Calculation of Entailed Rank Constraints in Partially Non-Linear and Cyclic Models

Peter Spirtes
Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence, PMLR R11:658-667, 2013.

Abstract

The Trek Separation Theorem (Sullivant et al. 2010) states necessary and sufficient conditions for a linear directed acyclic graphical model to entail for all possible values of its linear coefficients that the rank of various sub-matrices of the covariance matrix is less than or equal to n, for any given n. In this paper, I extend the Trek Separation Theorem in two ways: I prove that the same necessary and sufficient conditions apply even when the generating model is partially non-linear and contains some cycles. This justifies application of constraint-based causal search algorithms to data generated by a wider class of causal models that may contain non-linear and cyclic relations among the latent variables.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR11-spirtes13a, title = {Calculation of Entailed Rank Constraints in Partially Non-Linear and Cyclic Models}, author = {Spirtes, Peter}, booktitle = {Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence}, pages = {658--667}, year = {2013}, editor = {Nicholson, Ann and Smyth, Padhraic}, volume = {R11}, series = {Proceedings of Machine Learning Research}, month = {12--14 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r11/main/assets/spirtes13a/spirtes13a.pdf}, url = {https://proceedings.mlr.press/r11/spirtes13a.html}, abstract = {The Trek Separation Theorem (Sullivant et al. 2010) states necessary and sufficient conditions for a linear directed acyclic graphical model to entail for all possible values of its linear coefficients that the rank of various sub-matrices of the covariance matrix is less than or equal to n, for any given n. In this paper, I extend the Trek Separation Theorem in two ways: I prove that the same necessary and sufficient conditions apply even when the generating model is partially non-linear and contains some cycles. This justifies application of constraint-based causal search algorithms to data generated by a wider class of causal models that may contain non-linear and cyclic relations among the latent variables.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Calculation of Entailed Rank Constraints in Partially Non-Linear and Cyclic Models %A Peter Spirtes %B Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2013 %E Ann Nicholson %E Padhraic Smyth %F pmlr-vR11-spirtes13a %I PMLR %P 658--667 %U https://proceedings.mlr.press/r11/spirtes13a.html %V R11 %X The Trek Separation Theorem (Sullivant et al. 2010) states necessary and sufficient conditions for a linear directed acyclic graphical model to entail for all possible values of its linear coefficients that the rank of various sub-matrices of the covariance matrix is less than or equal to n, for any given n. In this paper, I extend the Trek Separation Theorem in two ways: I prove that the same necessary and sufficient conditions apply even when the generating model is partially non-linear and contains some cycles. This justifies application of constraint-based causal search algorithms to data generated by a wider class of causal models that may contain non-linear and cyclic relations among the latent variables. %Z Reissued by PMLR on 04 October 2026.
APA
Spirtes, P.. (2013). Calculation of Entailed Rank Constraints in Partially Non-Linear and Cyclic Models. Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R11:658-667 Available from https://proceedings.mlr.press/r11/spirtes13a.html. Reissued by PMLR on 04 October 2026.

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