Posterior Consistency for Gaussian Process Approximations of Bayesian Posterior Distributions

TL;DR

This paper investigates the use of Gaussian process emulators to approximate Bayesian posterior distributions, providing theoretical error bounds and numerical validation for the accuracy of these approximations.

Contribution

It establishes rigorous error bounds on the Hellinger distance between true and approximate posteriors using Gaussian process emulators in Bayesian inverse problems.

Findings

Error bounds depend on emulator moments

Both mean-based and full Gaussian process approximations analyzed

Numerical results validate theoretical error bounds

Abstract

We study the use of Gaussian process emulators to approximate the parameter-to-observation map or the negative log-likelihood in Bayesian inverse problems. We prove error bounds on the Hellinger distance between the true posterior distribution and various approximations based on the Gaussian process emulator. Our analysis includes approximations based on the mean of the predictive process, as well as approximations based on the full Gaussian process emulator. Our results show that the Hellinger distance between the true posterior and its approximations can be bounded by moments of the error in the emulator. Numerical results confirm our theoretical findings.