Random forward models and log-likelihoods in Bayesian inverse problems

TL;DR

This paper investigates how randomised forward models and log-likelihoods affect Bayesian inverse problems, providing bounds on the divergence between true and approximate posteriors and demonstrating applications in large data and probabilistic ODE solutions.

Contribution

It introduces stability bounds for Bayesian posteriors when using stochastic surrogates for forward models and log-likelihoods, applicable to large-scale and probabilistic numerical problems.

Findings

Hellinger distance is bounded by moments of log-likelihood differences.

Stability results apply to randomized misfit models in large data contexts.

Applicable to probabilistic solutions of ordinary differential equations.

Abstract

We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of these stability results are given for randomised misfit models in large data applications and the probabilistic solution of ordinary differential equations.