Analysis of a class of Multi-Level Markov Chain Monte Carlo algorithms based on Independent Metropolis-Hastings

TL;DR

This paper introduces and analyzes a class of Multi-Level Markov Chain Monte Carlo algorithms based on independent Metropolis-Hastings proposals, demonstrating improved efficiency and robustness in Bayesian inverse problems with complex models.

Contribution

It extends existing ML-MCMC methods to a broader class of proposals, provides a thorough convergence analysis, and introduces a self-tuning algorithm with demonstrated robustness.

Findings

The proposed ML-MCMC algorithms have a unique invariant measure.

Coupled chains are proven to be uniformly ergodic.

Numerical experiments confirm robustness on challenging posteriors.

Abstract

In this work, we present, analyze, and implement a class of Multi-Level Markov chain Monte Carlo (ML-MCMC) algorithms based on independent Metropolis-Hastings proposals for Bayesian inverse problems. In this context, the likelihood function involves solving a complex differential model, which is then approximated on a sequence of increasingly accurate discretizations. The key point of this algorithm is to construct highly coupled Markov chains together with the standard Multi-level Monte Carlo argument to obtain a better cost-tolerance complexity than a single-level MCMC algorithm. Our method extends the ideas of Dodwell, et al. "A hierarchical multilevel Markov chain Monte Carlo algorithm with applications to uncertainty quantification in subsurface flow," \textit{SIAM/ASA Journal on Uncertainty Quantification 3.1 (2015): 1075-1108,} to a wider range of proposal distributions. We present a thorough convergence analysis of the ML-MCMC method proposed, and show, in particular, that (i) under some mild conditions on the (independent) proposals and the family of posteriors, there exists a unique invariant probability measure for the coupled chains generated by our method, and (ii) that such coupled chains are uniformly ergodic. We also generalize the cost-tolerance theorem of Dodwell et al., to our wider class of ML-MCMC algorithms. Finally, we propose a self-tuning continuation-type ML-MCMC algorithm (C-ML-MCMC). The presented method is tested on an array of academic examples, where some of our theoretical results are numerically verified. These numerical experiments evidence how our extended ML-MCMC method is robust when targeting some \emph{pathological} posteriors, for which some of the previously proposed ML-MCMC algorithms fail.