Prevention Science

, Volume 20, Issue 3, pp 431–441 | Cite as

Non-Gaussian Methods for Causal Structure Learning

  • Shohei ShimizuEmail author


Causal structure learning is one of the most exciting new topics in the fields of machine learning and statistics. In many empirical sciences including prevention science, the causal mechanisms underlying various phenomena need to be studied. Nevertheless, in many cases, classical methods for causal structure learning are not capable of estimating the causal structure of variables. This is because it explicitly or implicitly assumes Gaussianity of data and typically utilizes only the covariance structure. In many applications, however, non-Gaussian data are often obtained, which means that more information may be contained in the data distribution than the covariance matrix is capable of containing. Thus, many new methods have recently been proposed for using the non-Gaussian structure of data and inferring the causal structure of variables. This paper introduces prevention scientists to such causal structure learning methods, particularly those based on the linear, non-Gaussian, acyclic model known as LiNGAM. These non-Gaussian data analysis tools can fully estimate the underlying causal structures of variables under assumptions even in the presence of unobserved common causes. This feature is in contrast to other approaches. A simulated example is also provided.


Causal structure discovery Observational data Non-Gaussianity Structural causal models 



The author thanks the guest editor Wolfgang Wiedermann and two reviewers for their helpful comments.

Funding Information

This work was supported by JSPS KAKENHI Grant Number 16K00045.

Compliance with Ethical Standards

Conflict of Interest

The author declares that there is no conflict of interest.

Ethical Approval

This article does not contain any studies with human participants or animals performed by the author.

Informed Consent

Informed consent was not required for this study.

Supplementary material

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  1. Bach, F.R., & Jordan, M.I. (2002). Kernel independent component analysis. Journal of Machine Learning Research, 3, 1–48.Google Scholar
  2. Billingsley, P. (1986). Probability and measure. New York: Wiley-Interscience.Google Scholar
  3. Bollen, K. (1989). Structural equations with latent variables. New York: Wiley.CrossRefGoogle Scholar
  4. Darmois, G. (1953). Analyse générale des liaisons stochastiques. Review of the International Statistical Institute, 21, 2–8.CrossRefGoogle Scholar
  5. Demidenko, E. (2004). Mixed models: Theory and applications. New York: Wiley-Interscience.CrossRefGoogle Scholar
  6. Gretton, A., Bousquet, O., Smola, A.J., Schölkopf, B. (2005). Measuring statistical dependence with Hilbert-Schmidt norms. In Proceedings of 16th international conference on algorithmic learning theory (ALT2005) (pp. 63–77).Google Scholar
  7. Hoyer, P.O., Shimizu, S., Kerminen, A., Palviainen, M. (2008). Estimation of causal effects using linear non-Gaussian causal models with hidden variables. International Journal of Approximate Reasoning, 49, 362–378.CrossRefGoogle Scholar
  8. Hoyer, P.O., Janzing, D., Mooij, J., Peters, J., Schölkopf, B. (2009). Nonlinear causal discovery with additive noise models. Advances in Neural Information Processing Systems, 21, 689–696.Google Scholar
  9. Hyvärinen, A., Karhunen, J., Oja, E. (2001). Independent component analysis. New York: Wiley.CrossRefGoogle Scholar
  10. Hyvärinen, A., Zhang, K., Shimizu, S., Hoyer, P. (2010). Estimation of a structural vector autoregression model using non-Gaussianity. Journal of Machine Learning Research, 11, 1709–1731.Google Scholar
  11. Imbens, G.W., & Rubin, D.B. (2015). Causal inference in statistics, social, and biomedical sciences. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  12. Kass, R.E., & Raftery, A.E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795.CrossRefGoogle Scholar
  13. Kraskov, A., Stögbauer, H., Grassberger, P. (2004). Estimating mutual information. Physical Review E, 69, 066138.CrossRefGoogle Scholar
  14. Lacerda, G., Spirtes, P., Ramsey, J., Hoyer, P.O. (2008). Discovering cyclic causal models by independent components analysis. In Proceedings of the 24th conference on uncertainty in artificial intelligence (UAI2008) (pp. 366–374).Google Scholar
  15. Mills-Finnerty, C., Hanson, C., Hanson, S.J. (2014). Brain network response underlying decisions about abstract reinforcers. NeuroImage, 103, 48–54.CrossRefGoogle Scholar
  16. Moneta, A., Entner, D., Hoyer, P., Coad, A. (2013). Causal inference by independent component analysis: theory and applications. Oxford Bulletin of Economics and Statistics, 75, 705–730.CrossRefGoogle Scholar
  17. Pearl, J. (2000). Causality: models, reasoning, and inference. Cambridge: Cambridge University Press.Google Scholar
  18. Pearl, J., & Verma, T. (1991). A theory of inferred causation. In Allen, J., Fikes, R., Sandewall, E. (Eds.) Proceedings of the 2nd international conference on principles of knowledge representation and reasoning (pp. 441–452). San Mateo: Morgan Kaufmann.Google Scholar
  19. Raitakari, O.T., Juonala, M., Rönnemaa, T., Keltikangas-Järvinen, L., Räsänen, L., Pietikäinen, M., Hutri-Kähönen, N., Taittonen, L., Jokinen, E., Marniemi, J., et al. (2008). Cohort profile: The cardiovascular risk in young finns study. International Journal of Epidemiology, 37, 1220–1226.CrossRefGoogle Scholar
  20. Rosenström, T., Jokela, M., Puttonen, S., Hintsanen, M., Pulkki-Råback, L., Viikari, J.S., Raitakari, O.T., Keltikangas-Järvinen, L. (2012). Pairwise measures of causal direction in the epidemiology of sleep problems and depression. PLoS ONE, 7, e50841.CrossRefGoogle Scholar
  21. Rubin, D.B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology, 66, 688–701.CrossRefGoogle Scholar
  22. Shimizu, S. (2014). LiNGAM: Non-gaussian methods for estimating causal structures. Behaviormetrika, 41, 65–98.CrossRefGoogle Scholar
  23. Shimizu, S., & Bollen, K. (2014). Bayesian estimation of causal direction in acyclic structural equation models with individual-specific confounder variables and non-Gaussian distributions. Journal of Machine Learning Research, 15, 2629–2652.Google Scholar
  24. Shimizu, S., Hoyer, P.O., Hyvärinen, A., Kerminen, A. (2006). A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7, 2003–2030.Google Scholar
  25. Shimizu, S., Inazumi, T., Sogawa, Y., Hyvärinen, A., Kawahara, Y., Washio, T., Hoyer, P.O., Bollen, K. (2011). DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model. Journal of Machine Learning Research, 12, 1225–1248.Google Scholar
  26. Skitovitch, W.P. (1953). On a property of the normal distribution. Doklady Akademii Nauk SSSR, 89, 217–219.Google Scholar
  27. Spirtes, P., & Zhang, K. (2016). Causal discovery and inference: concepts and recent methodological advances. Applied Informatics, 3.
  28. Spirtes, P., Glymour, C., Scheines, R. (1993). Causation, prediction, and search. Berlin: Springer. (2nd edn. MIT Press 2000).CrossRefGoogle Scholar
  29. Zhang, K., & Chan, L. (2008). Minimal nonlinear distortion principle for nonlinear independent component analysis. Journal of Machine Learning Research, 9, 2455–2487.Google Scholar
  30. Zhang, K., & Hyvärinen, A. (2009). On the identifiability of the post-nonlinear causal model. In Proceedings of the 25th conference on uncertainty in artificial intelligence (UAI2009) (pp. 647–655).Google Scholar
  31. Zhang, K., & Hyvärinen, A. (2016). Nonlinear functional causal models for distinguishing causes form effect. In Wiedermann, W., & von Eye, A. (Eds.) Statistics and causality: methods for applied empirical research. Wiley.Google Scholar

Copyright information

© Society for Prevention Research 2018

Authors and Affiliations

  1. 1.Faculty of Data ScienceShiga UniversityHikoneJapan
  2. 2.The RIKEN Center for Advanced Intelligence ProjectTokyoJapan

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