Markov Categories, Causal Theories, and the Do-Calculus

We give a category-theoretic treatment of causal models that formalizes the syntax for causal reasoning over a directed acyclic graph (DAG) by associating a free Markov category with the DAG in a canonical way. This framework enables us to define and study important concepts in causal reasoning from an abstract and “purely causal” point of view, such as causal independence/separation, causal conditionals, and decomposition of intervention effects. Our results regarding these concepts abstract away from the details of the commonly adopted causal models such as (recursive) structural equation models or causal Bayesian networks. They are therefore more widely applicable and in a way conceptually clearer. Our results are also intimately related to Judea Pearl’s celebrated do-calculus, and yield a syntactic version of a core part of the calculus that is inherited in all causal models. In particular, it induces a simpler and specialized version of Pearl’s do-calculus in the context of causal Bayesian networks, which we show is as strong as the full version.

本文循范畴论的方式处理基于有向无环图的因果模型，用一种自由马尔可夫范畴（名曰“因果理论”）作为因果推理的语形范畴，并在此框架下定义因果推理中的一些重要概念，比如因果独立性、因果条件化，以及干预作用的分解。本文的结果不依赖于常见的一些因果模型的特性，因此适用的范围更广，概念上也更加清晰。它们与朱迪亚·珀尔的著名的干预演算关系密切，相当于一个语形层面的干预演算，在各种基于有向无环图的因果模型上都有效。对因果贝叶斯网这种概率因果模型来说，这个语形层面的干预演算对应于珀尔演算的一个相对简单的特殊形式。此特殊形式实际上是干预演算的“因果核心”，加上概率演算就足以推导出标准的珀尔演算。