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Identifiability of Gaussian structural equation models with equal error variances

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  • J. Peters
  • P. Bühlmann

Abstract

We consider structural equation models in which variables can be written as a function of their parents and noise terms, which are assumed to be jointly independent. Corresponding to each structural equation model is a directed acyclic graph describing the relationships between the variables. In Gaussian structural equation models with linear functions, the graph can be identified from the joint distribution only up to Markov equivalence classes, assuming faithfulness. In this work, we prove full identifiability in the case where all noise variables have the same variance: the directed acyclic graph can be recovered from the joint Gaussian distribution. Our result has direct implications for causal inference: if the data follow a Gaussian structural equation model with equal error variances, then, assuming that all variables are observed, the causal structure can be inferred from observational data only. We propose a statistical method and an algorithm based on our theoretical findings.

Suggested Citation

  • J. Peters & P. Bühlmann, 2014. "Identifiability of Gaussian structural equation models with equal error variances," Biometrika, Biometrika Trust, vol. 101(1), pages 219-228.
  • Handle: RePEc:oup:biomet:v:101:y:2014:i:1:p:219-228.
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    File URL: http://hdl.handle.net/10.1093/biomet/ast043
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    Cited by:

    1. Castelletti, Federico & Peluso, Stefano, 2021. "Equivalence class selection of categorical graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 164(C).
    2. Xiao Guo & Hai Zhang, 2020. "Sparse directed acyclic graphs incorporating the covariates," Statistical Papers, Springer, vol. 61(5), pages 2119-2148, October.
    3. Federico Castelletti & Guido Consonni, 2020. "Discovering causal structures in Bayesian Gaussian directed acyclic graph models," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 183(4), pages 1727-1745, October.
    4. Park, Gunwoong & Kim, Yesool, 2021. "Learning high-dimensional Gaussian linear structural equation models with heterogeneous error variances," Computational Statistics & Data Analysis, Elsevier, vol. 154(C).
    5. Shi, Chengchun & Li, Lexin, 2022. "Testing mediation effects using logic of Boolean matrices," LSE Research Online Documents on Economics 108881, London School of Economics and Political Science, LSE Library.
    6. Federico Castelletti, 2020. "Bayesian Model Selection of Gaussian Directed Acyclic Graph Structures," International Statistical Review, International Statistical Institute, vol. 88(3), pages 752-775, December.
    7. Wang, Bingling & Zhou, Qing, 2021. "Causal network learning with non-invertible functional relationships," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).
    8. Fangting Zhou & Kejun He & Yang Ni, 2023. "Individualized causal discovery with latent trajectory embedded Bayesian networks," Biometrics, The International Biometric Society, vol. 79(4), pages 3191-3202, December.
    9. Li, Lexin & Shi, Chengchun & Guo, Tengfei & Jagust, William J., 2022. "Sequential pathway inference for multimodal neuroimaging analysis," LSE Research Online Documents on Economics 111904, London School of Economics and Political Science, LSE Library.
    10. Nikolaos Petrakis & Stefano Peluso & Dimitris Fouskakis & Guido Consonni, 2020. "Objective methods for graphical structural learning," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 74(3), pages 420-438, August.
    11. Jonas Peters & Peter Bühlmann & Nicolai Meinshausen, 2016. "Causal inference by using invariant prediction: identification and confidence intervals," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(5), pages 947-1012, November.
    12. Choi, Semin & Kim, Yesool & Park, Gunwoong, 2023. "Densely connected sub-Gaussian linear structural equation model learning via ℓ1- and ℓ2-regularized regressions," Computational Statistics & Data Analysis, Elsevier, vol. 181(C).
    13. Federico Castelletti & Guido Consonni, 2021. "Bayesian inference of causal effects from observational data in Gaussian graphical models," Biometrics, The International Biometric Society, vol. 77(1), pages 136-149, March.
    14. Fangting Zhou & Kejun He & Kunbo Wang & Yanxun Xu & Yang Ni, 2023. "Functional Bayesian networks for discovering causality from multivariate functional data," Biometrics, The International Biometric Society, vol. 79(4), pages 3279-3293, December.
    15. C Schultheiss & P Bühlmann, 2023. "Ancestor regression in linear structural equation models," Biometrika, Biometrika Trust, vol. 110(4), pages 1117-1124.

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