Conditioning Gaussian Processes on Almost Anything
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arXiv
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| Autori principali: | , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913148401876992 |
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| author | Moss, Henry Astfalck, Lachlan Cowperthwaite, Thomas Doumont, Colin Willis, Sam Hennig, Philipp Nemeth, Christopher Zammit-Mangion, Andrew |
| author_facet | Moss, Henry Astfalck, Lachlan Cowperthwaite, Thomas Doumont, Colin Willis, Sam Hennig, Philipp Nemeth, Christopher Zammit-Mangion, Andrew |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21041 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Conditioning Gaussian Processes on Almost Anything Moss, Henry Astfalck, Lachlan Cowperthwaite, Thomas Doumont, Colin Willis, Sam Hennig, Philipp Nemeth, Christopher Zammit-Mangion, Andrew Machine Learning Methodology 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. |
| title | Conditioning Gaussian Processes on Almost Anything |
| topic | Machine Learning Methodology |
| url | https://arxiv.org/abs/2605.21041 |