The major difference with the talks I gave in Montréal, Edinburgh, Warwick or Rimini is the additional experiment we ran with Darren Wraith on the banana targets already used in the “PMC for cosmologist” paper. As for the mixture benchmark used in the paper with Nicolas Chopin, we found that implementing nested sampling by the book, ie based on the lighthouse code provided in the original papers, led to the right value on average but with a lot more variability than for importance sampling solutions (with a comparable number of iterations).
When running this morning on the campus (i.e. wading through water particles without a snorkel), I thought—helped by comments from Olivier Cappé—that the best explanation for nested sampling is one of importance sampling, the weight of points being the prior mass of the current upper likelihood region. Obviously, this is not the whole story since those constrained priors have a smaller support than the posterior, but this may help in a better evaluation of nested sampling.