Advanced random field models and practical Bayesian estimation

涉及高斯过程和随机场的贝叶斯方法通常用于模拟生态学、地学、医学成像和流行病学中的广泛问题的空间和时空现象。然而，现实的模型可能需要非平稳、非高斯和非线性的行为，而目前的方法无法有效地处理这些行为。在本项目中，目前在R-INLA软件(http://r-inla.org和http://inlabru.org))中实现的基于高斯随机偏微分方程组、数值优化和积分的现有模型和方法将扩展到非平稳和非高斯模型。计算效率方法的一个重要方面是评估数值结果与理论精确值的接近程度，以及评估估计的模型模拟真实观测现象的程度。这方面将涉及为近似和评估概率空间和时间预测制定实用的诊断措施。

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.