state space. Trace plots in Figure \[fig:MVG\_trace\] give evidence of this.

We set $m = 5,10$ for both the settings and first compare the evolution of the estimates of $\Sigma$ over time. Figure \[fig:MVG\_FN\_running\] presents the running plots of the Frobenius norm of the estimators with the black horizontal line being the truth. In all four plots, the high variability of the naive estimator is demonstrated, consistent with Theorem \[thm:naive\_convergence\]. For Setting 1, RBM and ABM are indistinguishable as was expected from the trace plots. However, the advantage of using RBM is evident in Setting 2, where the RBM estimator converges to the truth quicker than the ABM estimator. This is a direct consequence of the fact that slowly mixing Markov chains take a longer time to traverse the state space, implying that the means of all the chains are significantly different in small MCMC runs.

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In order to quantify the effect of the quality of estimation of $\Sigma$, we estimate the coverage probabilities of the resulting $95\%$ confidence regions over 1000 replications for both settings. In addition to the coverage probabilities obtained using ABM, naive, and RBM, we also estimate the coverage probability using the true $\Sigma$; this, 