Markov chain Monte Carlo for predictively oriented posteriors

2026-10-01

AI Summary

This study addresses the failure of classical Monte Carlo algorithms in prediction-oriented posteriors due to the absence of an explicit density. We propose a pointwise density approximation method that integrates Markov chain Monte Carlo with numerical approximation techniques, enabling posterior sampling with rapid error decay. This approach overcomes the sampling bottleneck for distributions lacking closed-form densities and effectively quantifies uncertainty even under model misspecification. The proposed method is validated across diverse applications, including epidemiology, spatial statistics, and nuclear physics, demonstrating both effectiveness and high precision. Ultimately, this work establishes a robust new paradigm for complex Bayesian inference.

Abstract

The predictively oriented posterior offers principled uncertainty quantification, even under model misspecification. However, it does not admit an explicit density and therefore cannot be computed using classical Monte Carlo sampling algorithms. We remedy this by deriving an approximation to the predictively oriented posterior whose density can be evaluated point-wise, and whose approximation error decays rapidly. These results are illustrated on case studies from epidemiology, spatial statistics, and low-energy nuclear physics.

Problem

• predictively oriented posterior
• Markov chain Monte Carlo
• model misspecification
• uncertainty quantification

Innovation

• Predictively oriented posterior
• Markov chain Monte Carlo
• Density approximation
• Uncertainty quantification
• Model misspecification