Property Elicitation on Imprecise Probabilities

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.