Gaussian random fields (GFs) are fundamental tools in spatial modeling and can be represented flexibly and efficiently as solutions to stochastic partial differential equations (SPDEs). The SPDEs depend on specific parameters, which enforce various field behaviors and can be estimated using Bayesian inference. However, even under in-fill asymptotics, the likelihood only provides limited insights into the covariance structure. In response, it is essential to leverage priors to achieve appropriate, meaningful covariance structures in the posterior. This study introduces a smooth, invertible parameterization of the correlation length and diffusion matrix of an anisotropic GF and constructs penalized complexity (PC) priors for the model when the parameters are constant in space. The formulated prior is weakly informative, effectively penalizing complexity by pushing the correlation range toward infinity and the anisotropy to zero.

翻译：高斯随机场是空间建模的基础工具，可通过随机偏微分方程灵活高效地表示。随机偏微分方程依赖于特定参数，这些参数可施加多种场行为并通过贝叶斯推断估计。然而，即使在填充渐近条件下，似然函数对协方差结构的解读仍十分有限。为此，需充分利用先验信息在后验中获得恰当且具实际意义的协方差结构。本研究针对各向异性高斯场的相关长度和扩散矩阵提出一种光滑可逆的参数化方法，并在参数空间恒定的条件下构建了模型的惩罚复杂性先验。该先验具有弱信息性，通过将相关范围推至无穷大并将各向异性推至零，有效惩罚了模型复杂性。