Variable Complexity in the Univariate and Multivariate Structural Causal Model

We show that by comparing the individual complexities of univariante cause and effect in the Structural Causal Model, one can identify the cause and the effect, without considering their interaction at all. The entropy of each variable is ineffective in measuring the complexity, and we propose to capture it by an autoencoder that operates on the list of sorted samples. Comparing the reconstruction errors of the two autoencoders, one for each variable, is shown to perform well on the accepted benchmarks of the field. In the multivariate case, where one can ensure that the complexities of the cause and effect are balanced, we propose a new method that mimics the disentangled structure of the causal model. We extend the results of~\cite{Zhang:2009:IPC:1795114.1795190} to the multidimensional case, showing that such modeling is only likely in the direction of causality. Furthermore, the learned model is shown theoretically to perform the separation to the causal component and to the residual (noise) component. Our multidimensional method obtains a significantly higher accuracy than the literature methods.