Conditioning Gaussian Processes on Almost Anything
Abstract
Gaussian processes (GPs) offer a principled probabilistic model over functions, but exact inference is restricted to the linear-Gaussian regime. We establish an explicit equivalence between GPs and a class of linear diffusion models, recasting predictive sampling as an ODE with closed-form Gaussian dynamics and a likelihood-dependent guidance term that admits a simple Monte Carlo approximation. In the linear-Gaussian setting, we recover standard GP conditioning exactly; beyond conjugacy, the same machinery handles any conditioning statement admitting point-wise likelihood evaluation - including non-linear physics, and, for the first time, natural language via large language models. Whitening isolates the irreducible non-Gaussian dynamics, minimising Wasserstein-2 transport cost and eliminating numerical stiffness. The result is a general-purpose GP inference scheme requiring no bespoke derivations. Together, these results provide a general mechanism for incorporating the full richness of real-world knowledge as conditioning information, opening a new frontier for the probabilistic modelling of real-world problems.
Related
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BibTeX
@misc{moss2026conditioning,
title = {{Conditioning Gaussian Processes on Almost Anything}},
author = {Henry Moss and Lachlan Astfalck and Thomas Cowperthwaite and Colin Doumont and Sam Willis and Philipp Hennig and Christopher Nemeth and Andrew Zammit-Mangion},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.21041},
eprint = {2605.21041},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.21041},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Conditioning Gaussian Processes on Almost Anything
AU - Henry Moss
AU - Lachlan Astfalck
AU - Thomas Cowperthwaite
AU - Colin Doumont
AU - Sam Willis
AU - Philipp Hennig
AU - Christopher Nemeth
AU - Andrew Zammit-Mangion
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2605.21041
UR - https://arxiv.org/abs/2605.21041
AB - Gaussian processes (GPs) offer a principled probabilistic model over functions, but exact inference is restricted to the linear-Gaussian regime. We establish an explicit equivalence between GPs and a class of linear diffusion models, recasting predictive sampling as an ODE with closed-form Gaussian dynamics and a likelihood-dependent guidance term that admits a simple Monte Carlo approximation. In the linear-Gaussian setting, we recover standard GP conditioning exactly; beyond conjugacy, the same machinery handles any conditioning statement admitting point-wise likelihood evaluation - including non-linear physics, and, for the first time, natural language via large language models. Whitening isolates the irreducible non-Gaussian dynamics, minimising Wasserstein-2 transport cost and eliminating numerical stiffness. The result is a general-purpose GP inference scheme requiring no bespoke derivations. Together, these results provide a general mechanism for incorporating the full richness of real-world knowledge as conditioning information, opening a new frontier for the probabilistic modelling of real-world problems.
ER -