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There are a few strategies for selecting the values of the temperatures (or betas, where $\beta=1/T$) in a parallel tempering MCMC (geometric, adaptive, aimed at a 0.234 temperature swap acceptance rate). What I have not found is a strategy for selecting the number of temperatures (replicas) that one should use.

Naively one could assume that increasing the number of temperatures as much as possible is better. However in Atchadé et al. (2011) the authors show an example where using either too few or too many temperatures is almost equally inefficient (with the addition that using too many increases the computation time). So there is a number of temperatures that gives better results, at least better mixing.

Is there such a strategy?

Gabriel
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    Temperature leves must be "close enough" to allow reasonable movement between neighbour tempered targets, which in turn determines the number of temperature levels. Too many temperatures restrict _in fine_ the move of the Markov chain associated with the original target because it multiplies the opportunities for random walk tracing back behaviour. – Xi'an Dec 20 '18 at 09:46
  • Thanks @Xi'an. You say "*which in turn determines the number of temperature levels*" but I don't see how I would go about fixing this value. To ensure that temperatures are "close enough", I could just increase their number. But as you said, this hurts the sampler's efficiency. Assuming I now the maximum temperature value I should use, and assuming I use one of the strategies mentioned in the question to select the remaining values, is there a recipe/strategy for how many temperatures I should use? – Gabriel Dec 20 '18 at 15:19
  • Start with the highest temperature, compute the effective sample size as a function of the second highest temperature, select this second highest temperature to keep the ESS above a lower bound like 20%, resample and move with this second highest temperature, compute the effective sample size as a function of the third highest temperature, select this third highest temperature to keep the ESS above a lower bound like 20%, &tc... – Xi'an Dec 21 '18 at 07:15

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