Although nonstationary data are more common in the real world, most existing causal discovery methods do not take nonstationarity into consideration. In this letter, we propose a kernel embedding–based approach, ENCI, for nonstationary causal model inference where data are collected from multiple domains with varying distributions. In ENCI, we transform the complicated relation of a cause-effect pair into a linear model of variables of which observations correspond to the kernel embeddings of the cause-and-effect distributions in different domains. In this way, we are able to estimate the causal direction by exploiting the causal asymmetry of the transformed linear model. Furthermore, we extend ENCI to causal graph discovery for multiple variables by transforming the relations among them into a linear nongaussian acyclic model. We show that by exploiting the nonstationarity of distributions, both cause-effect pairs and two kinds of causal graphs are identifiable under mild conditions. Experiments on synthetic and real-world data are conducted to justify the efficacy of ENCI over major existing methods.

Causal discovery via the asymmetry between the cause and the effect has proved to be a promising way to infer the causal direction from observations. The basic idea is to assume that the mechanism generating the cause distribution p ( x ) and that generating the conditional distribution p ( y | x ) correspond to two independent natural processes and thus p ( x ) and p ( y | x ) fulfill some sort of independence condition. However, in many situations, the independence condition does not hold for the anticausal direction; if we consider p ( x , y ) as generated via p ( y ) p ( x | y ), then there are usually some contrived mutual adjustments between p ( y ) and p ( x | y ). This kind of asymmetry can be exploited to identify the causal direction. Based on this postulate, in this letter, we define an uncorrelatedness criterion between p ( x ) and p ( y | x ) and, based on this uncorrelatedness, show asymmetry between the cause and the effect in terms that a certain complexity metric on p ( x ) and p ( y | x ) is less than the complexity metric on p ( y ) and p ( x | y ). We propose a Hilbert space embedding-based method EMD (an abbreviation for EMbeDding) to calculate the complexity metric and show that this method preserves the relative magnitude of the complexity metric. Based on the complexity metric, we propose an efficient kernel-based algorithm for causal discovery. The contribution of this letter is threefold. It allows a general transformation from the cause to the effect involving the noise effect and is applicable to both one-dimensional and high-dimensional data. Furthermore it can be used to infer the causal ordering for multiple variables. Extensive experiments on simulated and real-world data are conducted to show the effectiveness of the proposed method.

LiNGAM has been successfully applied to some real-world causal discovery problems. Nevertheless, causal sufficiency is assumed; that is, there is no latent confounder of the observations, which may be unrealistic for real-world problems. Taking into the consideration latent confounders will improve the reliability and accuracy of estimations of the real causal structures. In this letter, we investigate a model called linear nongaussian acyclic models in the presence of latent gaussian confounders (LiNGAM-GC) which can be seen as a specific case of lvLiNGAM. This model includes the latent confounders, which are assumed to be independent gaussian distributed and statistically independent of the disturbances. To tackle the causal discovery problem of this model, first we propose a pairwise cumulant-based measure of causal directions for cause-effect pairs. We prove that in spite of the presence of latent gaussian confounders, the causal direction of the observed cause-effect pair can be identified under the mild condition that the disturbances are simultaneously supergaussian or subgaussian. We propose a simple and efficient method to detect the violation of this condition. We extend our work to multivariate causal network discovery problems. Specifically we propose algorithms to estimate the causal network structure, including causal ordering and causal strengths, using an iterative root finding-removing scheme based on pairwise measure. To address the redundant edge problem due to the finite sample size effect, we develop an efficient bootstrapping-based pruning algorithm. Experiments on synthetic data and real-world data have been conducted to show the applicability of our model and the effectiveness of our proposed algorithms.