Testing whether linear equations are causal: A free probability theory approach

Jakob Zscheischler, Dominik Janzing, Kun Zhang
Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence, PMLR R9:920-927, 2011.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR9-zscheischler11a, title = {Testing whether linear equations are causal: A free probability theory approach}, author = {Zscheischler, Jakob and Janzing, Dominik and Zhang, Kun}, booktitle = {Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence}, pages = {920--927}, year = {2011}, editor = {Cozman, Fabio and Pfeffer, Avi}, volume = {R9}, series = {Proceedings of Machine Learning Research}, month = {14--17 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r9/main/assets/zscheischler11a/zscheischler11a.pdf}, url = {https://proceedings.mlr.press/r9/zscheischler11a.html}, abstract = {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.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Testing whether linear equations are causal: A free probability theory approach %A Jakob Zscheischler %A Dominik Janzing %A Kun Zhang %B Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2011 %E Fabio Cozman %E Avi Pfeffer %F pmlr-vR9-zscheischler11a %I PMLR %P 920--927 %U https://proceedings.mlr.press/r9/zscheischler11a.html %V R9 %X 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. %Z Reissued by PMLR on 04 October 2026.
APA
Zscheischler, J., Janzing, D. & Zhang, K.. (2011). Testing whether linear equations are causal: A free probability theory approach. Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R9:920-927 Available from https://proceedings.mlr.press/r9/zscheischler11a.html. Reissued by PMLR on 04 October 2026.

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