see also part I.
Nested sampling as an algorithm has a few papers presenting convergence proofs. It’s interesting from how many angles people have looked at this problem. See the relevant section in the literature review.
Diagnostics (ongoing)
Andrew Fowlie proposed an elegant test for diagnosing live whether nested sampling is sampling in an unbiased fashion, by tallying where in the likelihood-ordered list of live points the replacement point lands (I tend to confuse the wods order and rank). A variant is the statistically more powerful U-test ( section 4.5.2. of Buchner 2023). But ultimately, neither of these are as sensitive as you would wish.
In practice, the most powerful technique is still observing the log(Z) value across reruns, e.g., when increasing the number of steps of the slice sampling procedure.
More research is still needed on diagnostics. I suspect that the gradients of the dead points can be compared to the growth of the L-V curve and that could be more powerful.
In the meantime, I transferred MCMC diagnostics to nested sampling, specifically the Jump Distance which traces the first-order autocorrelation length of a random walk. This research appears to be trivial for statistics journals and too non-astronomy for astronomy journals.
Plauteaus (solved?)
When the likelihood has plateaus, the original algorithm had an issue. In practice, this mostly occurs if your log-likelihood function returns a fixed number such as -1e300 when the parameters are invalid. Pragmatically, one can make that penalty slant towards the good region. More generally, an auxiliary with a tiny likelihood slant is a solution to plateaus, but in any case the nested sampling variant where all points with the same lowest likelihood value are removed has been proposed and should be the standard nested sampling method people implement (it is in UltraNest).
Live point inheritance
In most implementations, the dead point is replaced by sampling a new point, starting from another live point. This is true for ellipsoidal nested sampling and MCMC-based replacement schemes.
In multi-modal settings, removing points can remove all points of a mode, and this was discussed early in the literature, mostly casually in conference proceedings (which I cannot find anymore, help!).
Recently, I used analysis techniques from genetics to analyse the occurrence probability of mode die-out in nested sampling. The result is not analytic, but gives a simple rule of thumb, which I think is basically always fulfilled. So: do not worry about this!
End-to-end proofs
A limitation of current theoretical analyses is that most of these assume faithful likelihood-restricted prior sampling (LRPS). How can we go beyond this?
Brendon Brewer solved this by embedding nested sampling within an MCMC framework. Fine, but most people do not use nested sampling that way.
For nested sampling with step samplers, a key paper is “Unbiased and Consistent Nested Sampling via Sequential Monte Carlo” by Queensland University of Technology researchers Robert Salomone, Leah South et al., which present a proof by placing nested sampling within the sequential nested sampling umbrella of well-built-out theoretical analysis. There is a hole in that paper, which they are clear about: In SMC, there is a kernel which refreshes all points, while in NS, only one specific point is replaced. And so the connection is as of yet almost, but not quite, there.
For nested sampling with region samplers, I just put out a theory paper analysing the replacement sampling space with Binomial point processes, which is the exact right tool for the job. This analysis is on the path to a end-to-end convergence proof of nested sampling, or at least bounding the errors.
The nice aspect of convergence analyses is that once you have them, you know what you can tune. It does not work the other way around …
By my recent count, there are over 6000 citations to the top 4 nested samplng packages. As machine learning becomes more prevalent, nested sampling analyses are used as the ground truth to compare against. Both of these facts should be justification to thoroughly understand nested sampling. Now we just need astronomers, statisticians and computer scientists to allocate funding for this research.
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