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, the likelihood typically only provides limited insights into the covariance structure under in-fill asymptotics. 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中的基本工具，可灵活高效地表示为随机偏微分方程的解。随机偏微分方程依赖于特定参数，这些参数决定了场的不同行为特征，并可通过贝叶斯推断进行估计。然而，在填充渐近条件下，似然函数通常仅能提供关于协方差结构的有限信息。为此，必须利用先验分布在后验中实现恰当且有意义的协方差结构。本研究提出了各向异性高斯场相关长度与扩散矩阵的光滑可逆参数化方法，并为空间恒定参数情形构建了惩罚复杂度先验。所构建的先验具有弱信息性，通过将相关范围推向无穷大、各向异性趋近于零的方式，有效实现了对复杂度的惩罚。