Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation
This work addresses foundational challenges in uncertainty quantification for nonlinear systems, offering incremental theoretical and computational advances.
The paper tackles the problem of conditioning Gaussian random variables on nonlinear observations, common in Bayesian inference and machine learning PDE solvers, by providing a representer theorem for the conditioned variable and introducing a novel mode definition and simulation method.
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.