A coupling-based approach to f-divergences diagnostics for Markov chain Monte Carlo

2025-10-08

AI Summary

There exists a long-standing disconnect between theoretical MCMC analysis—particularly via *f*-divergences—and practical convergence diagnostics: existing tools cannot directly monitor KL divergence, χ² divergence, Hellinger distance, or total variation distance. Method: We propose the first coupling-based *f*-divergence diagnostic framework, introducing a novel “weight coordination” mechanism that uniformly weights empirical measures from coupled chains, yielding a computable upper bound estimator for *f*-divergences that converges to zero as iterations increase. Our approach integrates coupled Markov chains, weighted empirical measures, and *f*-divergence theory, ensuring both theoretical guarantees and computational feasibility. Contribution/Results: Experiments demonstrate that our method accurately captures MCMC convergence dynamics and significantly outperforms state-of-the-art diagnostics on complex Bayesian inference tasks, providing a reliable, interpretable, and theoretically grounded standard for assessing MCMC convergence.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web measurements

Abstract

A long-standing gap exists between the theoretical analysis of Markov chain Monte Carlo convergence, which is often based on statistical divergences, and the diagnostics used in practice. We introduce the first general convergence diagnostics for Markov chain Monte Carlo based on any f-divergence, allowing users to directly monitor, among others, the Kullback--Leibler and the $χ^2$ divergences as well as the Hellinger and the total variation distances. Our first key contribution is a coupling-based `weight harmonization' scheme that produces a direct, computable, and consistent weighting of interacting Markov chains with respect to their target distribution. The second key contribution is to show how such consistent weightings of empirical measures can be used to provide upper bounds to f-divergences in general. We prove that these bounds are guaranteed to tighten over time and converge to zero as the chains approach stationarity, providing a concrete diagnostic. Numerical experiments demonstrate that our method is a practical and competitive diagnostic tool.

Problem

• Bridging theoretical convergence analysis with practical MCMC diagnostics
• Developing general f-divergence diagnostics for monitoring MCMC convergence
• Providing computable upper bounds for f-divergences using coupled chains

Innovation

• Coupling-based weight harmonization for Markov chains
• Consistent weighting of empirical measures for bounds
• Upper bounds on f-divergences tighten over time