Given a Bayesian network structure (directed acyclic graph), the celebrated d-separation algorithm efficiently determines whether the network structure implies a given conditional independence relation. We show that this changes drastically when we consider two Bayesian network structures instead. It is undecidable to determine whether two given network structures imply a given conditional independency, that is, whether every collection of random variables satisfying both network structures must also satisfy the conditional independency. Although the approximate combination of two Bayesian networks is a well-studied topic, our result shows that it is fundamentally impossible to accurately combine the knowledge of two Bayesian network structures, in the sense that no algorithm can tell what conditional independencies are implied by the two network structures. We can also explicitly construct two Bayesian network structures, such that whether they imply a certain conditional independency is unprovable in the ZFC set theory, assuming ZFC is consistent.

翻译：给定一个贝叶斯网络结构（有向无环图），著名的d-分离算法能够高效地判定该网络结构是否蕴含给定的条件独立关系。我们发现，当考虑两个贝叶斯网络结构时，情况发生了根本性变化。判定两个给定网络结构是否蕴含某个条件独立性——即同时满足这两个网络结构的任意随机变量集合是否也必然满足该条件独立性——是不可判定的。尽管两个贝叶斯网络的大致组合是一个被广泛研究的课题，但我们的结果表明，从本质上讲，准确组合两个贝叶斯网络结构的知识是不可能的，因为没有任何算法能够判定这两个网络结构蕴含哪些条件独立性。我们还可以显式构造出两个贝叶斯网络结构，使得在假设ZFC集合论一致的前提下，它们是否蕴含某个特定条件独立性在ZFC中是无法证明的。