Continuum Limits of Posteriors in Graph Bayesian Inverse Problems

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 $p\le n$ 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.