We define what it means for a joint probability distribution to be compatible with a set of independent causal mechanisms, at a qualitative level -- or, more precisely, with a directed hypergraph ${\mathcal{A}}$, which is the qualitative structure of a probabilistic dependency graph (PDG). When ${\mathcal{A}}$ represents a qualitative Bayesian network, QIM-compatibility with ${\mathcal{A}}$ reduces to satisfying the appropriate conditional independencies. But giving semantics to hypergraphs using QIM-compatibility lets us do much more. For one thing, we can capture functional dependencies. For another, we can capture important aspects of causality using compatibility: we can use compatibility to understand cyclic causal graphs, and to demonstrate structural compatibility, we must essentially produce a causal model. Finally, QIM-compatibility has deep connections to information theory. Applying our notion to cyclic structures helps to clarify a longstanding conceptual issue in information theory.

翻译：我们定义了联合概率分布与一组独立因果机制在定性层面上的兼容性含义——更准确地说，是与有向超图${\mathcal{A}}$的兼容性，该超图是概率依赖图（PDG）的定性结构。当${\mathcal{A}}$表示一个定性贝叶斯网络时，与${\mathcal{A}}$的QIM兼容性简化为满足适当的条件独立性。但利用QIM兼容性为超图赋予语义使我们能够实现更多功能。一方面，我们可以捕捉函数依赖关系。另一方面，我们可以通过兼容性捕捉因果关系的重要方面：利用兼容性理解循环因果图，并且为了证明结构兼容性，本质上必须构建一个因果模型。最后，QIM兼容性与信息论存在深刻联系。将我们的概念应用于循环结构有助于澄清信息论中长期存在的概念性问题。