Abstract
Doubly robust estimators are typically constructed by combining out- come regression and propensity score models to satisfy moment restrictions that ensure consistent estimation of causal quantities provided at least one of the com- ponent models is correctly specified. Standard Bayesian methods are difficult to apply because restricted moment models do not imply fully specified likelihood functions. This paper proposes a Bayesian bootstrap approach to derive approx- imate posterior predictive distributions that are doubly robust for estimation of causal quantities. Simulations show that the approach performs well under various sources of misspecification of the outcome regression or propensity score models. The estimator is applied in a case study of the effect of area deprivation on the incidence of child pedestrian casualties in British cities.
Highlights
Typical targets of inference in causal studies include average potential outcomes (APOs) and average treatment effects (ATEs)
This paper has presented an approach that can be used to derive approximate Bayesian inference for doubly robust estimation of causal quantities
Doubly robust ATE estimation typically involves prediction and extrapolation over unobserved covariate distributions and a Bayesian approach provides a natural framework for prediction in which both the unobserved covariates and the parameters have random status
Summary
Typical targets of inference in causal studies include average potential outcomes (APOs) and average treatment effects (ATEs). By repeatedly estimating the AOR model weighted by standardised sets of iid unit exponential random variables, we approximate the posterior distribution of model parameters and use these to form posterior predictive distribution for ATEs and APOs. While our approach cannot offer the coherent framework for inference that a fully Bayesian analysis would provide, it does still offer two features of Bayesian inference which are useful for causal modelling. While our approach cannot offer the coherent framework for inference that a fully Bayesian analysis would provide, it does still offer two features of Bayesian inference which are useful for causal modelling It provides a natural framework for prediction: estimation of ATEs and APOs necessarily involves prediction over unobserved data and our posterior predictive distributions incorporate randomness originating both from estimation of the parameters of the DR model itself and from the random nature of the unobserved observations used to make predictions.
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