  Multifidelity methods are widely used for estimating quantities of interest
(QoI) in computational science by employing numerical simulations of differing
costs and accuracies. Many methods approximate numerical-valued statistics that
represent only limited information, e.g., scalar statistics, about the QoI.
Further quantification of uncertainty, e.g., for risk assessment, failure
probabilities, or confidence intervals, requires estimation of the full
distributions. In this paper, we generalize the ideas in [Xu et al., SIAM J.
Sci. Comput. 44.1 (2022), A150-A175] to develop a multifidelity method that
approximates the full distribution of scalar-valued QoI. The main advantage of
our approach compared to alternative methods is that we require no particular
relationships among the high and lower-fidelity models (e.g. model hierarchy),
and we do not assume any knowledge of model statistics including correlations
and other cross-model statistics before the procedure starts. Under suitable
assumptions in the framework above, we achieve provable 1-Wasserstein metric
convergence of an algorithmically constructed distributional emulator via an
exploration-exploitation strategy. We also prove that crucial policy actions
taken by our algorithm are budget-asymptotically optimal. Numerical experiments
are provided to support our theoretical analysis.
