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 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.