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When we say acceptance rate in a MH algorithm (for examples, suppose it's recommended to have 40% acceptance rate), do we mean overall acceptance rate or we mean acceptance rate after burn in? It's a very simple question but I just realized that I never thought about it.

Thanks very much,

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    $\begingroup$ After the burn in period. $\endgroup$
    – Glen_b
    May 12 '14 at 8:43
  • $\begingroup$ Asymptotically, it does not matter, Birkhoff's ergodic theorem applies to both rates. $\endgroup$
    – Xi'an
    Feb 26 '16 at 11:59
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I would say that in general, we want the realized acceptance rate after the burn-in period to correspond to the targeted acceptance rate. Consider for example using an adaptive Metropolis sampler where the scale (or generally covariance matrix) of the proposal distribution is adjusted during the burn-in phase to achieve the targeted acceptance rate. Why would you want the samples during burn-in to be included in the calculation of the acceptance rate though they are discarded and not part of the posterior?

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    $\begingroup$ What if you believe burn-in is unnecessary? See Charlie Geyer's MCMC rants: http://users.stat.umn.edu/~geyer/mcmc/burn.html $\endgroup$
    – Donnie
    May 27 '14 at 12:49
  • $\begingroup$ Interesting point. If the proposal distribution is not altered and stationarity given, I have no argument why the samples of the burn-in phase should not be used to compute the acceptance rate. $\endgroup$ May 31 '14 at 20:02

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