The Learning Rate Is Not a Constant: Sandwich-Adjusted Markov Chain Monte Carlo Simulation.

Area of Science:

• Statistics
• Computational Statistics
• Bayesian Inference

Background:

• Model misspecification is a key limitation in maximum likelihood and Bayesian methods.
• Standard methods yield biased covariance matrices under misspecification, leading to underestimated uncertainty.
• Existing robust posterior sampling methods have limitations.

Purpose of the Study:

• To develop a new sandwich-adjusted Markov chain Monte Carlo (MCMC) method for robust Bayesian inference.
• To introduce information-theoretic diagnostics for quantifying model misspecification.
• To provide asymptotically valid and robust uncertainty estimates under model misspecification.

Main Methods:

• Developed a novel sandwich-adjusted MCMC method with a parameter-dependent learning rate for direction-specific likelihood tempering.
• Proposed information-theoretic diagnostics including divergence scores and scalar summaries (Frobenius norm, Earth mover's distance, Herfindahl index).
• Applied methods to parametric distributions and a rainfall-runoff case study.

Main Results:

• The proposed MCMC method captures directional asymmetries and yields valid credible regions.
• Conventional Bayesian methods systematically underestimate uncertainty.
• The new method provides asymptotically valid and robust uncertainty estimates, outperforming standard approaches.

Conclusions:

• Sandwich-based adjustments are recommended for robust Bayesian practice and workflows.
• The developed diagnostics effectively assess model misspecification by evaluating sensitivity and variability matrices.
• The new MCMC method offers a principled way to handle model misspecification and improve uncertainty quantification.