  The estimation of heterogeneous treatment effects in the potential outcome
setting is biased when there exists model misspecification or unobserved
confounding. As these biases are unobservable, what model to use when remains a
critical open question. In this paper, we propose a novel Bayesian methodology
to mitigate misspecification and improve estimation via a synthesis of multiple
causal estimates, which we call Bayesian causal synthesis. Our development is
built upon identifying a synthesis function that correctly specifies the
heterogeneous treatment effect under no unobserved confounding, and achieves
the irreducible bias under unobserved confounding. We show that our proposed
method results in consistent estimates of the heterogeneous treatment effect;
either with no bias or with irreducible bias. We provide a computational
algorithm for fast posterior sampling. Several benchmark simulations and an
empirical study highlight the efficacy of the proposed approach compared to
existing methodologies, providing improved point and density estimation of the
heterogeneous treatment effect, even under unobserved confounding.
