Goodness-of-fit diagnostics for Bayesian hierarchical models

Area of Science:

• Statistics
• Computational Statistics
• Bayesian Inference

Background:

• Assessing goodness of fit is crucial for validating statistical models.
• Bayesian hierarchical models are widely used but require robust diagnostic tools.
• Existing methods for model checking may lack statistical power or be computationally intensive.

Purpose of the Study:

• To propose a novel methodology for assessing goodness of fit in Bayesian hierarchical models.
• To introduce pivotal discrepancy measures (PDMs) as a tool for model diagnostics.
• To evaluate the performance of PDMs against existing posterior-predictive checks.

Main Methods:

• The methodology involves computing pivotal discrepancy measures (PDMs) using posterior parameter samples.
• PDMs are compared to known reference distributions for assessing model fit.
• The approach leverages standard output from Markov chain Monte Carlo (MCMC) algorithms.

Main Results:

• Simulation studies indicate that diagnostics based on PDMs exhibit higher statistical power in detecting model departures compared to posterior-predictive checks.
• The proposed diagnostics are computationally efficient, requiring minimal additional resources beyond standard MCMC output.
• The methodology is demonstrated through a clinical application and an application to discrete data.

Conclusions:

• The proposed methodology using pivotal discrepancy measures offers a powerful and computationally efficient approach for assessing goodness of fit in Bayesian hierarchical models.
• PDMs provide a valuable alternative to traditional diagnostic checks, particularly for identifying model misspecification.
• The approach is broadly applicable and demonstrated in real-world scenarios.