Estimating Intractable Posterior Distributions through Gaussian Process regression and Metropolis-adjusted Langevin procedure

TL;DR

This paper introduces a sequential surrogate-based Bayesian calibration method combining Gaussian Process Regression and Langevin sampling to efficiently estimate complex posterior distributions with expensive likelihood evaluations.

Contribution

The paper presents a novel sequential surrogate approach that adaptively refines the likelihood approximation and integrates it with Langevin sampling for efficient Bayesian calibration.

Findings

Accelerates convergence in high-dimensional settings

Reduces computational costs of likelihood evaluations

Effective on synthetic and industrial calibration problems

Abstract

Numerical simulations are crucial for modeling complex systems, but calibrating them becomes challenging when data are noisy or incomplete and likelihood evaluations are computationally expensive. Bayesian calibration offers an interesting way to handle uncertainty, yet computing the posterior distribution remains a major challenge under such conditions. To address this, we propose a sequential surrogate-based approach that incrementally improves the approximation of the log-likelihood using Gaussian Process Regression. Starting from limited evaluations, the surrogate and its gradient are refined step by step. At each iteration, new evaluations of the expensive likelihood are added only at informative locations, that is to say where the surrogate is most uncertain and where the potential impact on the posterior is greatest. The surrogate is then coupled with the Metropolis-Adjusted Langevin Algorithm, which uses gradient information to efficiently explore the posterior. This approach accelerates convergence, handles relatively high-dimensional settings, and keeps computational costs low. We demonstrate its effectiveness on both a synthetic benchmark and an industrial application involving the calibration of high-speed train parameters from incomplete sensor data.