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We provide a unifying view of statistical information measures, multi-wayBayesian hypothesis testing, loss functions for multi-class classificationproblems, and multi-distribution $f$-divergences, elaborating equivalenceresults between all of these objects, and extending existing results for binaryoutcome spaces to more general ones. We consider a generalization of$f$-divergences to multiple distributions, and we provide a constructiveequivalence between divergences, statistical information (in the sense ofDeGroot), and losses for multiclass classification. A major application of ourresults is in multi-class classification problems in which we must both infer adiscriminant function $\gamma$---for making predictions on a label $Y$ fromdatum $X$---and a data representation (or, in the setting of a hypothesistesting problem, an experimental design), represented as a quantizer$\mathsf{q}$ from a family of possible quantizers $\mathsf{Q}$. In thissetting, we characterize the equivalence between loss functions, meaning thatoptimizing either of two losses yields an optimal discriminant and quantizer$\mathsf{q}$, complementing and extending earlier results of Nguyen et. al. tothe multiclass case. Our results provide a more substantial basis than standardclassification calibration results for comparing different losses: we describethe convex losses that are consistent for jointly choosing a datarepresentation and minimizing the (weighted) probability of error in multiclassclassification problems.

Multiclass Classification, Information, Divergence, and Surrogate Risk