A common phenomenon in spatial regression models is spatial confounding. This phenomenon occurs when spatially indexed covariates modeling the mean of the response are correlated with a spatial effect included in the model. spatial+ Dupont et al. (2022) is a popular approach to reducing spatial confounding. spatial+ is a two-stage frequentist approach that explicitly models the spatial structure in the confounded covariate, removes it, and uses the corresponding residuals in the second stage. In a frequentist setting, there is no uncertainty propagation from the first stage estimation determining the residuals since only point estimates are used. Inference can also be cumbersome in a frequentist setting, and some of the gaps in the original approach can easily be remedied in a Bayesian framework. First, a Bayesian joint model can easily achieve uncertainty propagation from the first to the second stage of the model. In a Bayesian framework, we also have the tools to infer the model's parameters directly. Notably, another advantage of using a Bayesian framework we thoroughly explore is the ability to use prior information to impose restrictions on the spatial effects rather than applying them directly to their posterior. We build a joint prior for the smoothness of all spatial effects that simultaneously shrinks towards a high smoothness of the response and imposes that the spatial effect in the response is a smoother of the confounded covariates' spatial effect. This prevents the response from operating at a smaller scale than the covariate and can help to avoid situations where there is insufficient variation in the residuals resulting from the first stage model. We evaluate the performance of the Bayesian spatial+ via both simulated and real datasets.

翻译：空间回归模型中常见的现象是空间混杂，即当建模响应均值的空间索引协变量与模型中包含的空间效应相关时发生。空间+（Dupont等人，2022）是减少空间混杂的流行方法。空间+是一种两阶段频率学派方法，它显式建模混杂协变量中的空间结构，将其移除，并在第二阶段使用相应的残差。在频率学派设定中，由于仅使用点估计，因此不存在从第一阶段估计确定残差的不确定性传播。频率学派框架下的推断也可能较为繁琐，而原始方法中的一些不足可通过贝叶斯框架轻松弥补。首先，贝叶斯联合模型可以轻松实现从模型第一阶段到第二阶段的不确定性传播。在贝叶斯框架中，我们也有直接推断模型参数的工具。值得注意的是，我们深入探讨的另一个贝叶斯框架优势是能够利用先验信息对空间效应施加约束，而非直接对后验施加约束。我们为所有空间效应的光滑性构建了一个联合先验，该先验同时将响应的高光滑性方向收缩，并要求响应中的空间效应是混杂协变量空间效应的更光滑版本。这可以防止响应以比协变量更小的尺度运行，并有助于避免第一阶段模型产生的残差变异性不足的情况。我们通过模拟数据集和真实数据集评估了贝叶斯空间+的性能。