Risk-Based Comparison of Classification Systems

Performance measures for families of classification system families that rely upon the analysis of receiver operating characteristics (ROCs), such as area under the ROC curve (AUC), often fail to fully address the issue of risk, especially for classification systems involving more than two classes. For the general case, we denote matrices of class prevalences, costs, and class-conditional probabilities, and assume costs are subjectively fixed, acceptable estimates for expected values of class-conditional probabilities exist, and mutual independence between a variable in one such matrix and those of any other matrix. The ROC Risk Functional (RRF), valid for any finite number of classes, has an associated parameter argument, that which specifies a member of a family of classification systems, and which system minimizes Bayes risk over the family. We typify joint distributions for class prevalences over standard simplices by means of uniform and beta distributions, and create a family of classification systems using actual data, testing independence assumptions under two such class prevalence distributions. We minimize risk under two different sets of costs.