Predictive posterior sampling from non-stationnary Gaussian process priors via Diffusion models with application to climate data

TL;DR

This paper introduces a novel method using diffusion generative models to efficiently sample from complex non-stationary Gaussian process posteriors, enabling improved environmental data predictions.

Contribution

It proposes a two-step approach replacing GP priors with DGMs and leverages guidance algorithms for sampling, addressing intractability issues in non-stationary GP models.

Findings

Accurately mimics GP posterior distributions with statistical metrics.

Enables fine-tuning of DGMs for targeted posterior sampling.

Achieves state-of-the-art predictions in environmental inverse problems.

Abstract

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.