Functional causal models (fCMs) specify functional dependencies between random variables associated to the vertices of a graph. In directed acyclic graphs (DAGs), fCMs are well-understood: a unique probability distribution on the random variables can be easily specified, and a crucial graph-separation result called the d-separation theorem allows one to characterize conditional independences between the variables. However, fCMs on cyclic graphs pose challenges due to the absence of a systematic way to assign a unique probability distribution to the fCM's variables, the failure of the d-separation theorem, and lack of a generalization of this theorem that is applicable to all consistent cyclic fCMs. In this work, we develop a causal modeling framework applicable to all cyclic fCMs involving finite-cardinality variables, except inconsistent ones admitting no solutions. Our probability rule assigns a unique distribution even to non-uniquely solvable cyclic fCMs and reduces to the known rule for uniquely solvable fCMs. We identify a class of fCMs, called averagely uniquely solvable, that we show to be the largest class where the probabilities admit a Markov factorization. Furthermore, we introduce a new graph-separation property, p-separation, and prove this to be sound and complete for all consistent finite-cardinality cyclic fCMs while recovering the d-separation theorem for DAGs. These results are obtained by considering classical post-selected teleportation protocols inspired by analogous protocols in quantum information theory. We discuss further avenues for exploration, linking in particular problems in cyclic fCMs and in quantum causality.

翻译：函数因果模型（fCM）规定了与图中顶点相关联的随机变量之间的函数依赖关系。在有向无环图（DAG）中，fCM已得到充分理解：可以轻松指定随机变量上的唯一概率分布，并且一个称为d-分离定理的关键图分离结果允许人们刻画变量之间的条件独立性。然而，循环图上的fCM带来了挑战，原因在于缺乏为fCM变量分配唯一概率分布的系统方法、d-分离定理的失效，以及缺乏适用于所有一致循环fCM的该定理的推广。在本工作中，我们开发了一个适用于所有涉及有限基数变量（除了无解的不一致模型）的循环fCM的因果建模框架。我们的概率规则即使对非唯一可解的循环fCM也分配了唯一分布，并简化为已知的唯一可解fCM规则。我们识别了一类称为平均唯一可解的fCM，并证明其是概率允许马尔可夫分解的最大类别。此外，我们引入了一种新的图分离性质——p-分离，并证明其对所有一致有限基数循环fCM是可靠且完备的，同时恢复了DAG的d-分离定理。这些结果是通过考虑受量子信息论中类似协议启发的经典后选择隐形传态协议而获得的。我们讨论了进一步探索的途径，特别是将循环fCM中的问题与量子因果性问题联系起来。