  A Bayesian coreset is a small, weighted subset of data that replaces the full
dataset during inference in order to reduce computational cost. However, state
of the art methods for tuning coreset weights are expensive, require nontrivial
user input, and impose constraints on the model. In this work, we propose a new
method -- Coreset MCMC -- that simulates a Markov chain targeting the coreset
posterior, while simultaneously updating the coreset weights using those same
draws. Coreset MCMC is simple to implement and tune, and can be used with any
existing MCMC kernel. We analyze Coreset MCMC in a representative setting to
obtain key insights about the convergence behaviour of the method. Empirical
results demonstrate that Coreset MCMC provides higher quality posterior
approximations and reduced computational cost compared with other coreset
construction methods. Further, compared with other general subsampling MCMC
methods, we find that Coreset MCMC has a higher sampling efficiency with
competitively accurate posterior approximations.
