In causal inference, the joint law of a set of counterfactual random variables is generally not identified. We show that a conservative version of the joint law - corresponding to the smallest treatment effect - is identified. Finding this law uses recent results from optimal transport theory. Under this conservative law we can bound causal effects and we may construct inferences for each individual's counterfactual dose-response curve. Intuitively, this is the flattest counterfactual curve for each subject that is consistent with the distribution of the observables. If the outcome is univariate then, under mild conditions, this curve is simply the quantile function of the counterfactual distribution that passes through the observed point. This curve corresponds to a nonparametric rank preserving structural model.

翻译：在因果推断中，一组反事实随机变量的联合分布通常不可识别。我们证明，保守版本的联合分布——对应于最小处理效应——是可以识别的。寻找该分布利用了最优输运理论的最新成果。在此保守分布下，我们可以限定因果效应，并可能为每个个体的反事实剂量-反应曲线构建推断。直观上，这是每个个体与观测数据分布一致的最平坦反事实曲线。若结果为单变量，则在温和条件下，该曲线仅是通过观测点的反事实分布的分位数函数。此曲线对应一个非参数秩保持结构模型。