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Representing the conditional independences present in a multivariate randomvector via graphs has found widespread use in applications, and suchrepresentations are popularly known as graphical models or Markov randomfields. These models have many useful properties, but their fundamentalattractive feature is their ability to reflect conditional independencesbetween blocks of variables through graph separation, a consequence of theequivalence of the pairwise, local and global Markov properties demonstrated byPearl and Paz (1985). Modern day applications often necessitate working witheither an infinite collection of variables (such as in a spatial-temporalfield) or approximating a large high-dimensional finite stochastic system withan infinite-dimensional system. However, it is unclear whether the conditionalindependences present in an infinite-dimensional random vector or stochasticprocess can still be represented by separation criteria in an infinite graph.In light of the advantages of using graphs as tools to represent stochasticrelationships, we undertake in this paper a general study of infinite graphicalmodels. First, we demonstrate that naive extensions of the assumptions requiredfor the finite case results do not yield equivalence of the Markov propertiesin the infinite-dimensional setting, thus calling for a more in-depth analysis.To this end, we proceed to derive general conditions which do allowrepresenting the conditional independence in an infinite-dimensional randomsystem by means of graphs, and our results render the result of Pearl and Pazas a special case of a more general phenomenon. We conclude by demonstratingthe applicability of our theory through concrete examples ofinfinite-dimensional graphical models.

Graphical Markov models for infinitely many variables