Property Elicitation on Imprecise Probabilities
James Bailie, Rabanus Derr
TL;DR
This work generalizes property elicitation to imprecise probabilities, framing IP-elicitability via a Gamma-minimax loss framework and connecting the elicited IP-properties to the classical properties of the worst-case distribution within an IP. It develops necessary structural conditions for IP-elicitability, a first sufficient condition, and clarifies how minimax theory governs what can be learned under distributional uncertainty. The results bridge MDL/DRO perspectives with elicitation theory and pave the way for characterizing which imprecise-properties can be reliably learned in minimax settings. Overall, the paper provides foundational theory for determining when IP-based properties are learnable under maximin risk and identifies key open questions for a complete characterization.
Abstract
Property elicitation studies which attributes of a probability distribution can be determined by minimizing a risk. We investigate a generalization of property elicitation to imprecise probabilities (IP). This investigation is motivated by distributionally robust optimization and multi-distribution learning. Both those frameworks replace the minimization of a single risk over a (precise) probability by a maximin risk minimization over a set of probabilities -- i.e. an IP. We show what can be learned in those multi-distribution setups by providing necessary and sufficient conditions for the elicitability of an IP-property. Central to these conditions is the observation made in related literature that the elicited IP-property is the corresponding classical property of the probability in the IP with the maximum Bayes risk.