Continuously indexed graphical models

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2025 · Volume 87 · Issue 1

Abstract

Let X = { X u } u ∈ U be a real-valued Gaussian process indexed by a set U ⁠ . We show that X can be viewed as a graphical model with an uncountably infinite graph, where each X u is a vertex. This graph is characterized by the reproducing property of X ’ s covariance kernel, without restricting U to be finite or countable, allowing the modelling of stochastic processes in continuous time/space. Unlike traditional methods, this characterization is not based on zero entries of an inverse covariance, posing challenges for structure estimation. We propose a plug-in methodology that targets graph recovery up to a finite resolution and shows consistency for graphs which are sufficiently regular and that can be applied to virtually any measurement regime. Furthermore, we derive convergence rates and finite-sample guarantees for the method, and demonstrate its performance through a simulation study and two data analyses.

Keywords

graphical models, Gaussian processes, reproducing kernels, conditional independence

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