by [@mcmcpack], we set an improper prior for $\beta$, $f(\beta) \propto 1$. The resulting posterior distribution for $\beta$ is intractable and MCMC is needed to estimate the posterior mean. We use the `MCMCmnl` function in the `R` package `MCMCpack` which uses a random walk Metropolis-Hastings sampler with a Gaussian proposal distribution. We run 2 parallel Markov chains with starting values dispersed around the maximum likelihood estimator of $\beta$. The autocorrelation plots of randomly picked elements in Figure \[fig:multinomial\_acf\_plots\] shows significant autocorrelation in the Markov chain.

In this example, both the true posterior mean and the resulting asymptotic covariance matrix $\Sigma$ are unknown. Thus coverage probabilities are not estimable. We, therefore, focus on the quality of estimation of ESS. Figure \[fig:multinomial\_ESS\_running\] contains the running estimate of $\widehat{\text{ESS}}/mn$ using both ABM and RBM estimators. Since the target distribution is unimodal with moderate levels of posterior correlation, it does not take long for the two Markov chains to adequately explore the state space. Even then, there is a noticeable advantage of using RBM over ABM. RBM produces a systematically lower estimate of ESS whereas the ABM estimator may give a false sense of security in early small runs of the sampler. However, the 