Causality plays an important role in understanding intelligent behavior, and there is a wealth of literature on mathematical models for causality, most of which is focused on causal graphs. Causal graphs are a powerful tool for a wide range of applications, in particular when the relevant variables are known and at the same level of abstraction. However, the given variables can also be unstructured data, like pixels of an image. Meanwhile, the causal variables, such as the positions of objects in the image, can be arbitrary deterministic functions of the given variables. Moreover, the causal variables may form a hierarchy of abstractions, in which the macro-level variables are deterministic functions of the micro-level variables. Causal graphs are limited when it comes to modeling this kind of situation. In the presence of deterministic relationships there is generally no causal graph that satisfies both the Markov condition and the faithfulness condition. We introduce factored space models as an alternative to causal graphs which naturally represent both probabilistic and deterministic relationships at all levels of abstraction. Moreover, we introduce structural independence and establish that it is equivalent to statistical independence in every distribution that factorizes over the factored space. This theorem generalizes the classical soundness and completeness theorem for d-separation.

翻译：因果性在理解智能行为中扮演重要角色，现有大量关于因果性数学模型的文献，其中多数聚焦于因果图。因果图是适用于广泛应用的强大工具，尤其在相关变量已知且处于同一抽象层级时效果显著。然而，给定变量也可能是非结构化数据，如图像像素。同时，因果变量（如图中物体位置）可以是给定变量的任意确定性函数。此外，因果变量可能形成抽象层次结构，其中宏观变量是微观变量的确定性函数。因果图在建模此类情况时存在局限：当存在确定性关系时，通常无法找到同时满足马尔可夫条件与忠实性条件的因果图。我们提出因子空间模型作为因果图的替代方案，其能自然表征所有抽象层级的概率性与确定性关系。进一步，我们引入结构独立性概念，并证明其在所有基于因子空间分解的分布中均等价于统计独立性。该定理推广了经典d-分离的完备性定理。