  Bayesian inference with nested sampling requires a likelihood-restricted
prior sampling method, which draws samples from the prior distribution that
exceed a likelihood threshold. For high-dimensional problems, Markov Chain
Monte Carlo derivatives have been proposed. We numerically study ten algorithms
based on slice sampling, hit-and-run and differential evolution algorithms in
ellipsoidal, non-ellipsoidal and non-convex problems from 2 to 100 dimensions.
Mixing capabilities are evaluated with the nested sampling shrinkage test. This
makes our results valid independent of how heavy-tailed the posteriors are.
Given the same number of steps, slice sampling is outperformed by hit-and-run
and whitened slice sampling, while whitened hit-and-run does not provide as
good results. Proposing along differential vectors of live point pairs also
leads to the highest efficiencies, and appears promising for multi-modal
problems. The tested proposals are implemented in the UltraNest nested sampling
package, enabling efficient low and high-dimensional inference of a large class
of practical inference problems relevant to astronomy, cosmology, particle
physics and astronomy.
