10.9 Monte Carlo standard error

Even if we have a very low and a very high value of ESS. The samples from MCMC are still finite, and thus we are introducing an error in the estimation of the posterior parameters. Fortunately, we can estimate the error, and it is called the Monte Carlo Standard Error (MCSE). The estimation of the MCSE takes into account that the samples are not truly independent of each other. The precision we want in our results is limited by this value. If the MCSE for a parameter is 0.2, it does not make sense to report a parameter as 2.54. Instead, if we repeat the simulation (with a different random seed), we should expect that for 68% of the results, we obtain values in the range 2.54 ± 0.2. Similarly, for 95% of them, we should get values in the range 2.54 ± 0.4. Here, I am assuming the MCSE distributes normally and then using the fact that ≈ 68% of the value of a Gaussian is within one standard deviation and ≈ 95% is within two standard...