Conditioning Gaussian Processes on Almost Anything

TL;DR

This paper introduces a new general-purpose Gaussian process inference method that leverages linear diffusion models and ODEs, enabling conditioning on complex, real-world information including non-linear physics and language models.

Contribution

It establishes an explicit equivalence between GPs and linear diffusion models, enabling flexible conditioning beyond conjugacy without bespoke derivations.

Findings

Recovers standard GP conditioning exactly in linear-Gaussian cases

Handles non-linear physics and language models with the same machinery

Minimizes Wasserstein-2 transport cost to reduce numerical stiffness

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.