consists in determining a threshold alpha such that classifying in omega + when S greater than or equal to alpha, in omega - otherwise, leads to the highest percentage of well-classed elements. The second one consists in choosing the most probable group, given the observed value of S. We give mathematical formulas for the thresholds involved in these two strategies when the density functions, determined by the application of the maximum entropy principle, are those of normal distributions. These formulas prove that the two considered strategies are frequently equivalent, and we give simpler formulas when the partial variances of S on omega + and omega - are unknown or approximately equal. All the formulas are adapted to the case where a cost coefficient is introduced to display the unequal seriousness of the two possible errors (misclassification in omega + or omega -). Then we consider an example, where we see that the computed thresholds can be graphically validated from empirical curves and have the same performances on the learning sample and on a test sample.