Causal Inference on Multidimensional Data Using Free Probability Theory

In this paper, we deal with the problem of inferring causal relations for multidimensional data. Based on the postulate that the distribution of the cause and the conditional distribution of the effect given cause are generated independently, we show that the covariance matrix of the mean embedding of the cause in reproducing kernel Hilbert space (RKHS) is free independent with the covariance matrix of the conditional embedding of the effect given cause. This, called freeness condition, induces a cause-effect asymmetry that a designed measurement is 0 in the causal direction but smaller than 0 in the anticausal direction, and it uncovers the causal direction. One important novel aspect of this paper is that we interpret the independence as a freeness condition between covariance matrices of RKHS distribution embeddings, and it has a wide applicability. We show that our freeness condition-based inference method succeeds in scenarios like additive noise cases, where other methods fail, by theoretical analysis and experimental results.