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

Under mild assumptions, we show the sharp convergence rate in total variation is also sharp in weaker Wasserstein distances for the Metropolis-Hastings independence sampler. We derive exact convergence expressions for general Wasserstein distances when initialization is at a specific point. Using optimization, we construct a novel centered independent proposal to develop exact convergence rates in Bayesian quantile regression and many generalized linear model settings. We show the exact convergence rate can be upper bounded in Bayesian binary response regression (e.g. logistic and probit) when the sample size and dimension grow together.