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We introduce a novel procedure for obtaining cross-validated predictiveestimates for Bayesian hierarchical regression models (BHRMs). Bayesianhierarchical models are popular for their ability to model complex dependencestructures and provide probabilistic uncertainty estimates, but can becomputationally expensive to run. Cross-validation (CV) is therefore not acommon practice to evaluate the predictive performance of BHRMs. Our methodcircumvents the need to re-run computationally costly estimation methods foreach cross-validation fold and makes CV more feasible for large BHRMs. Byconditioning on the variance-covariance parameters, we shift the CV problemfrom probability-based sampling to a simple and familiar optimization problem.In many cases, this produces estimates which are equivalent to full CV. Weprovide theoretical results and demonstrate its efficacy on publicly availabledata and in simulations.

Approximate Cross-validated Mean Estimates for Bayesian Hierarchical Regression Models