Open Access
February 2018 Proper scoring rules and Bregman divergence
Evgeni Y. Ovcharov
Bernoulli 24(1): 53-79 (February 2018). DOI: 10.3150/16-BEJ857


Proper scoring rules measure the quality of probabilistic forecasts. They induce dissimilarity measures of probability distributions known as Bregman divergences. We survey the literature on both entities and present their mathematical properties in a unified theoretical framework. Score and Bregman divergences are developed as a single concept. We formalize the proper affine scoring rules and present a motivating example from robust estimation. And lastly, we develop the elements of the regularity theory of entropy functions and describe under what conditions a general convex function may be identified as the entropy function of a proper scoring rule and whether this association is unique.


Download Citation

Evgeni Y. Ovcharov. "Proper scoring rules and Bregman divergence." Bernoulli 24 (1) 53 - 79, February 2018.


Received: 1 August 2015; Revised: 1 March 2016; Published: February 2018
First available in Project Euclid: 27 July 2017

zbMATH: 06778321
MathSciNet: MR3706750
Digital Object Identifier: 10.3150/16-BEJ857

Keywords: Bregman divergence , characterisation , convex analysis , Entropy , proper scoring rule , robust estimation , subgradient

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 1 • February 2018
Back to Top