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.

翻译：本文系统研究了高斯随机变量 $\xi$ 在 $F \circ \phi(\xi)$ 形式非线性观测下的条件化问题，其中 $\phi: \mathcal{X} \to \mathbb{R}^N$ 为有界线性算子，$F$ 为非线性映射。此类问题常见于贝叶斯推断及近期受机器学习启发的偏微分方程求解器中。我们给出了条件随机变量 $\xi \mid F\circ \phi(\xi)$ 的表示定理，证明其可分解为一个无限维高斯测度（可解析表示）与一个有限维非高斯测度之和。通过取该问题自然松弛形式的极限，我们引入了条件测度众数的新定义，从而可应用现有后验测度最大后验估计的理论框架。最后，针对前述条件化高斯随机变量的高效模拟问题，我们提出了一种改进的拉普拉斯近似方法，以支持不确定性量化分析。