Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation
Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M Stuart
TL;DR
This work studies conditioning a Gaussian measure on nonlinear observations $F(\bm{\phi}(\xi))$, establishing that the posterior $\mu^{\mathbf y}_\beta$ decomposes into a Gaussian infinite-dimensional part and a finite-dimensional non-Gaussian component. It proves convergence of $\mu^{\mathbf y}_\beta$ to a conditional measure $\mu^{\mathbf y}_0$ as the noise vanishes and derives a representer-type finite-dimensional characterization for both the posterior and conditional MAP estimators. The authors develop Laplace and Gauss–Newton-type approximations to efficiently sample the non-Gaussian part, enabling scalable uncertainty quantification, and apply the framework to GP-PDE solvers, producing UQ-guided adaptive collocation strategies. The results offer a principled bridge between Bayesian inverse problems and numerical PDE solvers, highlighting practical algorithms that reduce computation to a finite-dimensional non-Gaussian core plus an analytically tractable Gaussian tail. Overall, the paper provides both rigorous structure for conditioned Gaussian measures and practical tools for UQ in nonlinear settings like GP-PDE and nonlinear elliptic/Burgers-type PDEs.
Abstract
The article presents a systematic study of the problem of conditioning a Gaussian random variable $ξ$ on nonlinear observations of the form $F \circ φ(ξ)$ where $φ: \mathcal{X} \to \mathbb{R}^N$ is a bounded linear operator and $F$ is nonlinear. Such problems arise in the context of Bayesian inference and recent machine learning-inspired PDE solvers. We give a representer theorem for the conditioned random variable $ξ\mid F\circ φ(ξ)$, stating that it decomposes as the sum of an infinite-dimensional Gaussian (which is identified analytically) as well as a finite-dimensional non-Gaussian measure. We also introduce a novel notion of the mode of a conditional measure by taking the limit of the natural relaxation of the problem, to which we can apply the existing notion of maximum a posteriori estimators of posterior measures. Finally, we introduce a variant of the Laplace approximation for the efficient simulation of the aforementioned conditioned Gaussian random variables towards uncertainty quantification.