Ideal observers and optimal ROC hypersurfaces in N-class classification

The likelihood ratio, or ideal observer, decision rule is known to be optimal for two-class classification tasks in the sense that it maximizes expected utility (or, equivalently, minimizes the Bayes risk). Furthermore, using this decision rule yields a receiver operating characteristic (ROC) curve which is never above the ROC curve produced using any other decision rule, provided the observer\´s misclassification rate with respect to one of the two classes is chosen as the dependent variable for the curve (i.e., an "inversion" of the more common formulation in which the observer\´s true-positive fraction is plotted against its false-positive fraction). It is also known that for a decision task requiring classification of observations into N classes, optimal performance in the expected utility sense is obtained using a set of N-1 likelihood ratios as decision variables. In the N-class extension of ROC analysis, the ideal observer performance is describable in terms of an (N²-N-1)-parameter hypersurface in an (N²-N)-dimensional probability space. We show that the result for two classes holds in this case as well, namely that the ROC hypersurface obtained using the ideal observer decision rule is never above the ROC hypersurface obtained using any other decision rule (where in our formulation performance is given exclusively with respect to between-class error rates rather than within-class sensitivities).