Acyclic Linear SEMs Obey the Nested Markov Property

Ilya Shpitser, Robin Evans, Thomas S. Richardson
Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence, PMLR R16:734-744, 2018.

Abstract

The conditional independence structure in- duced on the observed marginal distribution by a hidden variable directed acyclic graph (DAG) may be represented by a graphical model rep- resented by mixed graphs called maximal an- cestral graphs (MAGs). This model has a num- ber of desirable properties, in particular the set of Gaussian distributions can be parameterized by viewing the graph as a path diagram. Mod- els represented by MAGs have been used for causal discovery [22], and identification theory for causal effects [28]. In addition to ordinary conditional indepen- dence constraints, hidden variable DAGs also induce generalized independence constraints. These constraints form the nested Markov property [20]. We first show that acyclic linear SEMs obey this property. Further we show that a natural parameterization for all Gaussian dis- tributions obeying the nested Markov property arises from a generalization of maximal ances- tral graphs that we call maximal arid graphs (MArG). We show that every nested Markov model can be associated with a MArG; viewed as a path diagram this MArG parametrizes the Gaussian nested Markov model. This leads di- rectly to methods for ML fitting and computing BIC scores for Gaussian nested models.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR16-shpitser18b, title = {Acyclic Linear SEMs Obey the Nested {M}arkov Property}, author = {Shpitser, Ilya and Evans, Robin and Richardson, Thomas S.}, booktitle = {Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence}, pages = {734--744}, year = {2018}, editor = {Globerson, Amir and Silva, Ricardo}, volume = {R16}, series = {Proceedings of Machine Learning Research}, month = {06--10 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r16/main/assets/shpitser18b/shpitser18b.pdf}, url = {https://proceedings.mlr.press/r16/shpitser18b.html}, abstract = {The conditional independence structure in- duced on the observed marginal distribution by a hidden variable directed acyclic graph (DAG) may be represented by a graphical model rep- resented by mixed graphs called maximal an- cestral graphs (MAGs). This model has a num- ber of desirable properties, in particular the set of Gaussian distributions can be parameterized by viewing the graph as a path diagram. Mod- els represented by MAGs have been used for causal discovery [22], and identification theory for causal effects [28]. In addition to ordinary conditional indepen- dence constraints, hidden variable DAGs also induce generalized independence constraints. These constraints form the nested Markov property [20]. We first show that acyclic linear SEMs obey this property. Further we show that a natural parameterization for all Gaussian dis- tributions obeying the nested Markov property arises from a generalization of maximal ances- tral graphs that we call maximal arid graphs (MArG). We show that every nested Markov model can be associated with a MArG; viewed as a path diagram this MArG parametrizes the Gaussian nested Markov model. This leads di- rectly to methods for ML fitting and computing BIC scores for Gaussian nested models.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Acyclic Linear SEMs Obey the Nested Markov Property %A Ilya Shpitser %A Robin Evans %A Thomas S. Richardson %B Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2018 %E Amir Globerson %E Ricardo Silva %F pmlr-vR16-shpitser18b %I PMLR %P 734--744 %U https://proceedings.mlr.press/r16/shpitser18b.html %V R16 %X The conditional independence structure in- duced on the observed marginal distribution by a hidden variable directed acyclic graph (DAG) may be represented by a graphical model rep- resented by mixed graphs called maximal an- cestral graphs (MAGs). This model has a num- ber of desirable properties, in particular the set of Gaussian distributions can be parameterized by viewing the graph as a path diagram. Mod- els represented by MAGs have been used for causal discovery [22], and identification theory for causal effects [28]. In addition to ordinary conditional indepen- dence constraints, hidden variable DAGs also induce generalized independence constraints. These constraints form the nested Markov property [20]. We first show that acyclic linear SEMs obey this property. Further we show that a natural parameterization for all Gaussian dis- tributions obeying the nested Markov property arises from a generalization of maximal ances- tral graphs that we call maximal arid graphs (MArG). We show that every nested Markov model can be associated with a MArG; viewed as a path diagram this MArG parametrizes the Gaussian nested Markov model. This leads di- rectly to methods for ML fitting and computing BIC scores for Gaussian nested models. %Z Reissued by PMLR on 04 October 2026.
APA
Shpitser, I., Evans, R. & Richardson, T.S.. (2018). Acyclic Linear SEMs Obey the Nested Markov Property. Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R16:734-744 Available from https://proceedings.mlr.press/r16/shpitser18b.html. Reissued by PMLR on 04 October 2026.

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