Abstract
One of the important advantages of nested sampling as an MCMC technique
is its ability to draw representative samples from multimodal distributions
and distributions with other degeneracies. This coverage is accomplished
by maintaining a number of so-called live samples within a likelihood
constraint. In usual practice, at each step, only the sample with
the least likelihood is discarded from this set of live samples and
replaced. In skilling2012, Skilling shows that for a given
number of live samples, discarding only one sample yields the highest
precision in estimation of the log-evidence. However, if we increase
the number of live samples, more samples can be discarded at once
while still maintaining the same precision. For computer code running
only serially, this modification would considerably increase the
wall clock time necessary to reach convergence. However, if we use
a computer with parallel processing capabilities, and we write our
code to take advantage of this parallelism to replace multiple samples
concurrently, the performance penalty can be eliminated entirely
and possibly reversed. In this case, we must use the more general
equation in skilling2012 for computing the expectation of
the shrinkage distribution: \ E-t = (N_r - r + 1)^-1
+ (N_r - r + 2)^-1 + + N_r^-1 , \ for shrinkage
$t$ with $N_r$ live samples and $r$ samples discarded at each iteration.
The equation for the variance \ Var(-t) = (N_r-r+1)^-2
+ (N_r-r+2)^-2 + + N_r^-2 \ is used to find the appropriate
number of live samples $N_r$ to use with $r > 1$ to match the variance
achieved with $N_1$ live samples and $r = 1$. In this paper, we show
that by replacing multiple discarded samples in parallel, we are
able to achieve a more thorough sampling of the constrained prior
distribution, reduce runtime, and increase precision.
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