Paper:
Causal Inference via Kernel Deviance Measures
Discovering the causal structure among a set of variables is a fundamental problem in many areas of science. In this paper, we propose Kernel Conditional Deviance for Causal Inference (KCDC) a fully nonparametric causal discovery method based on purely observational data. From a novel interpretation of the notion of asymmetry between cause and effect, we derive a corresponding asymmetry measure using the framework of reproducing kernel Hilbert spaces. Based on this, we propose three decision rules for causal discovery. We demonstrate the wide applicability and robustness of our method across a range of diverse synthetic datasets. Furthermore, we test our method on real-world time series data and the real-world benchmark dataset Tübingen Cause-Effect Pairs where we outperform state-of-the-art approaches.

Review:
This paper proposes a novel approach to causal discovery based on kernel conditional deviance methods. I found the paper very well structured and well written. After setting out the problem and giving a nice visual motivation for their approach, the main contributions are clearly described, and the related work is very nicely summarized in the next section. In the description of the proposed approach, the notation was well introduced, the approach is quite intuitive and accessible through the provided description, and I didn’t find any problems with the mathematical description and derivations. Finally, the set-up of the simulation study was well described so that I would expect to be able to reproduce the results, and the results are quite impressive in that the proposed approach performs well in comparison to existing approaches.  A few minor comments:  - For me, the motivating example in the introduction would have been easier accessible with more details which values of x and y, respectively were used in generating the plots.  - In the simulation experiments in Table 3 in the supplements, it is nice that their proposed method achieves perfect performance in almost all scenarios, but that makes me wonder when it does not achieve perfect performance and how it compares in these scenarios to the alternative methods. I.e. I would be interested in more simulations with larger noise. Furthermore, I would be interested in seeing how KCDC performs for linear or quadratic association between X and Y.  - There were a couple of minor typos which could be removed from one more proofread.