What types of differences among causal structures with latent variables are impossible to distinguish by statistical data obtained by probing each visible variable? If the probing scheme is simply passive observation, then it is well-known that many different causal structures can realize the same joint probability distributions. Even for the simplest case of two visible variables, for instance, one cannot distinguish between one variable being a causal parent of the other and the two variables sharing a latent common cause. However, it is possible to distinguish between these two causal structures if we have recourse to more powerful probing schemes, such as the possibility of intervening on one of the variables and observing the other. Herein, we address the question of which causal structures remain indistinguishable even given the most informative types of probing schemes on the visible variables. We find that two causal structures remain indistinguishable if and only if they are both associated with the same mDAG structure (as defined by Evans (2016)). We also consider the question of when one causal structure dominates another in the sense that it can realize all of the joint probability distributions that can be realized by the other using a given probing scheme. (Equivalence of causal structures is the special case of mutual dominance.) Finally, we investigate to what extent one can weaken the probing schemes implemented on the visible variables and still have the same discrimination power as a maximally informative probing scheme.

翻译：对于含隐变量的因果结构，通过探测每个可见变量获得的统计数据，哪些类型的差异是无法区分的？如果探测方案仅为被动观测，众所周知，许多不同的因果结构可以实现相同的联合概率分布。例如，即使在最简单的两个可见变量情形下，也无法区分一个变量是另一个变量的因果父节点与两个变量共享一个潜在共同原因这两种因果结构。然而，若我们能借助更强大的探测方案（例如对其中一个变量进行干预并观测另一个变量的可能性），则有可能区分这两种因果结构。本文致力于探究：即使对可见变量实施最具信息量的探测方案，哪些因果结构仍然无法区分。我们发现，两个因果结构不可区分的充要条件是它们均与相同的mDAG结构相关联（依据Evans（2016）的定义）。我们还考虑了因果结构间的支配关系问题：在给定探测方案下，若某个因果结构能够实现另一个因果结构所能实现的所有联合概率分布，则称前者支配后者（因果结构的等价性是相互支配的特殊情形）。最后，我们研究了在保持与最具信息量的探测方案相同区分能力的前提下，对可见变量实施的探测方案可弱化至何种程度。