Error bounds for simultaneous Wasserstein contractive adaptive increasingly rare MCMC

2026-06-29

AI Summary

This work addresses the absence of explicit error bounds for expectation estimators in adaptive rare-event Markov chain Monte Carlo (MCMC) by establishing the first explicit mean squared error upper bound for the time-averaged estimator of adaptive incremental rare-event MCMC under a simultaneous Wasserstein contraction condition. The proposed method integrates normalizing flows, adaptive stereographic projection, and the Metropolis–Hastings algorithm, and is extended to a general adaptive framework applicable to doubly intractable problems. The theoretical analysis combines Wasserstein contraction properties with complexity assessments, yielding an error bound that informs computational resource allocation for achieving a desired target accuracy. Experimental evaluations across multiple adaptive schemes confirm both the tightness of the derived theoretical bound and the scalability of the algorithm.

Abstract

We investigate adaptive increasingly rare Markov chain Monte Carlo algorithms and the associated time-average estimator for approximating expectations. Under a simultaneous Wasserstein contraction assumption on the underlying family of Markov kernels we derive explicit bounds for the mean squared error. We illustrate the applicability of our estimate through adaptive stereographic algorithms and Metropolis-Hastings schemes that employ normalizing flows for adaptation. We also consider a generic adaptive algorithm for doubly intractable problems and provide a corresponding cost analysis to achieve a desired precision.

Problem

• Wasserstein contraction
• adaptive MCMC
• error bounds
• time-average estimator
• increasingly rare events

Innovation

• Wasserstein contraction
• adaptive MCMC
• normalizing flows
• doubly intractable problems
• error bounds