  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.
