Conditioning Gaussian Processes on Almost Anything

2026-05-20

AI Summary

This work addresses the long-standing limitation of Gaussian processes (GPs) in handling non-conjugate observations—such as those arising from nonlinear physical laws or natural language—by establishing an explicit equivalence between GPs and a class of linear diffusion models. The authors reformulate conditional prediction as an ordinary differential equation with closed-form Gaussian dynamics, augmented by a likelihood-dependent guidance term that accommodates arbitrary pointwise-evaluable conditioning information. By employing a whitening transformation to decouple irreducible non-Gaussian dynamics and minimizing the Wasserstein-2 transport cost to mitigate numerical stiffness, the framework enables, for the first time, general-purpose conditioning of GPs on nontraditional observations like natural language without task-specific derivations. While exactly recovering standard linear-Gaussian results, this approach substantially enhances both the representational capacity and numerical stability of GPs in nonlinear systems and when guided by large language models.

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.

Problem

• Gaussian processes
• non-conjugate conditioning
• probabilistic inference
• diffusion models
• likelihood evaluation

Innovation

• Gaussian processes
• diffusion models
• non-conjugate conditioning
• whitening
• likelihood-guided sampling