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# Vector-Valued Property Elicitation

*Proceedings of The 28th Conference on Learning Theory*, PMLR 40:710-727, 2015.

#### Abstract

The elicitation of a statistic, or property of a distribution, is the task of devising proper scoring rules, equivalently proper losses, which incentivize an agent or algorithm to truthfully estimate the desired property of the underlying probability distribution or data set. Leveraging connections between elicitation and convex analysis, we address the vector-valued property case, which has received little attention in the literature despite its applications to both machine learning and statistics. We first provide a very general characterization of linear and ratio-of-linear properties, the first of which resolves an open problem by unifying and strengthening several previous characterizations in machine learning and statistics. We then ask which vectors of properties admit nonseparable scores, which cannot be expressed as a sum of scores for each coordinate separately, a natural desideratum for machine learning. We show that linear and ratio-of-linear do admit nonseparable scores, and provide evidence for a conjecture that these are the only such properties (up to link functions). Finally, we give a general method for producing identification functions and address an open problem by showing that convex maximal level sets are insufficient for elicitability in general.