mlsanityData to 7 Oct 2026

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

arXiv 2405.13149 ↗First posted 21 May 2024stat.ML cs.LG cs.NAcited by 4

Not on the map. Why?

Abstract (arXiv)

The article presents a systematic study of the problem of conditioning a Gaussian random variable xi on nonlinear observations of the form F circ φ(xi) 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 xi mid Fcirc φ(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.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 29 of its 96 references that are in the map. First posted on arXiv 21 May 2024.

Builds on

29 of 96 references.

References · 29

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17 more
  1. 2014Dimension-independent likelihood-informed MCMCcited by 36
  2. 2023Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processescited by 31
  3. 2021Computational Graph Completioncited by 28
  4. 2021Deep Neural Networks as Point Estimates for Deep Gaussian Processescited by 26
  5. 2017Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexitycited by 24
  6. 2019On the well-posedness of Bayesian inverse problemscited by 20
  7. 2022AutoIP: A United Framework to Integrate Physics into Gaussian Processescited by 20
  8. 2017Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problemscited by 19
  9. 2022Fully probabilistic deep models for forward and inverse problems in parametric PDEscited by 19
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  17. 2022An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problemscited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 3; 2026: 1.

4 papers.

  1. 2025Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processescited by 3
  2. 2025Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fieldscited by 1
  3. 2025Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphscited by 1
  4. 2026Approximating Measures on Function Spaces: Transport and Truncationcited by 0