Faster algorithms for Markov equivalence

Zhongyi Hu, Robin Evans
Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI), PMLR 124:739-748, 2020.

Abstract

Maximal ancestral graphs (MAGs) have many desirable properties; in particular they can fully describe conditional independences from directed acyclic graphs (DAGs) in the presence of latent and selection variables. However, different MAGs may encode the same conditional independences, and are said to be \emph{Markov equivalent}. Thus identifying necessary and sufficient conditions for equivalence is essential for structure learning. Several criteria for this already exist, but in this paper we give a new non-parametric characterization in terms of the heads and tails that arise in the parameterization for discrete models. We also provide a polynomial time algorithm ($O(ne^{2})$, where $n$ and $e$ are the number of vertices and edges respectively) to verify equivalence. Moreover, we extend our criterion to ADMGs and summary graphs and propose an algorithm that converts an ADMG or summary graph to an equivalent MAG in polynomial time ($O(n^{2}e)$). Hence by combining both algorithms, we can also verify equivalence between two summary graphs or ADMGs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v124-hu20a, title = {Faster algorithms for Markov equivalence}, author = {Hu, Zhongyi and Evans, Robin}, booktitle = {Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI)}, pages = {739--748}, year = {2020}, editor = {Peters, Jonas and Sontag, David}, volume = {124}, series = {Proceedings of Machine Learning Research}, month = {03--06 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v124/hu20a/hu20a.pdf}, url = {https://proceedings.mlr.press/v124/hu20a.html}, abstract = {Maximal ancestral graphs (MAGs) have many desirable properties; in particular they can fully describe conditional independences from directed acyclic graphs (DAGs) in the presence of latent and selection variables. However, different MAGs may encode the same conditional independences, and are said to be \emph{Markov equivalent}. Thus identifying necessary and sufficient conditions for equivalence is essential for structure learning. Several criteria for this already exist, but in this paper we give a new non-parametric characterization in terms of the heads and tails that arise in the parameterization for discrete models. We also provide a polynomial time algorithm ($O(ne^{2})$, where $n$ and $e$ are the number of vertices and edges respectively) to verify equivalence. Moreover, we extend our criterion to ADMGs and summary graphs and propose an algorithm that converts an ADMG or summary graph to an equivalent MAG in polynomial time ($O(n^{2}e)$). Hence by combining both algorithms, we can also verify equivalence between two summary graphs or ADMGs.} }
Endnote
%0 Conference Paper %T Faster algorithms for Markov equivalence %A Zhongyi Hu %A Robin Evans %B Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI) %C Proceedings of Machine Learning Research %D 2020 %E Jonas Peters %E David Sontag %F pmlr-v124-hu20a %I PMLR %P 739--748 %U https://proceedings.mlr.press/v124/hu20a.html %V 124 %X Maximal ancestral graphs (MAGs) have many desirable properties; in particular they can fully describe conditional independences from directed acyclic graphs (DAGs) in the presence of latent and selection variables. However, different MAGs may encode the same conditional independences, and are said to be \emph{Markov equivalent}. Thus identifying necessary and sufficient conditions for equivalence is essential for structure learning. Several criteria for this already exist, but in this paper we give a new non-parametric characterization in terms of the heads and tails that arise in the parameterization for discrete models. We also provide a polynomial time algorithm ($O(ne^{2})$, where $n$ and $e$ are the number of vertices and edges respectively) to verify equivalence. Moreover, we extend our criterion to ADMGs and summary graphs and propose an algorithm that converts an ADMG or summary graph to an equivalent MAG in polynomial time ($O(n^{2}e)$). Hence by combining both algorithms, we can also verify equivalence between two summary graphs or ADMGs.
APA
Hu, Z. & Evans, R.. (2020). Faster algorithms for Markov equivalence. Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI), in Proceedings of Machine Learning Research 124:739-748 Available from https://proceedings.mlr.press/v124/hu20a.html.

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