Testing whether linear equations are causal: A free probability theory approach
This work addresses the challenge of causal inference in high-dimensional settings for researchers in statistics and machine learning, representing an incremental extension of existing methods.
The paper tackles the problem of inferring causal directions between two high-dimensional variables X and Y using linear equations, extending the Trace Method to handle cases where dimensions exceed sample sizes. It presents a statistical test that rejects both causal directions in the presence of a common cause, showing promising results on simulated and real-world data.
We propose a method that infers whether linear relations between two high-dimensional variables X and Y are due to a causal influence from X to Y or from Y to X. The earlier proposed so-called Trace Method is extended to the regime where the dimension of the observed variables exceeds the sample size. Based on previous work, we postulate conditions that characterize a causal relation between X and Y. Moreover, we describe a statistical test and argue that both causal directions are typically rejected if there is a common cause. A full theoretical analysis is presented for the deterministic case but our approach seems to be valid for the noisy case, too, for which we additionally present an approach based on a sparsity constraint. The discussed method yields promising results for both simulated and real world data.