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

arXiv: https://arxiv.org/abs/2405.13149 · HTML: https://mlsanity.com/paper/2405.13149/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 96 references that are in the map.
- First posted on arXiv 21 May 2024.

## Builds on (most cited first, 29 of 29)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [GP-VAE: Deep Probabilistic Time Series Imputation](https://mlsanity.com/paper/1907.04155/) (2019)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [Gaussian Process Prior Variational Autoencoders](https://mlsanity.com/paper/1810.11738/) (2018)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Sparse Cholesky factorization by Kullback-Leibler minimization](https://mlsanity.com/paper/2004.14455/) (2020)
- [Inferring solutions of differential equations using noisy multi-fidelity data](https://mlsanity.com/paper/1607.04805/) (2016)
- [Dimension-independent likelihood-informed MCMC](https://mlsanity.com/paper/1411.3688/) (2014)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Computational Graph Completion](https://mlsanity.com/paper/2110.10323/) (2021)
- [Deep Neural Networks as Point Estimates for Deep Gaussian Processes](https://mlsanity.com/paper/2105.04504/) (2021)
- [Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity](https://mlsanity.com/paper/1706.02205/) (2017)
- [Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems](https://mlsanity.com/paper/1701.04006/) (2017)
- [On the well-posedness of Bayesian inverse problems](https://mlsanity.com/paper/1902.10257/) (2019)
- [AutoIP: A United Framework to Integrate Physics into Gaussian Processes](https://mlsanity.com/paper/2202.12316/) (2022)
- [Fully probabilistic deep models for forward and inverse problems in parametric PDEs](https://mlsanity.com/paper/2208.04856/) (2022)
- [Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimation](https://mlsanity.com/paper/2005.11375/) (2020)
- [Do ideas have shape? Idea registration as the continuous limit of artificial neural networks](https://mlsanity.com/paper/2008.03920/) (2020)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [Deep Gaussian Processes: A Survey](https://mlsanity.com/paper/2106.12135/) (2021)
- [Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dots](https://mlsanity.com/paper/2311.17007/) (2023)
- [Diffeomorphic Measure Matching with Kernels for Generative Modeling](https://mlsanity.com/paper/2402.08077/) (2024)
- [Generalized modes in Bayesian inverse problems](https://mlsanity.com/paper/1806.00519/) (2018)
- [An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems](https://mlsanity.com/paper/2209.11517/) (2022)

## Built on by (most cited first, 4 of 4)

- [Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processes](https://mlsanity.com/paper/2510.05568/) (2025)
- [Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields](https://mlsanity.com/paper/2503.08343/) (2025)
- [Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs](https://mlsanity.com/paper/2506.02337/) (2025)
- [Approximating Measures on Function Spaces: Transport and Truncation](https://mlsanity.com/paper/2609.17802/) (2026)

## 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.