E-variables for hypotheses generated by constraints

Abstract: E-variables are nonnegative random variables with expected value at most one under any distribution from a given null hypothesis. E-variables have been recently recognized as fundamental objects in hypothesis testing, and a key open problem is to characterize their form. We provide a complete solution to this problem for hypotheses generated by constraints, a broad and natural framework that encompasses many hypothesis classes occurring in practice. Our main result is an abstract representation theorem that describes all e-variables for any hypothesis defined by an arbitrary collection of measurable constraints. We instantiate this general theory for three important classes: hypotheses generated by finitely many constraints, one-sided sub-$\psi$ distributions (including sub-Gaussian distributions), and distributions constrained by group symmetries. In each case, we explicitly characterize all e-variables as well as all admissible e-variables. Building on these results we prove existence and uniqueness of optimal e-variables under a large class of expected utility-based objective functions, covering all criteria studied in the e-variable literature to date.