In MCMC, how is burn-in time chosen? In other words, how long do you need to wait before you think the Markov chain has reached its limiting distribution? Thanks!
There are several diagnostics, including the Geweke Diagnostic, the Heidelberg and Welch Diagnostic, the Raftery and Lewis Diagnostic, and the Gelman and Rubin Multiple Sequence Diagnostic. Also, visual examination of the trace plot can help. All of these are only indications, not guarantees.
You might check out:
EDIT: Also, you cannot determine the burn-in length in advance. You look at your run -- as suggested above -- and if it looks like things have converged by the end of your burn-in, the burn-in you did is long enough.
I would run the MCMC many times (with different starting values) and plot the log-likelihood along with parameter estimates across time (or iteration number). Hopefully you see a trend for what the iteration number is for the chain to enter the stationary distribution. I would then use this value (and add a little more to be conservative) as the burn-in time.
Of course there is no guarantee this will work across all scenarios, or that you have entered the true stationary distributions in your simulations. Therefore this advice should be taken with a grain of salt.