Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation

TL;DR

This paper studies how to condition Gaussian measures on nonlinear observations, providing theoretical insights, a representer theorem, a new mode concept, and an efficient Laplace approximation for uncertainty quantification.

Contribution

It introduces a representer theorem for conditioned Gaussian measures, a novel mode definition, and a Laplace approximation method for nonlinear observation conditioning.

Findings

Decomposition of conditioned Gaussian into Gaussian and non-Gaussian parts

Definition of a new mode concept for conditional measures

Development of a Laplace approximation for simulation

Abstract

The article presents a systematic study of the problem of conditioning a Gaussian random variable $\xi$ on nonlinear observations of the form $F \circ \phi(\xi)$ where $\phi: \mathcal{X} \to \mathbb{R}^N$ 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 $\xi \mid F\circ \phi(\xi)$, 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.