提出新型因果模型,可统一处理抽象层级间的随机与确定性关系。
Factored space models: Towards causality between levels of abstraction
- 用因子空间模型替代因果图,支持多层级抽象中的确定性关联
- 证明结构独立性等价于所有可分解分布下的统计独立性
- 适用于图像等复杂数据的层次化因果分析,适合做理论推导的研究者
因果关系在理解智能行为中至关重要,现有数学模型多基于因果图,但其仅适用于已知变量且处于相同抽象层级的情况。当输入为像素等原始数据时,真实因果变量(如物体位置)可能是这些数据的任意确定性函数,且可能形成自下而上的抽象层次。此时,由于存在确定性关系,通常不存在同时满足马尔可夫条件和忠实性条件的因果图。本文提出因子空间模型作为替代方案,能自然表达各抽象层级中的概率与确定性关系。此外,引入结构独立性概念,并证明其在所有因子化分布中均等价于统计独立性。该定理推广了经典d-分离的完备性与正确性定理。
原文摘要 · Abstract (English)
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.
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