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)=E[Y(1) 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) X=x); plausible bounds on ρ(x) enable a principled approach to this otherwise unidentified problem. We introduce retrospective counterfactual estimators μ_ρ(x,y) and prediction intervals C_ρ(x,y) that asymptotically satisfy P[Y(1)∈ C_ρ(x,y) X=x, Y(0)=y]≥1-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.