2024/05/21 by Yifan Chen, Bamdad Hosseini, Chen, Yifan +5 · 1 citation
Environmental Science · #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #Probability (math.PR) #Soil Geostatistics and Mapping
paper · pdf · doi:10.48550/arxiv.2405.13149
openalex publication_date 2024/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The article presents a systematic study of the problem of conditioning a Gaussian random variable ξ on nonlinear observations of the form F ∘ ϕ(ξ) where ϕ: X → ℝN 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| F∘ 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.