Inference on counterfactual distributions using martingale posteriors

2026-07-27

AI Summary

This study addresses the limitation of traditional causal inference methods, which primarily focus on average treatment effects and often fail to fully characterize the distributional features and uncertainty of counterfactual outcomes. The authors propose a nonparametric Bayesian approach based on martingale posteriors, leveraging Dirichlet process mixture models and predictive recursion algorithms to flexibly infer counterfactual densities, distribution functions, and quantiles. The method offers both computational efficiency and theoretical convergence guarantees, and naturally extends to settings involving conditional distributions and instrumental variables. Empirical evaluations on simulated data and real-world applications—including the effect of zinc lozenges on common cold duration and vitamin A supplementation on child survival—demonstrate the method’s effectiveness and practical utility.

Abstract

Causal inference is often focused on average effects, which can hide important aspects of the effect distributions. Here we consider the entire posterior effects distribution by estimating full counterfactual outcome distributions. We propose a methodology for inference on counterfactual distributions which builds upon the martingale posterior framework of Fong et al. (2023). This provides a highly flexible approach to estimating densities, distribution functions, and derived quantities such as quantiles, which coherently quantifies the epistemic uncertainty on any target estimand of interest. As the predictive recursions are based on an underlying nonparametric model (a Dirichlet process mixture model), our method naturally inherits robustness with respect to restrictive parametric assumptions. In addition, implementation of our method is typically very fast. This approach can be applied to marginal or conditional counterfactual distributions and is easily extended to an instrumental variables setup. Using the concept of almost conditionally identically distributed random variables, we prove convergence of the martingale posterior inference on the counterfactual outcome distributions for the causal models considered in the paper. We illustrate our approach on both simulated and real data. Using the latter, we investigate the effect of zinc lozenges on common cold duration, the impact of vitamin A supplementation on children's survival rates with one-sided non-compliance (analysed in Imbens and Rubin, 1997a) and the effect of job training (LaLonde, 1986).

Problem

• counterfactual distributions
• causal inference
• effect heterogeneity
• nonparametric Bayesian
• martingale posteriors

Innovation

• martingale posterior
• counterfactual distribution
• nonparametric Bayesian inference
• causal inference
• Dirichlet process mixture