# Pathwise Conditioning of Gaussian Processes

arXiv: https://arxiv.org/abs/2011.04026 · HTML: https://mlsanity.com/paper/2011.04026/

## Key facts

- Cited by 64 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2020.

## Built on by (most cited first, 30 of 64)

- [Joint Entropy Search for Multi-objective Bayesian Optimization](https://mlsanity.com/paper/2210.02905/) (2022)
- [Bayesian Optimization with High-Dimensional Outputs](https://mlsanity.com/paper/2106.12997/) (2021)
- [Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent](https://mlsanity.com/paper/2306.11589/) (2023)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [Posterior and Computational Uncertainty in Gaussian Processes](https://mlsanity.com/paper/2205.15449/) (2022)
- [The Behavior and Convergence of Local Bayesian Optimization](https://mlsanity.com/paper/2305.15572/) (2023)
- [Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels](https://mlsanity.com/paper/2110.14423/) (2021)
- [Stochastic Gradient Descent for Gaussian Processes Done Right](https://mlsanity.com/paper/2310.20581/) (2023)
- [Sparse Uncertainty Representation in Deep Learning with Inducing Weights](https://mlsanity.com/paper/2105.14594/) (2021)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces](https://mlsanity.com/paper/2301.13088/) (2023)
- [Interactive Segmentation as Gaussian Process Classification](https://mlsanity.com/paper/2302.14578/) (2023)
- [Linearization Turns Neural Operators into Function-Valued Gaussian Processes](https://mlsanity.com/paper/2406.05072/) (2024)
- [Stopping Bayesian Optimization with Probabilistic Regret Bounds](https://mlsanity.com/paper/2402.16811/) (2024)
- [The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs](https://mlsanity.com/paper/2407.08086/) (2024)
- [Uncertainty Estimation for Computed Tomography with a Linearised Deep Image Prior](https://mlsanity.com/paper/2203.00479/) (2022)
- [Scaling Gaussian Processes for Learning Curve Prediction via Latent Kronecker Structure](https://mlsanity.com/paper/2410.09239/) (2024)
- [Exploring Exploration in Bayesian Optimization](https://mlsanity.com/paper/2502.08208/) (2025)
- [Bayesian Experimental Design for Computed Tomography with the Linearised Deep Image Prior](https://mlsanity.com/paper/2207.05714/) (2022)
- [Isotropic Gaussian Processes on Finite Spaces of Graphs](https://mlsanity.com/paper/2211.01689/) (2022)
- [Implicit Gaussian process representation of vector fields over arbitrary latent manifolds](https://mlsanity.com/paper/2309.16746/) (2023)
- [Efficient Exploration in Binary and Preferential Bayesian Optimization](https://mlsanity.com/paper/2110.09361/) (2021)
- [Numerically Stable Sparse Gaussian Processes via Minimum Separation using Cover Trees](https://mlsanity.com/paper/2210.07893/) (2022)
- [Improving Linear System Solvers for Hyperparameter Optimisation in Iterative Gaussian Processes](https://mlsanity.com/paper/2405.18457/) (2024)
- [Optimizing Posterior Samples for Bayesian Optimization via Rootfinding](https://mlsanity.com/paper/2410.22322/) (2024)
- [Scalable Gaussian Processes with Latent Kronecker Structure](https://mlsanity.com/paper/2506.06895/) (2025)
- [Graph Random Features for Scalable Gaussian Processes](https://mlsanity.com/paper/2509.03691/) (2025)
- [BayeSQP: Bayesian Optimization through Sequential Quadratic Programming](https://mlsanity.com/paper/2602.03232/) (2026)
- [Implicit Manifold Gaussian Process Regression](https://mlsanity.com/paper/2310.19390/) (2023)
- [Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanics](https://mlsanity.com/paper/2409.10972/) (2024)

## Abstract (arXiv)

As Gaussian processes are used to answer increasingly complex questions, analytic solutions become scarcer and scarcer. Monte Carlo methods act as a convenient bridge for connecting intractable mathematical expressions with actionable estimates via sampling. Conventional approaches for simulating Gaussian process posteriors view samples as draws from marginal distributions of process values at finite sets of input locations. This distribution-centric characterization leads to generative strategies that scale cubically in the size of the desired random vector. These methods are prohibitively expensive in cases where we would, ideally, like to draw high-dimensional vectors or even continuous sample paths. In this work, we investigate a different line of reasoning: rather than focusing on distributions, we articulate Gaussian conditionals at the level of random variables. We show how this pathwise interpretation of conditioning gives rise to a general family of approximations that lend themselves to efficiently sampling Gaussian process posteriors. Starting from first principles, we derive these methods and analyze the approximation errors they introduce. We, then, ground these results by exploring the practical implications of pathwise conditioning in various applied settings, such as global optimization and reinforcement learning.