Solving Bayesian Inverse Problems With Expensive Likelihoods Using Constrained Gaussian Processes and Active Learning

TL;DR

This paper introduces a novel constrained Gaussian process surrogate model with active learning for efficient Bayesian inverse problem solving involving expensive likelihoods, achieving high accuracy with limited data.

Contribution

It develops a constrained Gaussian process approach that incorporates prior knowledge and epistemic uncertainty, improving surrogate accuracy and efficiency in high-dimensional Bayesian inverse problems.

Findings

Fast convergence demonstrated on a benchmark problem

Achieved parameter inference with only about 1000 model evaluations

Successfully calibrated a complex lung model using limited data

Abstract

Solving inverse problems using Bayesian methods can become prohibitively expensive when likelihood evaluations involve complex and large scale numerical models. A common approach to circumvent this issue is to approximate the forward model or the likelihood function with a surrogate model. But also there, due to limited computational resources, only a few training points are available in many practically relevant cases. Thus, it can be advantageous to model the additional uncertainties of the surrogate in order to incorporate the epistemic uncertainty due to limited data. In this paper, we develop a novel approach to approximate the log likelihood by a constrained Gaussian process based on prior knowledge about its boundedness. This improves the accuracy of the surrogate approximation without increasing the number of training samples. Additionally, we introduce a formulation to integrate the epistemic uncertainty due to limited training points into the posterior density approximation. This is combined with a state of the art active learning strategy for selecting training points, which allows to approximate posterior densities in higher dimensions very efficiently. We demonstrate the fast convergence of our approach for a benchmark problem and infer a random field that is discretized by 30 parameters using only about 1000 model evaluations. In a practically relevant example, the parameters of a reduced lung model are calibrated based on flow observations over time and voltage measurements from a coupled electrical impedance tomography simulation.