specialized MCMC algorithms available that traverse the contours of the Rosenbrock density more efficiently, we implement a random walk Metropolis-Hastings algorithm with a Gaussian proposal. Trace plots for one component of two runs are presented in Figure \[fig:rosenbrock\_contour\].

We set $m = 5,10$ with evenly dispersed starting points. Since $\Sigma$ is unknown, we compare ABM and RBM by analyzing running plots of the estimated effective sample size (ESS). ESS is a popular tool for assessing when to terminate MCMC simulations by determining the number of iid samples that would yield the same standard error for the Monte Carlo estimator. [@vats:fleg:jon:2019] define $$\text{ESS} = mn \left(\dfrac{|{\text{Var}}_F(h(X_{11}))|}{|\Sigma|} \right)^{1/p}\,,$$ where $|\cdot|$ is the determinant. Simulation terminates when an estimated ESS, $\widehat{\text{ESS}} > M$, for a pre-specified $M$. To avoid early termination, it is critical that the estimate of $\Sigma$ is not underestimated.

Figure \[fig:rosenbrock\_ESS\_running\] presents the estimated $\widehat{\text{ESS}}/mn$ over 100 replications using the average sample covariance matrix estimate of ${\text{Var}}_F(h(X_{11}))$ and ABM and RBM estimates of $\Sigma$. We observe that for both $m = 5,10$, ABM grossly overestimates $\widehat{\text{ESS}}/mn$ and converges from above, whereas RBM converges from below. As a consequence, using RBM will help safeguard against early termination, whereas ABM is significantly 