Robust Bayesian Inference of Causal Effects via Randomization Distributions

TL;DR

This paper introduces a robust Bayesian framework for causal inference based on treatment randomization, which requires fewer assumptions and aligns with classical estimators, demonstrated through simulations and a nutrition case study.

Contribution

It develops a general, design-based Bayesian method for causal effect inference that is robust, theoretically grounded, and compatible with classical estimators and model checking.

Findings

Posterior mean aligns with Hodges-Lehmann estimators.

Framework uncovers effect heterogeneity in case study.

Method requires fewer assumptions than traditional Bayesian approaches.

Abstract

We present a general framework for Bayesian inference of causal effects that delivers provably robust inferences founded on design-based randomization of treatments. The framework involves fixing the observed potential outcomes and forming a likelihood based on the randomization distribution of a statistic. The method requires specification of a treatment effect model; in many cases, however, it does not require specification of marginal outcome distributions, resulting in weaker assumptions compared to Bayesian superpopulation-based methods. We show that the framework is compatible with posterior model checking in the form of posterior-averaged randomization tests. We prove several theoretical properties for the method, including a Bernstein-von Mises theorem and large-sample properties of posterior expectations. In particular, we show that the posterior mean is asymptotically equivalent to Hodges-Lehmann estimators, which provides a bridge to many classical estimators in causal inference, including inverse-probability-weighted estimators and H\'ajek estimators. We evaluate the theory and utility of the framework in simulation and a case study involving a nutrition experiment. In the latter, our framework uncovers strong evidence of effect heterogeneity despite a lack of evidence for moderation effects. The basic framework allows numerous extensions, including the use of covariates, sensitivity analysis, estimation of assignment mechanisms, and generalization to nonbinary treatments.