Predictive posterior sampling from non-stationnary Gaussian process priors via Diffusion models with application to climate data
This addresses a computational bottleneck for researchers using Bayesian models with complex spatial patterns, though it appears incremental as an application of existing diffusion methods to a specific problem.
The paper tackles the computational intractability of sampling from predictive posterior distributions with non-stationary Gaussian process priors by proposing a diffusion generative model approach, which yields state-of-the-art predictions for inverse problems in environmental sciences.
Bayesian models based on Gaussian processes (GPs) offer a flexible framework to predict spatially distributed variables with uncertainty. But the use of nonstationary priors, often necessary for capturing complex spatial patterns, makes sampling from the predictive posterior distribution (PPD) computationally intractable. In this paper, we propose a two-step approach based on diffusion generative models (DGMs) to mimic PPDs associated with non-stationary GP priors: we replace the GP prior by a DGM surrogate, and leverage recent advances on training-free guidance algorithms for DGMs to sample from the desired posterior distribution. We apply our approach to a rich non-stationary GP prior from which exact posterior sampling is untractable and validate that the issuing distributions are close to their GP counterpart using several statistical metrics. We also demonstrate how one can fine-tune the trained DGMs to target specific parts of the GP prior. Finally we apply the proposed approach to solve inverse problems arising in environmental sciences, thus yielding state-of-the-art predictions.