# Random forward models and log-likelihoods in Bayesian inverse problems

arXiv: https://arxiv.org/abs/1712.05717 · HTML: https://mlsanity.com/paper/1712.05717/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Dec 2017.

## Built on by (most cited first, 4 of 4)

- [Gaussian processes for Bayesian inverse problems associated with linear partial differential equations](https://mlsanity.com/paper/2307.08343/) (2023)
- [Introduction To Gaussian Process Regression In Bayesian Inverse Problems, With New ResultsOn Experimental Design For Weighted Error Measures](https://mlsanity.com/paper/2302.04518/) (2023)
- [Neural network surrogates with uncertainty quantification for inverse problems in partial differential equations](https://mlsanity.com/paper/2606.20417/) (2026)
- [A Convex Approximation Framework for Neural Likelihood-Based Bayesian Inverse Problems](https://mlsanity.com/paper/2607.06252/) (2026)

## Abstract (arXiv)

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.