Optimization of an ROC hypersurface constructed only from an observer's within-class sensitivities

We have shown in previous work that an ideal observer in a classification task with N classes achieves the optimal receiver operating characteristic (ROC) hypersurface in a Neyman-Pearson sense. That is, the hypersurface obtained by taking one of the ideal observer's misclassification probabilities as a function of the other N2-N-1 misclassification probabilities is never above the corresponding hypersurface obtained by any other observer. Due to the inherent complexity of evaluating observer performance in an N-class classification task with N>2, some researchers have suggested a generally incomplete but more tractable evaluation in terms of a hypersurface plotting only the N "sensitivities" (the probabilities of correctly classifying observations in the various classes). An N-class observer generally has up to N2-N-1 degrees of freedom, so a given sensitivity will still vary when the other N-1 are held fixed; a well-defined hypersurface can be constructed by considering only the maximum possible value of one sensitivity for each achievable value of the other N-1. We show that optimal performance in terms of this generally incomplete performance descriptor, in a Neyman-Pearson sense, is still achieved by the N-class ideal observer. That is, the hypersurface obtained by taking the maximal value of one of the ideal observer's correct classification probabilities as a function of the other N-1 is never below the corresponding hypersurface obtained by any other observer.