  We consider the optimal decision-making problem in a primary sample of
interest with multiple auxiliary sources available. The outcome of interest is
limited in the sense that it is only observed in the primary sample. In
reality, such multiple data sources may belong to different populations and
thus cannot be combined directly. This paper proposes a novel calibrated
optimal decision rule (CODR) to address the limited outcome, by leveraging the
shared pattern in multiple data sources. Under a mild and testable assumption
that the conditional means of intermediate outcomes in different samples are
equal given baseline covariates and the treatment information, we can show that
the calibrated mean outcome of interest under the CODR is unbiased and more
efficient than using the primary sample solely. Extensive experiments on
simulated datasets demonstrate empirical validity and improvement of the
proposed CODR, followed by a real application on the MIMIC-III as the primary
sample with auxiliary data from eICU.
