Year: 2018
Type: article
Abstract: 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 $\gamma$—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 $\mathsf{q}$ from a family of possible quantizers $\mathsf{Q}$. In this setting, we characterize the equivalence between loss functions, meaning that optimizing either of two losses yields an optimal discriminant and quantizer $\mathsf{q}$, 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.
Source: The Annals of Statistics
DOI: 10.1214/17-aos1657
Authors: John Duchi, Khashayar Khosravi, Feng Ruan
Institution: Stanford University
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2962762853