Statistical Decision Theory with Counterfactual Loss

Abstract: Classical statistical decision theory evaluates treatment choices based solely on observed outcomes. However, by ignoring counterfactual outcomes, it cannot assess the quality of decisions relative to feasible alternatives. For example, the quality of a physician's decision may depend not only on patient survival, but also on whether a less invasive treatment could have produced a similar result. To address this limitation, we extend standard decision theory to incorporate counterfactual losses--criteria that evaluate decisions using all potential outcomes. The central challenge in this generalization is identification: because only one potential outcome is observed for each unit, the associated risk under a counterfactual loss is generally not identifiable. We show that under the assumption of strong ignorability, a counterfactual risk is identifiable if and only if the counterfactual loss function is additive in the potential outcomes. Moreover, we demonstrate that additive counterfactual losses can yield treatment recommendations that differ from those based on standard loss functions, provided that the decision problem involves more than two treatment options.