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.

该文章系统研究了在非线性观测条件下，对高斯随机变量进行条件化的问题，提出了一个有关条件随机变量的表示定理，还引入了一种条件度量的模式概念，最后介绍了一种Laplace近似用于高效模拟条件高斯随机变量来进行不确定性量化。

基于非线性观测的高斯测度：一致性、MAP估计和模拟