Default Bayesian Analysis of Spatial Data

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The collection and analysis of spatial data are ubiquitous in most natural and earth sciences, such as ecology, epidemiology, geology and hydrology. This research project will develop statistical methodology for automatic Bayesian analysis of Gaussian models that does not require any subjective input. These models play a prominent role due to their versatility to model spatially varying phenomena, and because they serve as building blocks for the construction of more elaborate models. The Bayesian approach is especially appealing when the main goal is spatial interpolation, but implementing it for such models faces two big challenges: (i) Automatic formulation of sensible prior distributions that are adapted to the scale of the data under investigation and (ii) Analysis of massive data sets that are the norm nowadays. The project aims at developing theory and practice to overcome both challenges, which will make practicably feasible the automatic Bayesian analysis of large spatial data sets. In addition, the project will train graduate students in spatial statistics in general, and the topics of this project in particular. The results derived from the project will be disseminated in diverse outlets, and software to implement the methodology will be made publicly available.The research project will make practicably feasible default Bayesian analyses of large spatial data sets, by contributing innovations to the two parts of the Bayesian model. First, approximate reference priors for the parameters of covariance models will be developed that allow carrying out Bayesian analyses for these models in an automatic fashion, not requiring subjective elicitation. These will be based on the spectral approximation of stationary random fields. Second, likelihood approximations feasible for large spatial data sets will be elaborated by developing strategies to tune a recently proposed approximation for stationary covariance functions. The tuning of the approximation aims at striking a balance between accuracy and computational effort. Both approximations rely on the spectral density, rather than the covariance function, of the model. Together, the reference prior and likelihood approximations will make possible carrying out default Bayesian analyses that include model selection and assessment. In addition, the project will critically assess the common practice of fixing the smoothness of the random field at a value, chosen by convention or tradition, that bears no relation to the data under analysis. The project will investigate methods to quantify the information content in spatial data about smoothness parameters, and uncover how this depends on the sample design. The methodology will be tested on diverse data sets from the earth sciences, with special focus on spatial rainfall data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.