# Why is it desirable to have low auto-correlation in MCMC?

I keep reading about the need to check for autocorrelation in MCMC. Why is it important that the autocorrelation is low? What does it measure in the context of MCMC?

• In fact, if one could produce high negative autocorrelation in an MCMC sampler, this sampler would improve upon iid sampling. This is however a very rare occurrence... – Xi'an Dec 27 '14 at 16:47

• I don't have much background with MCMC but your last sentence doesn't seem oversimplifying. If you look at the effect of auto-correlations on your error estimates they change the value from $\Delta A^² = \frac{\text{Var} A}{N}$ to $\Delta A^² = \frac{\text{Var} A}{N}(1+2\tau)$ where $\tau$ is the autocorreltion time measured on the same $A$ observables. So it is like having only $\frac{N}{1+2\tau}$ 'effective measurement' instead of $N$. Is there still some oversimplification in this statement? – Learning is a mess Aug 11 '14 at 17:32