Conditioning Gaussian Processes on Almost Anything
For practitioners using Gaussian processes, this work provides a general-purpose inference scheme that extends conditioning to complex, real-world likelihoods, eliminating the need for problem-specific derivations.
The paper establishes an equivalence between Gaussian processes and linear diffusion models, enabling GP conditioning on arbitrary likelihoods (e.g., nonlinear physics, natural language) via an ODE with a Monte Carlo guidance term. The method recovers exact GP conditioning in linear-Gaussian cases and handles non-conjugate settings without bespoke derivations.
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.