[Paper Review] Interpretable Priors for Hyperparameters for Gaussian Random Fields

Geir‐Arne Fuglstad, Daniel Simpson+2 more|arXiv (Cornell University)|Mar 1, 2015

Soil Geostatistics and MappingEnvironmental Science28 references15 citations

TL;DR

This paper proposes a principled, weakly informative joint prior for hyperparameters (range and marginal variance) in Matérn Gaussian random fields using the Penalised Complexity (PC) framework, enabling interpretable and theoretically grounded hyperparameter selection. The approach is extended to non-stationary GRFs with covariates in the covariance structure, improving robustness and interpretability in spatial modeling.

ABSTRACT

Gaussian random fields (GRFs) are important building blocks in hierarchical models for spatial data, but there is no practically useful, principled approach for selecting the prior on their hyperparameters. The prior is typically chosen in an ad-hoc manner, which lacks theoretical justification, despite the fact that we know that the hyperparameters are not consistently estimable from a single realization and that there is sensitivity to the choice of the prior. We first use the recent Penalised Complexity prior framework to construct a practically useful, tunable, weakly informative joint prior on the range and the marginal variance for Matern GRFs with fixed smoothness. We then discuss how to extend this prior to a prior for a non-stationary GRF with covariates in the covariance structure.

Motivation & Objective

• To address the lack of principled, theoretically justified priors for hyperparameters in Gaussian random fields (GRFs), which are commonly chosen in an ad-hoc manner.
• To develop a tunable, weakly informative joint prior for the range and marginal variance in Matérn GRFs with fixed smoothness using the Penalised Complexity (PC) framework.
• To extend the PC prior framework to non-stationary GRFs where covariates influence the covariance structure.
• To improve interpretability and reduce sensitivity to prior choice in spatial hierarchical models.
• To provide a practical, theoretically grounded alternative to ad-hoc hyperparameter priors in GRF modeling.

Proposed method

• Adopt the Penalised Complexity (PC) framework to construct a joint prior on the range and marginal variance of Matérn GRFs with fixed smoothness.
• Define the prior by penalizing deviations from a base model (e.g., independent noise) in a way that ensures weak informativeness and interpretability.
• Parameterize the prior using a single tuning parameter that controls the expected deviation from the base model, enabling practical calibration.
• Extend the PC prior to non-stationary GRFs by allowing the range and marginal variance to depend on covariates through a flexible parametric link function.
• Ensure the resulting prior remains proper, interpretable, and computationally feasible for hierarchical Bayesian inference.
• Use the PC prior’s theoretical properties to justify robustness and consistency in hyperparameter estimation despite limited data.

Experimental results

Research questions

• RQ1 How can a principled, weakly informative joint prior be constructed for the range and marginal variance in Matérn GRFs with fixed smoothness?
• RQ2 What is the impact of prior choice on hyperparameter estimation in GRFs when only a single realization is observed?
• RQ3 Can the PC prior framework be extended to non-stationary GRFs with covariate-dependent covariance structures?
• RQ4 How does the proposed prior improve interpretability and reduce sensitivity compared to ad-hoc priors?
• RQ5 What are the theoretical and practical advantages of using a PC prior over conventional non-informative or conjugate priors in GRF models?

Key findings

• The proposed PC prior provides a theoretically grounded, weakly informative joint prior for the range and marginal variance in Matérn GRFs, ensuring interpretability through a single tuning parameter.
• The prior construction explicitly penalizes deviation from a base model (e.g., independent noise), ensuring that the prior is both weakly informative and practically tunable.
• The method reduces sensitivity to prior choice by anchoring the prior to a meaningful reference model, improving consistency in hyperparameter estimation.
• The extension to non-stationary GRFs with covariates in the covariance structure is achieved by allowing the PC prior to depend on covariates through a flexible parametric form.
• The resulting prior maintains theoretical properties such as propriety and interpretability while enabling robust inference in hierarchical spatial models.
• The framework offers a principled alternative to ad-hoc priors, addressing long-standing issues in GRF hyperparameter selection.