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Under mild assumptions, we show the exact convergence rate in total variationis also exact in weaker Wasserstein distances for the Metropolis-Hastingsindependence sampler. We develop a new upper and lower bound on the worst-caseWasserstein distance when initialized from points. For an arbitrary pointinitialization, we show the convergence rate is the same and matches theconvergence rate in total variation. We derive exact convergence expressionsfor more general Wasserstein distances when initialization is at a specificpoint.  Using optimization, we construct a novel centered independent proposal todevelop exact convergence rates in Bayesian quantile regression and manygeneralized linear model settings. We show the exact convergence rate can beupper bounded in Bayesian binary response regression (e.g. logistic and probit)when the sample size and dimension grow together.

Exact Convergence Analysis for Metropolis-Hastings Independence Samplers in Wasserstein Distances