Statistical Modeling Using Bayesian Latent Gaussian Models

Bayesian latent Gaussian models are Bayesian hierarchical models that assign Gaussian prior densities to the latent parameters. In this chapter, we present three subclasses within the class of Bayesian latent Gaussian models, namely, Bayesian Gaussian–Gaussian models, Bayesian latent Gaussian models with a univariate link function, and Bayesian latent Gaussian models with a multivariate link function. The structure of each subclass is described along with methods to infer the parameters of these models. The construction of prior densities for the latent parameters and the hyperparameters is described. Several examples are given to demonstrate how to apply models from these subclasses to real datasets.

Bayesian methods for modelling and inference are being increasingly used in the cryospheric sciences and glaciology in particular. Here, we present a review of recent works in glaciology that adopt a Bayesian approach when conducting an analysis. We organise the chapter into three categories: (i) Gaussian–Gaussian models, (ii) Bayesian hierarchical models, and (iii) Bayesian calibration approaches. In addition, we present two detailed case studies that involve the application of Bayesian hierarchical models in glaciology. The first case study is on the spatial prediction of surface mass balance across the Icelandic mountain glacier Langjökull, and the second is on the prediction of sea-level rise contributions from the Antarctic ice sheet. This chapter is presented in such a way that it is accessible to both statisticians and Earth scientists.

The ground-motion model (GMM) is a key tool of the engineering seismologist. It predicts peak seismic ground-motion parameters given primary independent variables such as earthquake magnitude, distance from the earthquake, site effects, and more. The empirical GMM is expressed as a simple mathematical equation containing regression parameters to be inferred through a calibration of the model to a given dataset. Engineering seismologists strive for improved GMM predictions through the systematic reduction of model variability by incorporating additional independent variables, preferably physics-based ones. That leads to greater confidence in probabilistic seismic hazard assessment (PSHA), which is the foundation of earthquake-resistant building design and the mitigation of seismic risk of our modern society. In this chapter, we show examples of the application of the Bayesian statistical framework in ground-motion modeling. We use a large dataset of seismic ground motions recorded on a small urban strong-motion array in Southwest Iceland to calibrate a simple GMM using the Bayesian hierarchical modeling (BHM) approach. The partitioning of the model residuals into source, path, and site terms and the BHM quantifying their posterior distributions facilitates a physics-based interpretation of residual behavior. Namely, the modeling approach quantifies the relative contribution of the terms to the total residual variability, thereby identifying the term that contributes most to the overall variability, in this case, the site term. Second, by systematically analyzing the behavior of the individual elements that constitute the site term, we show how they are the manifestation of site-specific seismic wave amplification effects due to the local geological structure. On this basis, the GMM can be improved through the incorporation of geological amplification effects into the GMM. Then, using a regional dataset of seismic motions, we show how prior distributions of GMM parameters, which otherwise would be poorly constrained by limited data, can guide the Bayesian inference such that the GMM provides physically meaningful predictions in the range of limited data. Finally, we show how the Bayesian statistical framework can be used to objectively select the most appropriate GMM for use in PSHA in Iceland.

The specification of the spatial characterisation of earthquake sources in a seismic region and their seismic activity are two of the three key elements of probabilistic seismic hazard assessment, the third one being ground motion modelling. The seismic activity rate is specified by a magnitude–frequency relationship that is usually modelled using an exponential distribution for the earthquake magnitudes. We propose a Bayesian latent Gaussian model for the earthquake magnitudes that assumes a generalised Pareto distribution with a spatially varying scale parameter as an alternative to the exponential distribution with a constant scale parameter. We apply it to estimate the spatial variations of earthquake magnitudes across Southwest Iceland including the South Iceland transform zone, which is the region with the highest earthquake hazard and seismic risk in Iceland. We show that the generalised Pareto distribution, with a spatially varying scale parameter, provides a substantially better fit to the earthquake magnitudes than the exponential distribution with a constant scale parameter. An analysis based on the proposed spatial model reveals that the scale parameter takes different values depending on the location along the seismic region. The spatial distribution of seismicity in the region and its tectonic characteristics indicate that the scale parameter can be correlated with physical parameters that characterise seismicity. The results indicate that modelling earthquake magnitudes with the generalised Pareto distribution with a spatially varying scale parameter can be useful in hazard assessment.

Numerical predictions of weather and climate pose interesting problems for spatial and spatio-temporal statistical modelling, especially to quantify and correct systematic differences between forecasts and observation. We review the state of the art of statistical postprocessing of predictions produced by atmospheric simulation models and report encouraging results on the application of Bayesian hierarchical models to improve on the existing statistical postprocessing methodology. In particular, we show that after fitting postprocessing parameters at each grid point by maximum likelihood estimation, a spatial smoothing of the parameter estimates is justified in a Bayesian hierarchical modelling context and offers improvements of out-of-sample forecasts.

In this chapter, we show how to efficiently model high-dimensional extreme peaks-over-threshold events over space in complex non-stationary settings, using extended latent Gaussian models (LGMs), and how to exploit the fitted model in practice for the computation of long-term return levels. The extended LGM framework assumes that the data follow a specific parametric distribution, whose unknown parameters are transformed using a multivariate link function and are then further modeled at the latent level in terms of fixed and random effects that have a joint Gaussian distribution. In the extremal context, we here assume that the response level distribution is described in terms of a Poisson point process likelihood, motivated by asymptotic extreme-value theory, and which conveniently exploits information from all threshold exceedances. This contrasts with the more common data-wasteful approach based on block maxima, which are typically modeled with the generalized extreme-value (GEV) distribution. When conditional independence can be assumed at the response level and latent random effects have a sparse probabilistic structure, fast approximate Bayesian inference becomes possible in very high dimensions, and we here present the recently proposed inference approach called “Max-and-Smooth,” which provides exceptional speedup compared to alternative methods. The proposed methodology is illustrated by application to satellite-derived precipitation data over Saudi Arabia, obtained from the Tropical Rainfall Measuring Mission, with 2738 grid cells and about 20 million spatio-temporal observations in total. Our fitted model captures the spatial variability of extreme precipitation satisfactorily, and our results show that the most intense precipitation events are expected near the southwestern part of Saudi Arabia, along the Red Sea coastline.