NonSENS: Non-Linear SEM Estimation using Non-Stationarity

We consider the bivariate causal discovery problem. While this problem has been extensively studied, the majority of current methods assume a linear causal relationship, and the few methods which consider non-linear dependencies usually make the assumption of additive noise. Here, we propose a framework through which we can perform causal discovery in the presence of general non-linear relationships. The proposed method exploits a correspondence between a piecewise stationary non-linear ICA model and non-linear causal models. We show that in the case of bivariate causal discovery, non-linear ICA can be used to infer the causal direction via a series of independence tests. A series of experiments on simulated data demonstrate the capabilities of the proposed method. Causal models play a fundamental role in modern scientific endeavor [Spirtes et al., 2000, Pearl, 2009]. While randomized control studies are the gold standard, such an approach is unfeasible or unethical in many scenarios [Spirtes and Zhang, 2016]. Furthermore, big data sets publicly available on the internet often try to be generic and thus cannot be strongly based on specific intervention. As such, it is both necessary and important to develop causal discovery methods through which to uncover causal structure from (potentially large-scale) passively observed data. Data collected without the explicit manipulation of certain variables is often termed observational data, in contrast to experimental data where certain variables are intervened upon, as in randomized controlled trials. In this work we focus on the bivariate causal discovery problem. This corresponds to recovering the causal structure using observations from two variables, which we denote by X1 and X2. While bivariate causal discovery is a (simplified) special case of the more general causal discovery problem, it remains a challenging task. To date, the identification of causal relationships between two variables has primarily been studied under the assumption of linear causal dependencies [Kano and Shimizu, 2003, Hyvärinen and Smith, 2013]. Several extensions to accommodate non-linear causal dependencies have also been proposed, however, the majority of such methods typically assume additive noise [Hoyer et al., 2009, Peters et al., 2014] Here, we propose a general method for bivariate causal discovery in the presence of general nonlinearities. The proposed method is able to uncover non-linear causal relationships without requiring assumptions such as linear causal structure or additive noise. Our approach is based on the correspondence between a piece-wise stationary non-linear ICA model and a non-linear structural equation model (SEM). The proposed method therefore shares similarities with linear-ICA based causal discovery methods [Shimizu et al., 2006]. Under the assumption that we observed bivariate data generated under a specific non-linear ICA model utilizing non-stationarity, we demonstrate that if latent sources can be recovered via non-linear ICA, then a series of independence tests can be employed to uncover causal structure.