Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909208134287360 |
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| author | Chen, Yifan Hosseini, Bamdad Owhadi, Houman Stuart, Andrew M |
| author_facet | Chen, Yifan Hosseini, Bamdad Owhadi, Houman Stuart, Andrew M |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13149 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation Chen, Yifan Hosseini, Bamdad Owhadi, Houman Stuart, Andrew M Machine Learning Numerical Analysis Probability Computation 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. |
| title | Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation |
| topic | Machine Learning Numerical Analysis Probability Computation |
| url | https://arxiv.org/abs/2405.13149 |