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In causal inference confounding may be controlled either through regressionadjustment in an outcome model, or through propensity score adjustment orinverse probability of treatment weighting, or both. The latter approaches,which are based on modelling of the treatment assignment mechanism and theirdoubly robust extensions have been difficult to motivate using formal Bayesianarguments, in principle, for likelihood-based inferences, the treatmentassignment model can play no part in inferences concerning the expectedoutcomes if the models are assumed to be correctly specified. On the otherhand, forcing dependency between the outcome and treatment assignment models byallowing the former to be misspecified results in loss of the balancingproperty of the propensity scores and the loss of any double robustness. Inthis paper, we explain in the framework of misspecified models why doublyrobust inferences cannot arise from purely likelihood-based arguments, anddemonstrate this through simulations. As an alternative to Bayesian propensityscore analysis, we propose a Bayesian posterior predictive approach forconstructing doubly robust estimation procedures. Our approach appropriatelydecouples the outcome and treatment assignment models by incorporating theinverse treatment assignment probabilities in Bayesian causal inferences asimportance sampling weights in Monte Carlo integration.

A Bayesian view of doubly robust causal inference