# Introduction To Gaussian Process Regression In Bayesian Inverse Problems, With New ResultsOn Experimental Design For Weighted Error Measures

arXiv: https://arxiv.org/abs/2302.04518 · HTML: https://mlsanity.com/paper/2302.04518/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 50 references that are in the map.
- First posted on arXiv 9 Feb 2023.

## Builds on (most cited first, 9 of 9)

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Convergence Guarantees for Gaussian Process Means With Misspecified Likelihoods and Smoothness](https://mlsanity.com/paper/2001.10818/) (2020)
- [Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems](https://mlsanity.com/paper/1701.04006/) (2017)
- [On the well-posedness of Bayesian inverse problems](https://mlsanity.com/paper/1902.10257/) (2019)
- [Parallel Gaussian process surrogate Bayesian inference with noisy likelihood evaluations](https://mlsanity.com/paper/1905.01252/) (2019)
- [Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations](https://mlsanity.com/paper/1402.1694/) (2014)
- [On the Sampling Problem for Kernel Quadrature](https://mlsanity.com/paper/1706.03369/) (2017)
- [Iterative Updating of Model Error for Bayesian Inversion](https://mlsanity.com/paper/1707.04246/) (2017)
- [Random forward models and log-likelihoods in Bayesian inverse problems](https://mlsanity.com/paper/1712.05717/) (2017)

## Built on by (most cited first, 6 of 6)

- [Gaussian processes for Bayesian inverse problems associated with linear partial differential equations](https://mlsanity.com/paper/2307.08343/) (2023)
- [Enhancing Gaussian Process Surrogates for Optimization and Posterior Approximation via Random Exploration](https://mlsanity.com/paper/2401.17037/) (2024)
- [Bayesian Optimization with Inexact Acquisition: Is Random Grid Search Sufficient?](https://mlsanity.com/paper/2506.11831/) (2025)
- [Efficient optimization of expensive black-box simulators via marginal means, with application to neutrino detector design](https://mlsanity.com/paper/2508.01834/) (2025)
- [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)

Bayesian posterior distributions arising in modern applications, including inverse problems in partial differential equation models in tomography and subsurface flow, are often computationally intractable due to the large computational cost of evaluating the data likelihood. To alleviate this problem, we consider using Gaussian process regression to build a surrogate model for the likelihood, resulting in an approximate posterior distribution that is amenable to computations in practice. This work serves as an introduction to Gaussian process regression, in particular in the context of building surrogate models for inverse problems, and presents new insights into a suitable choice of training points. We show that the error between the true and approximate posterior distribution can be bounded by the error between the true and approximate likelihood, measured in the L2-norm weighted by the true posterior, and that efficiently bounding the error between the true and approximate likelihood in this norm suggests choosing the training points in the Gaussian process surrogate model based on the true posterior.