I use MCMC (Metropolis-Hastings) to sample posterior distributions of three parameters using a nonlinear least-squares objective function to calculate the likelihood of a parameter sets.
I want posteriors for about 500 different conditions, each time I calculate a posterior with MCMC.
Since it is hard to check all 500 runs by hand for convergence, I would like to have a simple rule based on autocorrelation which indicates for each run if the chain converged or not, checking all conditions individually would be very cumbersome.
Each chain has length 40000 (burn-in already removed) and I calculated autocorrelations for lags 1 to 10000.
I then calculated the autocorrelation time as the time constant for the autocorrelation series (the first lag when the autocorrelation drops below 1/exp(1)).
I get autocorrelation times between 16 and about 7000.
I suppose that with an autocorrelation time of 16 I can be sure that the Markov chain converged, but is there maybe some "rule of thumb" which tells me, given a chain of length 40000, which autocorrelation time is still "acceptable"?
I searched and looked for instance here, however, the approaches I found require a visual investigation.
Best Answer
Have you looked at the standard tests for convergence?
The Geweke's convergence diagnostic works with a single chain and provides a z-score. There is also the Heidelberger and Welch's convergence diagnostic, which also works with a single chain and gives a p-value of the null hypothesis that the sampled values come from a stationary distribution. If you run multiple chains, there's Gelman and Rubin's convergence diagnostic. If you use R, all of these are implemented in the
codapackage.