Structural Causal Models: Cycles, Marginalizations, Exogenous Reparametrizations and Reductions
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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.
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