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

arXiv:2603.27320v1 Announce Type: cross
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 $C_{rho}(x,y)$ that asymptotically satisfy $P[Y(1)in C_{rho}(x,y)mid X=x, Y(0)=y]ge1-alpha$ 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.