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Nested sampling (nest)
Nested sampling, introduced by Skilling (2004), is
fundamentally different from the three MCMC samplers above. Rather
than drawing from the posterior directly, it maintains a set of
“live points” distributed according to the prior and
iteratively replaces the lowest-likelihood live point with a new draw
from the prior, conditioned on having a higher likelihood than the
one removed. As the algorithm proceeds the live points climb up
the likelihood surface and converge on the posterior modes. The
posterior samples are recovered as a weighted re-sampling of the
removed (“dead”) points; the evidence
 |
(3.3) |
Strengths. Nested sampling is the natural choice when
the goal includes model comparison via Bayes factors. It explores
multi-modal posteriors much more reliably than any MCMC method
because the live points span the entire prior at the start.
It does not require a starting fit; running fit before
nest run has no effect on the sampler at all. The
XSPEC implementation supports multi-process parallelization via
parallel nest, which uses the K-point-removal generalization
of Buchner (2014) to remove live points per iteration
and parallelize the resamplings.
Weaknesses. The per-iteration cost is one or more
constrained prior samples, which can themselves require many
likelihood evaluations. Nested sampling is slower than HMC on smooth
unimodal posteriors but much more reliable on hard ones. The number of
iterations scales as roughly
where 
is the
information gained in going from the prior to the posterior — the
Kullback–Leibler divergence between the two distributions, measured
in nats (the unit of information based on the natural logarithm, as
the bit is based on log base 2; roughly, counts how many
-foldings the posterior volume is compressed relative to the prior
volume). For typical X-ray problems
this is a few thousand iterations and 
live points is
reasonable.
The strongest constraint nested sampling places on the user is that the prior must be properly normalized on a bounded domain. Parameters with default flat priors over many decades will cause nest run to refuse to start; the user must apply jeffreys (with the parameter's hard lower limit pushed above zero by newpar) or another integrable prior before invoking nest.
Tips. Set nest live 
as a default; lower
values risk under-exploring multi-modal regions, higher values cost
proportionally more. Use parallel nest 
where is
roughly the number of physical cores; the speedup is sub-linear but
significant. Allow an elapsed-time budget of order an hour for
6 parameter problems on a single CCD; multi-instrument joint
fits with response convolution dominate by an order of magnitude.