**H**ere is a response from Robert Salomone following my comments of the earlier day (and pointing out I already commented the paper two years ago):

You may be interested to know that we are at the tail end of carrying out a major revision of the paper, which we hope will be done in the near future — there will be some new theory (we are in the final stages for a consistency proof of the ANS-SMC algorithm with new co-author Adam Johansen), as well as new numerics (including comparisons to Nested Sampling), and additional discussion that clarifies the overall narrative.A few comments relating your post that may clear some things up:

- The method you describe with the auxiliary variable is actually one of three proposed algorithms. We call this one “Improved Nested Sampling” as it is the algorithm most similar to the original Nested Sampling. Two further extensions are the adaptive SMC sampler, and the fixed SMC sampler – the latter of which is provably consistent and unbiased for the model evidence (we also often see improvements over standard NS for similar computational effort when MCMC is used).
- Regarding computational effort – it is the same for Improved NS (in fact, you can obtain the standard Nested Sampling evidence estimate from the same computational run!). For the adaptive variant, the computational effort is roughly the same for ρ = e⁻¹. In the current version of the paper this is only discussed briefly (last page of p.23). However, in the revision we will include additional experiments comparing the practical performance.
- Regarding the question of “why not regular SMC”; we chose to focus more on why SMC is a good way to do Nested Sampling rather than why Nested Sampling is a good way to do SMC. Our main priority was to show there is a lot of opportunity to develop new nested sampling style algorithms by approaching it from a different angle. That said, Nested Sampling’s primary advantage over standard SMC seems to be in problems involving “phase transitions’’ such as our first example, for which temperature based methods are inherently ill-suited (and will often fail to detect so!).