## Abstract

Causal feature learning (CFL) (Chalupka et al., Proceedings of the Thirty-First Conference on Uncertainty in Artificial Intelligence. AUAI Press, Edinburgh, pp 181–190, 2015) is a causal inference framework rooted in the language of causal graphical models (Pearl J, Reasoning and inference. Cambridge University Press, Cambridge, 2009; Spirtes et al., Causation, Prediction, and Search. Massachusetts Institute of Technology, Massachusetts, 2000), and computational mechanics (Shalizi, PhD thesis, University of Wisconsin at Madison, 2001). CFL is aimed at discovering high-level causal relations from low-level data, and at reducing the experimental effort to understand confounding among the high-level variables. We first review the scientific motivation for CFL, then present a detailed introduction to the framework, laying out the definitions and algorithmic steps. A simple example illustrates the techniques involved in the learning steps and provides visual intuition. Finally, we discuss the limitations of the current framework and list a number of open problems.

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## Notes

- 1.
As discussed in some detail in Chalupka et al. (2016a), a causal interpretation of purely observational data is not possible without further assumptions.

- 2.
Python code that implements the learning algorithms and reproduces all the figures and experimental results is available online at http://vision.caltech.edu/~kchalupk/code.html.

- 3.
It is possible that this \(\gamma \) is not in \(P'[\gamma ;\alpha , \beta ]\). However, it is guaranteed to be in \(P[\gamma ;\alpha , \beta ]\). Since a subset of measure zero in \(P[\gamma ;\alpha , \beta ]\) is also measure zero in \(P'[\gamma ; \alpha , \beta ]\), this does not influence the proof.

- 4.
It is possible that this \(\gamma \) is not in \(P'[\gamma ;\alpha , \beta ]\). However, it is guaranteed to be in \(P[\gamma ;\alpha , \beta ]\). Since a subset of measure zero in \(P[\gamma ;\alpha , \beta ]\) is also measure zero in \(P'[\gamma ; \alpha , \beta ]\), this does not influence the proof.

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## Acknowledgements

We thank an anonymous reviewer for pointing out an error in our original theorem. This work was supported by NSF Award #1564330.

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On behalf of all authors, the corresponding author states that there is no conflict of interest.

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## Additional information

Communicated by Shohei Shimizu.

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Chalupka, K., Eberhardt, F. & Perona, P. Causal feature learning: an overview.
*Behaviormetrika* **44, **137–164 (2017). https://doi.org/10.1007/s41237-016-0008-2

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### Keywords

- Causal discovery
- Causal inference
- Graphical models
- Bayesian networks
- Macrovariables
- Multiscale modeling