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

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 $P[Y(1)\in C_{\rho }(x,y)\mid X=x,Y(0)=y]\ge 1-\alpha$ 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.