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I have a Gibbs sampling algorithm, for which I would like to estimate burn-in time. The model isn't hugely complex, and I run sampling for 1000 iterations.

One approach I took was tracking the running mean/standard deviation with increasing samples. This showed me that approximately 100 samples were needed before the mean/standard deviation converged/stabilised, suggesting a burn in of 100 samples.

I then turned to some more formal measures, such as the Geweke diagnostic and the Gelman-Rubin diagnostic. Both of those seem to tell me that the even without dropping any burn-in observations, we have convergence.

This leaves me with two questions. Firstly, how can MCMC converge instantaneously so that I don't need any burn-in? Secondly, how can I reconcile the observations when tracking running mean/standard deviation with the results from the more formal convergence metrics?

And a side question - with Geweke's diagnostic, I find that some high burn-in values sometimes have a smaller z-score than lower burn-in values. For example, burning the first 500 samples yields worse convergence than say burning the first 200 samples. Why might this be the case?

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    $\begingroup$ 1) If your MCMC initializes near the distribution, then no burn-in time is needed. Burn-in time is required to move towards high-density areas of your distribution. 2) Tracking the running mean/standard deviation seems a bit odd to determine burn-in time. Let's say the the burn-in time requires 100 iterations. The running mean/standard deviation will not stabilize until well beyond the 100th iteration. Lastly, there's no need to determine the exact number of burn-in iterations required. There's nothing wrong with being more conservative and throwing out more burn-in iterations. $\endgroup$
    – JLinsta
    Commented Nov 13, 2023 at 20:53
  • $\begingroup$ These various criteria are mostly ad-hoc and should not be taken at face value, as they tend to be widely conservative. Unless they aren't for missing some modes of the target distribution. As we joke in our book, "what you get is what you see", in that it is usually impossible to infer beyond the region explored by the MCMC algorithm. $\endgroup$
    – Xi'an
    Commented Nov 13, 2023 at 21:06
  • $\begingroup$ Your monitoring of cumulated sums is the CUSUM criterion of Cowles et al. (1996). $\endgroup$
    – Xi'an
    Commented Nov 13, 2023 at 21:09
  • $\begingroup$ @JLinsta Thanks for the reply. 1) How might it be the case that MCMC initialises near the distribution each time? For example, with Gelman-Rubin, I used 4 different chains with different random seeds, and yet the convergence was practically immediate (even a burn-in of 1 sample yielded a GR = 1.004). 2) Ah yes of course I was overlooking that fact. However, if as GR/Geweke predict, the convergence is almost instantaneous, then surely the mean and st.dev. convergence should be very quick too? $\endgroup$
    – Blahblahblacksheep
    Commented Nov 13, 2023 at 22:52
  • $\begingroup$ @Xi'an I'm not quite sure what you mean by the various criteria being widely conservative, since GR and Geweke seem to suggest almost instantaneous convergence for me? $\endgroup$
    – Blahblahblacksheep
    Commented Nov 13, 2023 at 22:57

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