Typo in Bayesian Core

Francisco García from Santiago, Chile, sent me the following email about Bayesian Core:

I’m trying to implement algorithm 6.5 “Simulated Tempering by Pumping” from your book Bayesian Core.

x(t)1 is generated in line 1 from MCMC(x(t),1), x(t)2 is generated in line 2 from MCMC(x(t),2),  and so on.  There is a correspondence between the line number and the order of the generated value x(t)i.

But in the line 2p-1 it says “generate x(t)2p from MCMC(x(t)2p1,1)”.  It seems it should say “generate x(t)2p-1 from MCMC(x(t)2p2,1)”

Am I right? If so, Is line 2p: “generate x(t)2p from MCMC(x(t)2p1, )”?

Indeed, this is a typo, the last item of the algorithm should be 2p not 2p-1. In addition, the last ratio in the acceptance probability should be

\frac{\displaystyle{\pi(x^{(t)}_{2p-1})}}{\displaystyle{\pi_{\alpha_1}(x^{(t)}_{2p-1})}}

the power used on page 169 being a special case of indexing in the tempering sequence. We surely should not have made the mistake pointed out by Francisco as tempering requires an even number of steps to cancel the normalising constants of the tempered targets.

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