# Why do we want low autocorrelation for MCMC convergence?

Usually, autocorrelation is one diagnostical tool for judging the convergence of a MCMC trail. Low autocorrelation is desired as this would mean that the parameter space is well explored.

I have a real struggle with this. Assume that we have a lot of data and the highest density interval of the posterior is very narrow. Thus, most of the density would fall onto a small range of parameters. If we derive this from a MCMC chain, this would mean that the parameters of the steps of the chain would not differ much. From my understanding, this would also imply that the autocorrelation is high.

What am I misunderstanding here?

• Suppose my posterior is very concentrated on $\theta=0.5$, then my samples drift towards this parameter and don't change much anylonger. Maybe I am really misunderstanding something completely, but the auotocrrelation for such a sequence would be high instead of low. Commented Jul 8, 2015 at 21:11
• @ManuHaq: The easiest way to see this is looking at the variance of the sum of two random variables: $V[X + Y] = V[X] + V[Y] + Cov[X, Y]$. Note that the variance is higher if they are positively correlated, the variance of the sum is higher than if they were not. Since we use the average (which is the sum times a constant) of all the MCMC samples, if they are positively correlated, the average will have a higher variance than if they were independent. Commented Oct 29, 2021 at 1:32