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 M. We assume to have access to a discrete domain M-n = {x(1),...,x(n)} subset of M and to noisy measurements of the output solution at p <= n of those points. We introduce a graph-based Bayesian inverse problem and show that the graph-posterior measures over functions in M-n converge, in the large n limit, to a posterior over functions in 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.