Constructing Priors that Penalize the Complexity of Gaussian Random Fields

By Geir-Arne Fuglstad, Daniel P. Simpson, Finn Lindgren, Håvard Rue. Journal of the American Statistical Association, 2015

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ABSTRACT Priors are important for achieving proper posteriors with physically meaningful covariance structures for Gaussian random fields (GRFs) since the likelihood typically only provides limited information about the covariance structure under in-fill asymptotics. We extend the recent penalized complexity prior framework and develop a principled joint prior for the range and the marginal variance of one-dimensional, two-dimensional, and three-dimensional Matérn GRFs with fixed smoothness. The prior is weakly informative and penalizes complexity by shrinking the range toward infinity and the marginal variance toward zero. We propose guidelines for selecting the hyperparameters, and a simulation study shows that the new prior provides a principled alternative to reference priors that can leverage prior knowledge to achieve shorter credible intervals while maintaining good coverage. We extend the prior to a nonstationary GRF parameterized through local ranges and marginal standard deviations, and introduce a scheme for selecting the hyperparameters based on the coverage of the parameters when fitting simulated stationary data. The approach is applied to a dataset of annual precipitation in southern Norway and the scheme for selecting the hyperparameters leads to conservative estimates of nonstationarity and improved predictive performance over the stationary model. Supplementary materials for this article are available online.

Subfields: Atmospheric Science, Environmental Engineering, Global and Planetary Change

View Paper: https://www.semanticscholar.org/paper/9dd30416786a287ab6e386eacf19f005860c05d3

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Citations in the corpus

The Prior Can Often Only Be Understood in the Context of the Likelihood (2017, methodology): “[16] recommend specifying the prior on (κ, σ) using coordinates parameterizing motion parallel and orthogonal to the the ridge.”
Bayesian Workflow (2020, background): “Zero-avoiding priors can make sense when such prior information is available—such as for the length-scale parameter of a Gaussian process (Fuglstad et al., 2019)—but we want to be careful when using such a restriction merely to make the algorithm run faster.”
Spatial modelling with R-INLA: A review (2018, methodology): “The SPDE approach is most easily described through the parameters τ and κ, where τ is the parameter for the Matérn covariance function that can be consistently estimated under infill-asymptotics (Zhang, 2004), κ shows up naturally in the differential operator, and, under infill-asymptotics the GRFs ”
Spatiotemporal wildfire modeling through point processes with moderate and extreme marks (2021, methodology): “…effect (in superscript): gCOX-BET(si) ∼ GP2D-SPDE(ω1), nCOX-BET = 1114, gCOX-BIN(si) ∼ GP2D-SPDE(ω2), nCOX-BIN = 1114, gBIN-GPD(si) ∼ GP2D-SPDE(ω3), nBIN-GPD = 1114, where ω1, ω2 and ω3 consist of separate Matérn range r and standard deviation σ parameters with PC priors (Fuglstad et al. 2018).”
Validating Bayesian Inference Algorithms with Simulation-Based Calibration (2018, methodology): “We use penalized complexity priors (Simpson et al., 2017; Fuglstad et al., 2017) on the remaining parameters tuned to ensure Pr( ρ < 0 .”
Maximum likelihood estimation of spatially varying coefficient models for large data with an application to real estate price prediction (2020, background): “(2017) and extensions thereof by Fuglstad et al. (2018) introduced penalizing complexity (PC) priors for Gaussian random fields of Matérn class.”
Bayesian Computing with INLA: A Review (2016, methodology): “…P-splines (Ventrucci and Rue, 2016), structured additive distributional regression (Klein and Kneib, 2016), Gaussian ﬁelds in spatial statistics (Fuglstad et al., 2016), modeling monthly maxima of instantaneous ﬂow (Ferkingstad et al., 2016) and autoregressive processes (Sørbye and Rue, 2016).”