Post-Processing of MCMC.

Area of Science:

• Computational Statistics
• Bayesian Inference
• Markov Chain Monte Carlo Methods

Background:

• Markov chain Monte Carlo (MCMC) is fundamental to modern Bayesian statistics for approximating posterior distributions.
• Post-processing of MCMC output, including convergence diagnostics and bias control, is often inadequately addressed.
• Limited computational resources can create a bias-variance trade-off, which standard convergence diagnostics do not account for.

Purpose of the Study:

• To review state-of-the-art techniques for post-processing Markov chain output.
• To highlight methods that manage the bias-variance trade-off inherent in computational constraints.
• To provide an overview of advanced strategies for approximating expected quantities of interest from MCMC samples.

Main Methods:

• Review of discrepancy minimisation techniques for direct bias-variance trade-off management.
• Examination of general-purpose control variate methods for approximating expected values.
• Analysis of current practices in post-processing Markov chain output.

Main Results:

• Identified discrepancy minimisation methods that directly tackle the bias-variance trade-off.
• Highlighted the utility of control variate methods for efficient approximation of target quantities.
• Emphasized the need for advanced post-processing beyond simple burn-in removal.

Conclusions:

• Advanced post-processing techniques are essential for reliable MCMC output interpretation.
• Methods addressing the bias-variance trade-off are crucial, especially under computational limitations.
• The reviewed techniques offer improved accuracy and efficiency in Bayesian statistical analyses.