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!
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2 Answers
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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:
http://www.people.fas.harvard.edu/~plam/teaching/methods/convergence/convergence_print.pdf or
http://www.stat.duke.edu/courses/Fall10/sta290/Lectures/Diagnostics/param-diag.pdf
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.
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1$\begingroup$ Those diagnostics do not tell you what you really want to know. They say that if the chain converges, then the diagnostic will probably say it converges, but they do not say that if the chain pseudo-converges, then the diagnostic will probably say that the chain did not converge. Those that claim to reliably diagnose pseudo-convergence have unverifiable conditions that make them useless. $\endgroup$– GlenJul 26, 2013 at 16:43
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$\begingroup$ As I said, they are only indicators, not guarantees. I actually don't see any way to actually diagnose pseudo-convergence. To do that, you'd have to know the true distribution already, but then why do MCMC? $\endgroup$– WayneJul 26, 2013 at 18:52
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1$\begingroup$ Correct, just wanted to emphasize that point. $\endgroup$– GlenJul 26, 2013 at 23:44
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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.
