In many applications of AI for Social Impact (e.g., when allocating spots in support programs for underserved communities), resources are scarce and an allocation policy is needed to decide who receives a resource. Before being deployed at scale, a rigorous evaluation of an AI-powered allocation policy is vital. In this paper, we introduce the methods necessary to evaluate index-based allocation policies, which allocate a limited number of resources to those who need them the most. Such policies create dependencies between agents, rendering standard statistical tests invalid and ineffective. Addressing the arising practical and technical challenges, we describe an efficient estimator and methods for drawing valid statistical conclusions. Our extensive experiments validate our methodology in practical settings while also showcasing its statistical power. We conclude by proposing and empirically verifying extensions of our methodology that enable us to reevaluate a past randomized control trial conducted with 10000 beneficiaries for a mHealth program for pregnant women. Our new methodology allows us to draw previously invisible conclusions when comparing two different ML allocation policies.

Uncertainty quantification (UQ) for adaptively collected data, such as that coming from adaptive experiments, bandits, or reinforcement learning, is necessary for critical elements of data collection such as ensuring safety and conducting after-study inference. The data's adaptivity creates significant challenges for frequentist UQ, yet Bayesian UQ remains the same as if the data were independent and identically distributed (i.i.d.), making it an appealing and commonly used approach. Bayesian UQ requires the (correct) specification of a prior distribution while frequentist UQ does not, but for i.i.d. data the celebrated Bernstein–von Mises theorem shows that as the sample size grows, the prior `washes out' and Bayesian UQ becomes frequentist-valid, implying that the choice of prior need not be a major impediment to Bayesian UQ as it makes no difference asymptotically. This paper for the first time extends the Bernstein–von Mises theorem to adaptively collected data, proving asymptotic equivalence between Bayesian UQ and Wald-type frequentist UQ in this challenging setting. Our results do not require the standard stability condition for validity of Wald-type frequentist UQ, and thus provide positive results on frequentist validity of Bayesian UQ under stability. Counterintuitively however, they also provide a negative result that Bayesian UQ is not asymptotically frequentist valid when stability fails, despite the fact that the prior washes out and Bayesian UQ asymptotically matches standard Wald-type frequentist UQ. We empirically validate our theory (positive and negative) via a range of simulations.