  Frequentist inference has a well-established supporting theory for doubly
robust causal inference based on the potential outcomes framework, which is
realized via outcome regression (OR) and propensity score (PS) models. The
Bayesian counterpart, however, is not obvious as the PS model loses its
balancing property in joint modeling. In this paper, we propose a natural and
formal Bayesian solution by bridging loss-type Bayesian inference with a
utility function derived from the notion of a pseudo-population via the change
of measure. Consistency of the posterior distribution is shown with correctly
specified and misspecified OR models. Simulation studies suggest that our
proposed method can estimate the true causal effect more efficiently and
achieve the frequentist coverage if either the OR model is correctly specified
or fit with a flexible function of the confounders, compared to the previous
Bayesian approach via the Bayesian bootstrap. Finally, we apply this novel
Bayesian method to assess the impact of speed cameras on the reduction of car
collisions in England.
