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

2026-03-28

AI Summary

This study addresses the problem of retrospective counterfactual prediction: estimating the expected potential outcome for an individual under a different intervention, given their observed covariates and realized outcome. To this end, the authors propose a unified framework based on a cross-world correlation parameter ρ(x), which links observable and unobservable potential outcomes within the Neyman–Rubin superpopulation model, thereby moving beyond the restrictive assumptions of ρ = 0 or ρ = 1 commonly adopted in existing methods. By incorporating correlation-aware identification strategies, they construct asymptotically efficient estimators and valid prediction intervals. Theoretical analysis and empirical experiments demonstrate that the proposed approach achieves nominal coverage under standard causal assumptions and substantially outperforms current baselines.

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 $μ(x,y)=\mathbb{E}[Y(1)\mid X=x,Y(0)=y]$, 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 $ρ(x)=cor(Y(1),Y(0)\mid X=x)$; plausible bounds on $ρ(x)$ enable a principled approach to this otherwise unidentified problem. We introduce retrospective counterfactual estimators $\hatμ_ρ(x,y)$ and prediction intervals $C_ρ(x,y)$ that asymptotically satisfy $P[Y(1)\in C_ρ(x,y)\mid X=x, Y(0)=y]\ge1-α$ under standard causal assumptions. Many common baselines implicitly correspond to endpoint choices $ρ=0$ or $ρ=1$ (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.

Problem

• counterfactual prediction
• retrospective causal inference
• cross-world correlation
• potential outcomes
• prediction intervals

Innovation

• retrospective counterfactual
• cross-world correlation
• potential outcomes
• prediction intervals
• causal inference