Efficient Online Estimation of Causal Effects by Deciding What to Observe

Researchers often face data fusion problems, where multiple data sources are available, each capturing a distinct subset of variables. While problem formulations typically take the data as given, in practice, data acquisition can be an ongoing process. In this paper, we introduce the problem of deciding, at each time, which data source to sample from. Our goal is to estimate a given functional of the parameters of a probabilistic model as efficiently as possible. We propose online moment selection (OMS), a framework in which structural assumptions are encoded as moment conditions. The optimal action at each step depends, in part, on the very moments that identify the functional of interest. Our algorithms balance exploration with choosing the best action as suggested by estimated moments. We propose two selection strategies: (1) explore-then-commit (ETC) and (2) explore-then-greedy (ETG), proving that both achieve zero asymptotic regret as assessed by MSE. We instantiate our setup for average treatment effect estimation, where structural assumptions are given by a causal graph and data sources include subsets of mediators, confounders, and instrumental variables.