Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation
<|reference_start|>Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation: 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.<|reference_end|>
@article{chen2024gaussian,
    title={Gaussian Measures Conditioned on Nonlinear Observations: Consistency,
  MAP Estimators, and Simulation},
    author={Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M Stuart},
    journal={arXiv preprint arXiv:2405.13149},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.13149},
    primaryClass={stat.ML cs.LG cs.NA math.NA math.PR stat.CO}
}