Gregor Steiner, Mark Steel
arXiv 27 Jul 2026 · Statistics — Methodology
arXiv:2607.24143 · PDF · Extracted main text
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).
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Fong, Edwin and Holmes, Chris and Walker, Stephen G (2023) Martingale posterior distributions | 1.000 | 16 | 4 | 100% |
| 2 | Imbens, Guido W. and Rubin, Donald B (1997) Bayesian inference for causal effects in randomized experiments with noncompliance | 0.928 | 4 | 3 | 100% |
| 3 | LaLonde, Robert J (1986) Evaluating the Econometric Evaluations of Training Programs with Experimental Data | 0.928 | 4 | 3 | 100% |
| 4 | Ng, Kenyon and Fong, Edwin and Frazier, David T. and Knoblauch, Jere… (2026) TabMGP: Martingale Posterior with TabPFN | 0.843 | 3 | 3 | 100% |
| 5 | Battiston, Marco and Cappello, Lorenzo (2025) Bayesian Predictive Inference Beyond Martingales | 0.721 | 8 | 3 | 38% |
| 6 | Berti, Patrizia and Pratelli, Luca and Rigo, Pietro (2004) LIMIT THEOREMS FOR A CLASS OF IDENTICALLY DISTRIBUTED RANDOM VARIABLES | 0.644 | 2 | 2 | 100% |
| 7 | Xu, Steven G. and Yang, Shu and Reich, Brian J (2022) A Bayesian Semiparametric Method For Estimating Causal Quantile Effects | 0.644 | 2 | 2 | 100% |
| 8 | Imbens, Guido W. and Xu, Yiqing (2025) Comparing Experimental and Nonexperimental Methods: What Lessons Have We Learned Four Decades after LaLonde (1986)? | 0.585 | 3 | 1 | 100% |
| 9 | Holovchak, Anastasiia and Saengkyongam, Sorawit and Meinshausen, Nic… (2025) Distributional Instrumental Variable Method | 0.511 | 2 | 2 | 50% |
| 10 | Ham, Daeyoung and Westling, Ted and Doss, Charles R (2024) Doubly robust estimation and inference for a log-concave counterfactual density | 0.511 | 2 | 1 | 100% |
Showing the top 10 of 38 scored citations.