Posterior Inference for Sparse Hierarchical Non-stationary Models

TL;DR

This paper introduces a scalable Bayesian framework for non-stationary Gaussian process models using hierarchical Gaussian Markov random fields with adaptive MCMC algorithms, enabling efficient inference in complex spatial settings.

Contribution

It proposes a novel hierarchical non-stationary Gaussian process model with sparse precision matrices and develops adaptive MCMC methods for efficient posterior sampling.

Findings

Efficient inference achieved with banded matrix operations.

The model accurately captures non-stationary spatial phenomena.

The approach scales well to multi-dimensional problems.

Abstract

Gaussian processes are valuable tools for non-parametric modelling, where typically an assumption of stationarity is employed. While removing this assumption can improve prediction, fitting such models is challenging. In this work, hierarchical models are constructed based on Gaussian Markov random fields with stochastic spatially varying parameters. Importantly, this allows for non-stationarity while also addressing the computational burden through a sparse banded representation of the precision matrix. In this setting, efficient Markov chain Monte Carlo (MCMC) sampling is challenging due to the strong coupling a posteriori of the parameters and hyperparameters. We develop and compare three adaptive MCMC schemes and make use of banded matrix operations for faster inference. Furthermore, a novel extension to multi-dimensional settings is proposed through an additive structure that retains the flexibility and scalability of the model, while also inheriting interpretability from the additive approach. A thorough assessment of the efficiency and accuracy of the methods in nonstationary settings is presented for both simulated experiments and a computer emulation problem.