The Relationship of the Bayes Risk to Certain Separability Measures in Normal Classification

For the problem of classifying an element (e.g., an unknown pattern) into one of two given categories where the associated observables are distributed according to one of two known multivariate normal populations having a common covariance matrix, it is shown that the minimum Bayes risk is a strict monotonic function of certain separability or statistical distance measures regardless of the a priori probabilities and the assigned loss function. However, for the associated conditional expected losses, strict monotonicity holds, if and only if a certain condition dependent on these probabilities and the given loss function is satisfied. These results remain valid for classification problems in which the observable can be transformed by a one-to-one differentiable mapping to normality.