  The counterfactual distribution models the effect of the treatment in the
untreated group. While most of the work focuses on the expected values of the
treatment effect, one may be interested in the whole counterfactual
distribution or other quantities associated to it. Building on the framework of
Bayesian conditional mean embeddings, we propose a Bayesian approach for
modeling the counterfactual distribution, which leads to quantifying the
epistemic uncertainty about the distribution. The framework naturally extends
to the setting where one observes multiple treatment effects (e.g. an
intermediate effect after an interim period, and an ultimate treatment effect
which is of main interest) and allows for additionally modelling uncertainty
about the relationship of these effects. For such goal, we present three novel
Bayesian methods to estimate the expectation of the ultimate treatment effect,
when only noisy samples of the dependence between intermediate and ultimate
effects are provided. These methods differ on the source of uncertainty
considered and allow for combining two sources of data. Moreover, we generalize
these ideas to the off-policy evaluation framework, which can be seen as an
extension of the counterfactual estimation problem. We empirically explore the
calibration of the algorithms in two different experimental settings which
require data fusion, and illustrate the value of considering the uncertainty
stemming from the two sources of data.
