Switching Regression Models and Causal Inference in the Presence of Latent Variables
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Given a response Y and a vector X = (X^1, ..., X^d) of d predictors, we investigate the problem of inferring direct causes of Y among the vector X. Models for Y that use its causal covariates as predictors enjoy the property of being invariant across different environments or interventional settings. Given data from such environments, this property has been exploited for causal discovery: one collects the models that show predictive stability across all environments and outputs the set of predictors that are necessary to obtain stability. If some of the direct causes are latent, however, there may not exist invariant models for Y based on variables from X, and the above reasoning breaks down. In this paper, we extend the principle of invariant prediction by introducing a relaxed version of the invariance assumption. This property can be used for causal discovery in the presence of latent variables if the latter's influence on Y can be restricted. More specifically, we allow for latent variables with a low-range discrete influence on the target Y. This assumption gives rise to switching regression models, where each value of the (unknown) hidden variable corresponds to a different regression coefficient. We provide sufficient conditions for the existence, consistency and asymptotic normality of the maximum likelihood estimator in switching regression models, and construct a test for the equality of such models. Our results on switching regression models allow us to prove that asymptotic false discovery control for the causal discovery method is obtained under mild conditions. We provide an algorithm for the overall method, make available code, and illustrate the performance of our method on simulated data.
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