Although the existing causal inference literature focuses on the forward-looking perspective by estimating effects of causes, the backward-looking perspective can provide insights into causes of effects. In backward-looking causal inference, the probability of necessity measures the probability that a certain event is caused by the treatment given the observed treatment and outcome. Most existing results focus on binary outcomes. Motivated by applications with ordinal outcomes, we propose a general definition of the probability of necessity. However, identifying the probability of necessity is challenging because it involves the joint distribution of the potential outcomes. We propose a novel assumption of monotonic incremental treatment effect to identify the probability of necessity with ordinal outcomes. We also discuss the testable implications of this key identification assumption. When it fails, we derive explicit formulas of the sharp large-sample bounds on the probability of necessity.

翻译：尽管现有因果推断文献主要通过估计因果效应来关注前向视角，但后向视角能为探究结果的原因提供洞见。在后向因果推断中，必要性概率衡量的是在观察到处理与结果的情况下，特定事件由处理引致的概率。现有研究大多聚焦于二值结果。受有序结果实际应用的启发，我们提出了必要性概率的通用定义。然而，识别必要性概率具有挑战性，因其涉及潜在结果的联合分布。我们提出了一种新颖的单调增量处理效应假设，用以识别有序结果下的必要性概率。同时，我们讨论了这一关键识别假设的可检验性含义。当该假设不成立时，我们推导了必要性概率的显式尖锐大样本边界公式。