Non-linear Causal Inference using Gaussianity Measures

In this paper we provide theoretical and empirical evidence for a type of asymmetry between causes and effects that is present when these are related via linear models contaminated with additive non-Gaussian noise. This asymmetry consists in the different levels of non-Gaussianity of the residuals of linear fits between the multivariate random variables in the causal and anti-causal directions: Under certain conditions, the distribution of the residuals is closer to a Gaussian distribution when the fit is made in the incorrect anti-causal direction. The method is closely related to causal inference techniques based on entropy estimation, extending their range of application to multivariate and nonlinear problems. The problem of non-linear causal inference is addressed by performing an embedding in an extended feature space, in which the relation between causes and effects can be assumed to be linear. In this extended space the required computations can be efficiently carried out using kernel techniques. The effectiveness of a method based on this type of asymmetry is illustrated in a variety of experiments in both synthetic and real-world cause-effect pairs. In the experiments performed one observes a Gaussianization of the residuals if the model is fitted in the anti-causal direction. In the problems investigated the method is competitive with state-of-the-art techniques for causal inference.