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

• URL: http://arxiv.org/abs/2405.13149v1
• Date: Tue, 21 May 2024 18:38:14 GMT
• Title: Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation
• Authors: Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M Stuart,
• Abstract summary: We give a representer theorem for the conditioned random variable $xi mid Fcirc phi(xi)$. We also introduce a novel notion of the mode of a conditional measure by taking the limit of the natural relaxation of the problem.
• Score: 6.5243065532527975
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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