  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 aim to estimate any functional of
a probabilistic model (e.g., a causal effect) as efficiently as possible, by
deciding, at each time, which data source to query. 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 current estimates of
the moments. We propose two selection strategies: (1) explore-then-commit
(OMS-ETC) and (2) explore-then-greedy (OMS-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 may include subsets of mediators, confounders, and
instrumental variables.
