In a previous question I asked if I could scale the likelihood as my MCMC process advanced, to keep the acceptance fraction within a reasonable range (~0.2-0.5). I was told that this is not a valid approach, since doing that meant that the "Markov chain is no longer time homogeneous".
But, what if I used the burn-in stage to find an appropriate scale factor for my likelihood such that the acceptance fraction is reasonable? By "scale factor", I mean simply a real value that multiplies (and thus scales) my likelihood:
lkl_scaled = scale_factor * lkl_original
In this case I wouldn't be changing this factor during the MCMC process from which I later obtain the distributions of the model parameters. I would only do so during the burn-in phase, which I later discard.
I've tried this already and the results are excellent (where the chains get stuck forever with my original likelihood, they properly explore the parameters space with the scaled likelihood). I can't really see nothing wrong with this approach, but I'd like to be sure.
Is this a valid approach? If so, are there any caveats I should be aware of?
PD: I am aware of the existence of parallel tempering MCMC, but this approach is far simpler and it allows me to use other MCMC methods that would otherwise be of no use since the acceptance fraction is generally below 1%.