Computation can pose a major challenge in Bayesian inverse problems when partial differential equations (PDEs) are involved. In particular, the effectiveness of Markov chain Monte Carlo (MCMC) can be quite limited. This talk presents a solution called Stein Thinning, that (1) automatically identifies when MCMC has or has not converged, (2) performs bias-removal for biased MCMC, and (3) provides compression of MCMC output. The challenge of scaling Stein Thinning to high-dimensional parameter spaces will be highlighted.