Suppose X and Y are binary exposure and outcome variables, and we have full knowledge of the distribution of Y, given application of X. We are interested in assessing whether an outcome in some case is due to the exposure. This “probability of causation” is of interest in comparative historical analysis where scholars use process tracing approaches to learn about causes of outcomes for single units by observing events along a causal path. The probability of causation is typically not identified, but bounds can be placed on it. Here, we provide a full characterization of the bounds that can be achieved in the ideal case that X and Y are connected by a causal chain of complete mediators, and we know the probabilistic structure of the full chain. Our results are largely negative. We show that, even in these very favorable conditions, the gains from positive evidence on mediators is modest.

假设X和Y是二元暴露和结果变量，并且我们完全了解Y的分布，给定X的应用。我们有兴趣评估某些情况下的结果是否是由于暴露造成的。这种“因果关系的概率”在比较历史分析中很有趣，在比较历史分析中，学者们使用过程追踪方法通过沿因果路径观察事件来了解单个单元结果的原因。因果关系的概率通常不会被识别，但可以对其设置界限。在这里，我们提供了在X和Y的理想情况下可以实现的边界的完整特征。由完整中介的因果链连接，我们知道完整链的概率结构。我们的结果在很大程度上是负面的。我们表明，即使在这些非常有利的条件下，来自中介的积极证据的收益也是有限的。