The spatial linear mixed model (SLMM) consists of fixed and spatial random effects that may be effected by unmeasured confounding and collinearity. Partially motivated as a means to address potential issues with confounding, the Restricted spatial regression (RSR) model restricts spatial random effects to be in the orthogonal column space of the covariates. Recent articles have shown that the misspecified Bayesian RSR generally performs worse than the SLMM when the data is generated from the SLMM. However, we show that the misspecified Bayesian RSR model's marginal posterior distribution is equivalent up to a reparameterization to that of the SLMM's marginal posterior distribution, under a certain prior assumption on the orthogonalized regression coefficients. These equivalence relationships show that when comparing the SLMM and RSR using the Bayes factor, stictly proper scoring rules, cross-validation, and many other standard impartial selection criteria we see that the Bayesian SLMM and the misspecified Bayesian RSR are seen as equivalent under certain prior specifications. We provide several new directions for the RSR including results that show that an augmented RSR can produce clear computational advantages via a direct sampler from the posterior distribution for all hyperparameters, fixed effects, and random effects leading to implementation strategies for the SLMM via these equivalence relationships. Additionally, a type of transfer learning approach offers a new interpretation to orthogonalized regression coefficients, which we show empirically can improve inference on linearly dependent regression coefficients, via a bias adjustment, in the presence of spatial confounding. Simulations and an illustration using COVID-19 mortality data are provided.

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