[论文解读] Structural Causal Models: Cycles, Marginalizations, Exogenous Reparametrizations and Reductions.

Stephan Bongers, Jonas Peters+2 more|arXiv (Cornell University)|Nov 18, 2016

Bayesian Modeling and Causal Inference参考文献 56被引用 18

一句话总结

本文对具有循环关系的结构因果模型（SCM）提供了严格的测度论处理，引入了边际化和外生变量重参数化作为操作，以将复杂SCM（尤其是含循环的SCM）简化为更简单且等价的子系统。证明了在温和条件下，此类简化能保持因果语义并降低模型复杂度，但也表明一般性简化并非总是存在，限制了现有估计方法向任意SCM的推广。

ABSTRACT

Structural causal models (SCMs), also known as non-parametric structural equation models (NP-SEMs), are widely used for causal modeling purposes. In this paper, we give a rigorous treatment of structural causal models, dealing with measure-theoretic complications that arise in the presence of cyclic relations. The central question studied in this paper is: given a (possibly cyclic) SCM defined on a large system (consisting of observable endogenous and latent exogenous variables), can we project it down to an SCM that describes a subsystem (consisting of a subset of the observed endogenous variables and possibly different latent exogenous variables) in order to obtain a more parsimonious but equivalent representation of the subsystem? We define a marginalization operation that effectively removes a subset of the endogenous variables from the model, and a class of mappings, exogenous reparameterizations, that can be used to reduce the space of exogenous variables. We show that both operations preserve the causal semantics of the model and that under mild conditions they can lead to a significant reduction of the model complexity, at least in terms of the number of variables in the model. We argue that for the task of estimating an SCM from data, the existence of smooth reductions would be desirable. We provide several conditions under which the existence of such reductions can be shown, but also provide a counterexample that shows that such reductions do not exist in general. The latter result implies that existing approaches to estimate linear or Markovian SCMs from data cannot be extended to general SCMs.

研究动机与目标

• 解决具有循环关系的结构因果模型（SCM）中的测度论挑战。
• 研究复杂SCM是否可被简化为更简单且等价的子系统表示。
• 定义在降低模型复杂度的同时保持因果语义的操作。
• 确定此类简化的条件，特别是针对数据驱动的SCM估计。
• 阐明将线性或马尔可夫SCM估计方法推广至一般SCM的局限性。

提出的方法

• 引入正式的边际化操作，可在移除SCM中部分内生变量的同时保持其因果语义。
• 将外生变量重参数化定义为降低外生变量空间维度的映射。
• 应用测度论工具以处理循环SCM中出现的数学复杂性。
• 建立边际化与重参数化产生等价且简化SCM的条件。
• 利用结构方程与结构反事实形式化简化模型的语义。
• 证明在温和正则性条件下，简化模型与原始模型保持相同的干预分布与反事实分布。

实验结果

研究问题

• RQ1 能否通过移除某些内生变量来简化一个具有循环关系的结构因果模型，同时保持其因果语义？
• RQ2 在何种条件下，存在一种外生变量重参数化，可在不改变模型行为的前提下降低外生变量的维度？
• RQ3 是否总能通过边际化与重参数化，找到一个与更大规模、可能含循环的SCM等价的简洁SCM？
• RQ4 现有线性或马尔可夫SCM估计方法在多大程度上可推广至任意SCM？
• RQ5 哪些根本性限制阻止了SCM被普遍简化为更简单的形式？

主要发现

• 边际化与外生变量重参数化被正式定义并证明可在存在循环的情况下保持SCM的因果语义。
• 在温和正则性条件下，两种操作均可通过减少变量数量显著降低模型复杂度。
• 一般情况下，光滑简化的存在性无法保证，反例表明此类简化并非总是存在。
• 该反例意味着依赖模型简化的估计方法无法普遍推广至一般SCM。
• 研究结果明确了将线性或马尔可夫SCM估计技术推广至更广泛结构因果模型类别的理论边界。