Advanced random field models and practical Bayesian estimation

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Bayesian methods involving Gaussian processes and random fields are commonly used when modelling spatial and spatiotemporal phenomena in a wide range of problems in ecology, geoscience, medical imaging, and epidemiology. However, realistic models can require non-stationary, non-Gaussian and non-linear behaviour that current methods are not equippedto handle efficiently. In this project, current models and methods based on Gaussian stochastic partial differential equations, numerical optimisation and integration, and currently implemented in the R-INLA software (http://r-inla.org and http://inlabru.org) will be extended to non-stationary and non-Gaussian models. An important aspect of computationallyefficient methods is to assess how close the numerical results are to the theoretically exact values, as well as to assess how well the estimated models mimic the real observed phenomena. This aspect will involve developing practical diagnostic measures for the approximations, and for assessing probabilistic spatial and temporal predictions.