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?
Autocorrelation is a measure of how much the value of a signal correlates to other values of that signal at different points in time. In the context of MCMC, autocorrelation is a measure of how independent different samples from your posterior distribution are – lower autocorrelation indicating more independent results.
When you have high autocorrelation the samples you've drawn don't accurately represent the posterior distribution and therefore don't provide as meaningful information for the solution to the problem. In other words, lower autocorrelation means higher efficiency in your chains and better estimates. A general rule would be that the lower your autocorrelation, the less samples you need for the method to be effective (but that might be oversimplifying).