in some sense, represents the oracle. The results are presented in Table \[table:MVG\_coverage\_m\_5\_10\]. For Setting 1 (for both $m = 5,10$), ABM and RBM yield similar coverage, as was expected. Here, the high variability of the naive estimator impacts the coverage significantly which does not improve as $n$ increases, demonstrating the effect of Theorem \[thm:naive\_convergence\]. For Setting 2, there is a clear separation of coverage probabilities between RBM and ABM, especially for smaller sample sizes. As the sample size increases, ABM and RBM yield more similar coverage as a consequence of less separation between the means from each chain.

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In the rest of the examples, we focus only on comparing the RBM to the ABM estimator, since it is evident that the naive estimator is unreliable.

Rosenbrock distribution
-----------------------

The Rosenbrock distribution is commonly used to serve as a benchmark for testing MCMC algorithms. As can be seen in Figure \[fig:rosenbrock\_contour\], the density takes positive values along a narrow parabola making it difficult for MCMC algorithms to take large steps. Consider the 2-dimensional Rosenbrock density discussed in [@goodman2010] and [@pagani2019ndimensional]: $$\pi (x_{1}, x_{2}) \propto \exp \left\{-\dfrac{1}{20}(x_{1} - 1)^{2} - 5 \left(x_{2} - x_{1}^{2} \right)^{2} \right\} \,.$$ Although there are 