Title: Continuum Limit of Posteriors in Graph Bayesian Inverse Problems,
Abstract: 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 $M_n=\{x_1, \dots, x_n\} \subset M$, 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 $M_n$ 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.