Retrospective Counterfactual Prediction by Conditioning on the Factual Outcome: A Cross-World Approach

Juraj Bodik
Proceedings of the Fifth Conference on Causal Learning and Reasoning, PMLR 323:50-83, 2026.

Abstract

Retrospective causal questions ask what would have happened to an observed individual had they received a different treatment. We study the problem of estimating $\mu(x,y)=\mathbb{E}[Y(1)\mid X=x,Y(0)=y]$, the expected counterfactual outcome for an individual with covariates $x$ and observed outcome $y$, and constructing valid prediction intervals under the Neyman-Rubin superpopulation model. This quantity is generally not identified without additional assumptions. To link the observed and unobserved potential outcomes, we work with a cross-world correlation $\rho(x)=cor(Y(1),Y(0)\mid X=x)$; plausible bounds on $\rho(x)$ enable a principled approach to this otherwise unidentified problem. We introduce retrospective counterfactual estimators $\hat{\mu}_{\rho}(x,y)$ and prediction intervals $C(x,y)$ that asymptotically satisfy $P[Y(1)\in C(x,y)\mid X=x, Y(0)=y]\ge1-\alpha$ under standard causal assumptions. Many common baselines implicitly correspond to endpoint choices $\rho=0$ or $\rho=1$ (ignoring the factual outcome or treating the counterfactual as a shifted factual outcome). Interpolating between these cases through cross-world dependence yields substantial gains in both theory and practice.

Cite this Paper

BibTeX
@InProceedings{pmlr-v323-bodik26a, title = {Retrospective Counterfactual Prediction by Conditioning on the Factual Outcome: A Cross-World Approach}, author = {Bodik, Juraj}, booktitle = {Proceedings of the Fifth Conference on Causal Learning and Reasoning}, pages = {50--83}, year = {2026}, editor = {Mazaheri, Bijan and Hanson, Niels Richard}, volume = {323}, series = {Proceedings of Machine Learning Research}, month = {06--08 Apr}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v323/main/assets/bodik26a/bodik26a.pdf}, url = {https://proceedings.mlr.press/v323/bodik26a.html}, abstract = {Retrospective causal questions ask what would have happened to an observed individual had they received a different treatment. We study the problem of estimating $\mu(x,y)=\mathbb{E}[Y(1)\mid X=x,Y(0)=y]$, the expected counterfactual outcome for an individual with covariates $x$ and observed outcome $y$, and constructing valid prediction intervals under the Neyman-Rubin superpopulation model. This quantity is generally not identified without additional assumptions. To link the observed and unobserved potential outcomes, we work with a cross-world correlation $\rho(x)=cor(Y(1),Y(0)\mid X=x)$; plausible bounds on $\rho(x)$ enable a principled approach to this otherwise unidentified problem. We introduce retrospective counterfactual estimators $\hat{\mu}_{\rho}(x,y)$ and prediction intervals $C(x,y)$ that asymptotically satisfy $P[Y(1)\in C(x,y)\mid X=x, Y(0)=y]\ge1-\alpha$ under standard causal assumptions. Many common baselines implicitly correspond to endpoint choices $\rho=0$ or $\rho=1$ (ignoring the factual outcome or treating the counterfactual as a shifted factual outcome). Interpolating between these cases through cross-world dependence yields substantial gains in both theory and practice.} }
Endnote
%0 Conference Paper %T Retrospective Counterfactual Prediction by Conditioning on the Factual Outcome: A Cross-World Approach %A Juraj Bodik %B Proceedings of the Fifth Conference on Causal Learning and Reasoning %C Proceedings of Machine Learning Research %D 2026 %E Bijan Mazaheri %E Niels Richard Hanson %F pmlr-v323-bodik26a %I PMLR %P 50--83 %U https://proceedings.mlr.press/v323/bodik26a.html %V 323 %X Retrospective causal questions ask what would have happened to an observed individual had they received a different treatment. We study the problem of estimating $\mu(x,y)=\mathbb{E}[Y(1)\mid X=x,Y(0)=y]$, the expected counterfactual outcome for an individual with covariates $x$ and observed outcome $y$, and constructing valid prediction intervals under the Neyman-Rubin superpopulation model. This quantity is generally not identified without additional assumptions. To link the observed and unobserved potential outcomes, we work with a cross-world correlation $\rho(x)=cor(Y(1),Y(0)\mid X=x)$; plausible bounds on $\rho(x)$ enable a principled approach to this otherwise unidentified problem. We introduce retrospective counterfactual estimators $\hat{\mu}_{\rho}(x,y)$ and prediction intervals $C(x,y)$ that asymptotically satisfy $P[Y(1)\in C(x,y)\mid X=x, Y(0)=y]\ge1-\alpha$ under standard causal assumptions. Many common baselines implicitly correspond to endpoint choices $\rho=0$ or $\rho=1$ (ignoring the factual outcome or treating the counterfactual as a shifted factual outcome). Interpolating between these cases through cross-world dependence yields substantial gains in both theory and practice.
APA
Bodik, J.. (2026). Retrospective Counterfactual Prediction by Conditioning on the Factual Outcome: A Cross-World Approach. Proceedings of the Fifth Conference on Causal Learning and Reasoning, in Proceedings of Machine Learning Research 323:50-83 Available from https://proceedings.mlr.press/v323/bodik26a.html.

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