# Use only the last sample as the posterior in MCMC

I am new to statistics.

After an MCMC sampler warmed up, the posterior is better estimated as the mean of several samples. (e.g. related question: https://stats.stackexchange.com//questions/56077)

However, in my MCMC sampler, after it reaches the equilibrium (because $\hat{R}=1.0$ in Stan), samples of the parameters still present a fairly large STD. Do the large STD means that the chain hasn't converged yet, even if $\hat{R}=1.0$?

In this case, I do not want to use the mean of the parameter samples, but only use one sample (say the 1000th) of the parameters as the posterior. Is it the right way to do?

P.S. "The $\hat{R}$ statistic measures the ratio of the average variance of samples within each chain to the variance of the pooled samples across chains, if all chains are at equilibrium, these will be the same and $\hat{R}$ will be one" -- Stan Manual

The key part from the definition of $\hat{R}$ is that this metric is a ratio of variance: we have evidence that the chain has converged to its stationary distribution because the ratio of variance in independent chains is roughly 1. The ratio can be 1 even if the variance is large in absolute terms.
• Thanks very much, this answer really helps! There is just one point I'm still a bit confused : $\hat{R}=1$ means samples of different chains have similar variances (but the absolute variance value could be large), so why we call such a state 'stationary'? I mean, can the absolute variance shrink if we continue sampling? – Shockley Aug 1 '14 at 18:01