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Continuum limits of posteriors in graph Bayesian inverse problems. (English) Zbl 1416.28003

Authors’ abstract: We consider the problem of recovering a function input of a differential equation formulated on an unknown domain \(\mathcal{M}\). We assume to have access to a discrete domain \(\mathcal{M}_n=\{\mathbf{x}_1, \dots, \mathbf{x}_n\} \subset \mathcal{M}\) and to noisy measurements of the output solution at \(p\leq n\) of those points. We introduce a graph-based Bayesian inverse problem and show that the graph-posterior measures over functions in \(\mathcal{M}_n\) converge, in the large \(n\) limit, to a posterior over functions in \(\mathcal{M}\) that solves a Bayesian inverse problem with known domain. The proofs rely on the variational formulation of the Bayesian update and on a new topology for the study of convergence of measures over functions on point clouds to a measure over functions on the continuum. Our framework, techniques, and results may serve to lay the foundations of robust uncertainty quantification of graph-based tasks in machine learning. The ideas are presented in the concrete setting of recovering the initial condition of the heat equation on an unknown manifold.

MSC:

28A33 Spaces of measures, convergence of measures
46N30 Applications of functional analysis in probability theory and statistics
62F15 Bayesian inference
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