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Transactions of the American Mathematical Society

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Graphical Markov models for infinitely many variables


Authors: David Montague and Bala Rajaratnam
Journal: Trans. Amer. Math. Soc. 370 (2018), 7557-7603
MSC (2010): Primary 60G05, 60G15, 60G60, 60K35
DOI: https://doi.org/10.1090/tran/7048
Published electronically: June 7, 2018
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Abstract: Representing the conditional independences present in a multivariate random vector via graphs has found widespread use in applications, and such representations are popularly known as graphical models or Markov random fields. These models have many useful properties, but their fundamental attractive feature is their ability to reflect conditional independences between blocks of variables through graph separation, a consequence of the equivalence of the pairwise, local, and global Markov properties demonstrated by Pearl and Paz (1985). Modern-day applications often necessitate working with either an infinite collection of variables (such as in a spatial-temporal field) or approximating a large high-dimensional finite stochastic system with an infinite-dimensional system. However, it is unclear whether the conditional independences present in an infinite-dimensional random vector or stochastic process can still be represented by separation criteria in an infinite graph. In light of the advantages of using graphs as tools to represent stochastic relationships, we undertake in this paper a general study of infinite graphical models. First, we demonstrate that naïve extensions of the assumptions required for the finite case results do not yield equivalence of the Markov properties in the infinite-dimensional setting, thus calling for a more in-depth analysis. To this end, we proceed to derive general conditions which do allow representing the conditional independence in an infinite-dimensional random system by means of graphs, and our results render the result of Pearl and Paz as a special case of a more general phenomenon. We conclude by demonstrating the applicability of our theory through concrete examples of infinite-dimensional graphical models.


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Additional Information

David Montague
Affiliation: Department of Statistics, University of California, Davis, California 95616
Email: davmont@gmail.com

Bala Rajaratnam
Affiliation: Department of Statistics, University of California, Davis, California 95616
Email: brajaratnam01@gmail.com

DOI: https://doi.org/10.1090/tran/7048
Keywords: Markov random fields, global Markov property, graph separation, general intersection property, graphical Gaussian processes, graphical discrete processes.
Received by editor(s): February 4, 2015
Received by editor(s) in revised form: August 14, 2016
Published electronically: June 7, 2018
Additional Notes: The authors were supported in part by the US NSF under grants DMS-CMG-1025465, AGS-1003823, DMS-1106642, and DMS-CAREER-1352656, and by the US Air Force Office of Scientific Research grant award FA9550-13-1-0043
Article copyright: © Copyright 2018 American Mathematical Society

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