“…the likelihoods of discarded points have interesting properties. In particular, the fraction of prior mass below the likelihood threshold is approximately 1/K [number of particles].”
I came across a newly arXived paper on nested sampling, written by Johannes Buchner, with a focus on sampling over the constrained space defined by the lower bound on the likelihood value, and promoting different manners to implement a slice sampler in this possibly complex space.
“After a number of Metropolis steps, by which points with lower likelihood than required are not visited, a useful independent prior sample is obtained. This is only the case if enough steps are made, such that the random walk can reach all of the relevant volume.”
A slice sampler for nested sampling means (1) picking a direction v and (2) deriving the length of the slice by bisection, before (3) sampling uniformly over the interval. I do not get why the slice should necessarily be connected, rather than made of several segments for multimodal likelihoods. Ten versions are opposed when selecting the direction! With some missing the detailed balance property.
The comparison between these different slice samplers makes use of a shrinkage test proposed by the author in Statistics & Computing (2014), monitoring convergence by evaluating the volume ratio distribution of a sequence of discarded samples produced by nested sampling. Namely a test (on which I had reservations, blogged at the time) for following the decrease in volume predicted by the Uniform order statistics. Now, I have trouble understanding the calibration figures (like the one above) that are at the core of the paper towards ranking the ten versions…
