%0 Journal Article %1 Henderson2014 %A Henderson, R. Wesley %A Goggans, Paul M. %B AIP Conference Proceedings %D 2014 %E Niven, Robert K. %E Brewer, Brendon %E Paull, David %E Shafi, Kamran %E Stokes, Barrie %J AIP Conference Proceedings %K Computing, MCMC Nested Parallel Sampling, %N 1 %P 100--105 %R 10.1063/1.4903717 %T Parallelized Nested Sampling %V 1636 %X 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.

@article{Henderson2014, 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 \citep{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 \citep{skilling2012} for computing the expectation of the shrinkage distribution: \[ \mathrm{E}[-\log t] = (N_r - r + 1)^{-1} + (N_r - r + 2)^{-1} + \cdots + N_r^{-1} \mathrm{,} \] for shrinkage $t$ with $N_r$ live samples and $r$ samples discarded at each iteration. The equation for the variance \[ \mathrm{Var}(-\log t) = (N_r-r+1)^{-2} + (N_r-r+2)^{-2} + \cdots + N_r^{-2} \] 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.}, added-at = {2019-03-04T22:26:50.000+0100}, author = {Henderson, R. Wesley and Goggans, Paul M.}, biburl = {https://www.bibsonomy.org/bibtex/2daa637bfefa851a14cbd962a91b5598c/rwhender}, booktitle = {AIP Conference Proceedings}, doi = {10.1063/1.4903717}, editor = {Niven, Robert K. and Brewer, Brendon and Paull, David and Shafi, Kamran and Stokes, Barrie}, file = {:Parallelized Nested Sampling Henderson Goggans Revised.pdf:PDF}, interhash = {cae95874f00b8c52954f4462730d1333}, intrahash = {daa637bfefa851a14cbd962a91b5598c}, journal = {AIP Conference Proceedings}, keywords = {Computing, MCMC Nested Parallel Sampling,}, number = 1, pages = {100--105}, timestamp = {2019-03-04T22:29:38.000+0100}, title = {Parallelized Nested Sampling}, volume = 1636, year = 2014 }