Sampling Methods for Bayesian Inference Involving Convergent Noisy Approximations of Forward Maps

TL;DR

This paper develops Bayesian methods for inverse problems with noisy, convergent approximations of forward maps, analyzing error propagation and comparing techniques through numerical experiments.

Contribution

It introduces a framework for Bayesian inverse problems with mean-square convergent random forward map approximations, and evaluates their performance.

Findings

Monte Carlo errors propagate predictably in these settings

Simple solution techniques can be effective for such inverse problems

Numerical experiments demonstrate the practical performance of the methods

Abstract

We present Bayesian techniques for solving inverse problems which involve mean-square convergent random approximations of the forward map. Noisy approximations of the forward map arise in several fields, such as multiscale problems and probabilistic numerical methods. In these fields, a random approximation can enhance the quality or the efficiency of the inference procedure, but entails additional theoretical and computational difficulties due to the randomness of the forward map. A standard technique to address this issue is to combine Monte Carlo averaging with Markov chain Monte Carlo samplers, as for example in the pseudo-marginal Metropolis--Hastings methods. In this paper, we consider mean-square convergent random approximations, and quantify how Monte Carlo errors propagate from the forward map to the solution of the inverse problems. Moreover, we review and describe simple techniques to solve such inverse problems, and compare performances with a series of numerical experiments.