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We introduce a framework for estimating causal effects of binary andcontinuous treatments in high dimensions. We show how posterior distributionsof treatment and outcome models can be used together with doubly robustestimators. We propose an approach to uncertainty quantification for the doublyrobust estimator which utilizes posterior distributions of model parameters and(1) results in good frequentist properties in small samples, (2) is based on asingle MCMC, and (3) improves over frequentist measures of uncertainty whichrely on asymptotic properties. We show that our proposed variance estimationstrategy is consistent when both models are correctly specified and that it isconservative in finite samples or when one or both models are misspecified. Weconsider a flexible framework for modeling the treatment and outcome processeswithin the Bayesian paradigm that reduces model dependence, accommodatesnonlinearity, and achieves dimension reduction of the covariate space. Weillustrate the ability of the proposed approach to flexibly estimate causaleffects in high dimensions and appropriately quantify uncertainty, and showthat it performs well relative to existing approaches. Finally, we estimate theeffect of continuous environmental exposures on cholesterol and triglyceridelevels. An R package is available at github.com/jantonelli111/DoublyRobustHD.

Causal Inference in high dimensions: A marriage between Bayesian modeling and good frequentist properties