Here is an email I received from Umberto:
I have a doubt regarding the tempered transitions method you considered in your JASA article with Celeux and Hurn.
On page 961 you detail the several steps for building a proposal for a given distribution by simulating through l tempered power densities. I am slightly confused regarding the interpretation of your MCMC(x,π) notation.
For example does means that an MCMC procedure starting at yl, say Metropolis-Hastings, is used to generate a single proposal yl+1 for ?
If this is the case, then yl+1 might be rejected or accepted and in the former case I would have yl+1=yl right? In other words I am not required to simulate proposals using until I finally accept yl+1.
By reading the last paragraph in page 962 it seems to me that, indeed, the y1,…,y2l-1 thus generated are not necessarily accepted proposals for the corresponding power densities.
In retrospect, I still like this MCMC(x,π) notation in the simulated tempering “up-and-down” scheme (and the paper!). Because it is generic, in the sense of an R function that would take the function MCMC(x,π) as its input. To clarify the notation in this light, MCMC(x,π) returns a value that is the outcome of the corresponding MCMC step. This value may be equal to x (MCMC rejection) or to another value (MCMC acceptance). So the sequence y1,…,y2l-1 is made of consecutive values that differ and of consecutive values that do not (it is even possible that all the terms in the sequence are equal). At the end of this “up-and-down” tempering, the value y2l-1 may be the next value of the Markov chain targeted at the original target π. Or the current value may be replicated. This depends on the overall acceptance probability (4) on page 961. (Following Neal, 1996, Statistics and Computing.) This is a very compelling idea, whose mileage may vary depending on the number of required steps and powers.