On Elicitation Complexity and Conditional Elicitation

Elicitation is the study of statistics or properties which are computable via empirical risk minimization.While several recent papers have approached the general question of which properties are elicitable, wesuggest that this is the wrong question—all properties are elicitable by first eliciting the entire distributionor data set, and thus the important question is how elicitable. Specifically, what is the minimum numberof regression parameters needed to compute the property?Building on previous work, we introduce a new notion of elicitation complexity and lay the foundationsfor a calculus of elicitation. We establish several general results and techniques for proving upper andlower bounds on elicitation complexity. These results provide tight bounds for eliciting the Bayes riskof any loss, a large class of properties which includes spectral risk measures and several new propertiesof interest. Finally, we extend our calculus to conditionally elicitable properties, which are elicitableconditioned on knowing the value of another property, giving a necessary condition for the elicitabilityof both properties together.