Approximate Cross-Validated Mean Estimates for Bayesian Hierarchical Regression Models

Area of Science:

• Statistical Modeling
• Computational Statistics

Background:

• Bayesian hierarchical regression models (BHRMs) are widely used for complex data structures.
• The computational cost of BHRMs often prohibits standard cross-validation (CV) for performance evaluation.

Purpose of the Study:

• To develop a computationally efficient method for obtaining cross-validated predictive estimates for BHRMs.
• To make CV a practical tool for assessing the predictive performance of large and complex BHRMs.

Main Methods:

• A novel procedure that reframes the CV problem as an optimization task by conditioning on variance-covariance parameters.
• The method provides approximations for leave-one-out CV and leave-one-cluster-out CV.

Main Results:

• The proposed method significantly reduces the computational burden of CV for BHRMs.
• Approximated CV estimates are shown to be equivalent to full CV estimates in many scenarios.
• The method's efficacy is demonstrated through theoretical results, simulations, and real-world data analysis.

Conclusions:

• This new procedure makes cross-validation a feasible and reliable method for evaluating Bayesian hierarchical regression models.
• The approach facilitates more robust model selection and performance assessment in complex statistical modeling.