two of them start agreeing after 5000 steps in the Markov chain, at which point RBM is no better than ABM. This agrees with our theoretical results which indicated that the large sample behavior of both methods is similar. Even in this simple Bayesian model, the finite sample gains of using the RBM estimator are undeniable.

Discussion
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Work in MCMC output analysis has so far not caught up to the ease with which users can implement parallel MCMC runs. This work is a first attempt at consistent estimation of $\Sigma$ for parallel MCMC. For fast mixing Markov chains, the RBM estimator is as good as the current state-of-the-art, and for slow mixing chains, the RBM estimator provides significant improvement and safeguards users from premature termination of the process. Choosing a batch size is a challenging problem for single-chain batch means, and the optimal batch size of [@liu2018batch] is a step in the right direction. Since each realization of a Markov chain can yield different optimal batch sizes, an important future extension would be to construct RBM estimators from pooling batches of different batch sizes, each tuned to its own Markov chain. Additionally, here we consider only non-overlapping batch means 