Multiclass classification, information, divergence and surrogate risk

We provide a unifying view of statistical information measures, multiway Bayesian hypothesis testing, loss functions for multiclass classification problems and multidistribution f-divergences, elaborating equivalence results between all of these objects, and extending existing results for binary outcome spaces to more general ones. We consider a generalization of f-divergences to multiple distributions, and we provide a constructive equivalence between divergences, statistical information (in the sense of DeGroot) and losses for multiclass classification. A major application of our results is in multiclass classification problems in which we must both infer a discriminant function γ —for making predictions on a label Y from datum X—and a data representation (or, in the setting of a hypothesis testing problem, an experimental design), represented as a quantizer q from a family of possible quantizers Q. In this setting, we characterize the equivalence between loss functions, meaning that optimizing either of two losses yields an optimal discriminant and quantizer q, complementing and extending earlier results of Nguyen et al. [Ann. Statist. 37 (2009) 876–904] to the multiclass case. Our results provide a more substantial basis than standard classification calibration results for comparing different losses: we describe the convex losses that are consistent for jointly choosing a data representation and minimizing the (weighted) probability of error in multiclass classification problems.