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Identifiability of latent-variable and structural-equation models: from linear to nonlinear

Author

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  • Aapo Hyvärinen

    (University of Helsinki)

  • Ilyes Khemakhem

    (UCL)

  • Ricardo Monti

    (UCL)

Abstract

An old problem in multivariate statistics is that linear Gaussian models are often unidentifiable. In factor analysis, an orthogonal rotation of the factors is unidentifiable, while in linear regression, the direction of effect cannot be identified. For such linear models, non-Gaussianity of the (latent) variables has been shown to provide identifiability. In the case of factor analysis, this leads to independent component analysis, while in the case of the direction of effect, non-Gaussian versions of structural equation modeling solve the problem. More recently, we have shown how even general nonparametric nonlinear versions of such models can be estimated. Non-Gaussianity is not enough in this case, but assuming we have time series, or that the distributions are suitably modulated by observed auxiliary variables, the models are identifiable. This paper reviews the identifiability theory for the linear and nonlinear cases, considering both factor analytic and structural equation models.

Suggested Citation

  • Aapo Hyvärinen & Ilyes Khemakhem & Ricardo Monti, 2024. "Identifiability of latent-variable and structural-equation models: from linear to nonlinear," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 76(1), pages 1-33, February.
    • Handle: RePEc:spr:aistmt:v:76:y:2024:i:1:d:10.1007_s10463-023-00884-4
    DOI: 10.1007/s10463-023-00884-4
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    References listed on IDEAS

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    1. Daniel D. Lee & H. Sebastian Seung, 1999. "Learning the parts of objects by non-negative matrix factorization," Nature, Nature, vol. 401(6755), pages 788-791, October.
    2. J. Peters & P. Bühlmann, 2014. "Identifiability of Gaussian structural equation models with equal error variances," Biometrika, Biometrika Trust, vol. 101(1), pages 219-228.
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