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|Posterior = Likelihood × Prior ÷ Evidence|
The nested sampling algorithm is a computational approach to the Bayesian statistics problems of comparing models and generating samples from posterior distributions. It was developed in 2004 by physicist John Skilling.
Bayes' theorem can be applied to a pair of competing models and for data , one of which may be true (though which one is unknown) but which both cannot be true simultaneously. The posterior probability for may be calculated as:
The prior probabilities and are already known, as they are chosen by the researcher ahead of time. However, the remaining Bayes factor is not so easy to evaluate, since in general it requires marginalizing nuisance parameters. Generally, has a set of parameters that can be grouped together and called , and has its own vector of parameters that may be of different dimensionality, but is still termed . The marginalization for is
and likewise for . This integral is often analytically intractable, and in these cases it is necessary to employ a numerical algorithm to find an approximation. The nested sampling algorithm was developed by John Skilling specifically to approximate these marginalization integrals, and it has the added benefit of generating samples from the posterior distribution . It is an alternative to methods from the Bayesian literature such as bridge sampling and defensive importance sampling.
Here is a simple version of the nested sampling algorithm, followed by a description of how it computes the marginal probability density where is or :
Start with points sampled from prior. for to do % The number of iterations j is chosen by guesswork. current likelihood values of the points; Save the point with least likelihood as a sample point with weight . Update the point with least likelihood with some Markov chain Monte Carlo steps according to the prior, accepting only steps that keep the likelihood above . end return ;
At each iteration, is an estimate of the amount of prior mass covered by the hypervolume in parameter space of all points with likelihood greater than . The weight factor is an estimate of the amount of prior mass that lies between two nested hypersurfaces and . The update step computes the sum over of to numerically approximate the integral
In the limit , this estimator has a positive bias of order  which can be removed by using instead of the in the above algorithm.
The idea is to subdivide the range of and estimate, for each interval , how likely it is a priori that a randomly chosen would map to this interval. This can be thought of as a Bayesian's way to numerically implement Lebesgue integration.
Example implementations demonstrating the nested sampling algorithm are publicly available for download, written in several programming languages.
Since nested sampling was proposed in 2004, it has been used in many aspects of the field of astronomy. One paper suggested using nested sampling for cosmological model selection and object detection, as it "uniquely combines accuracy, general applicability and computational feasibility." A refinement of the algorithm to handle multimodal posteriors has been suggested as a means to detect astronomical objects in extant datasets. Other applications of nested sampling are in the field of finite element updating where the algorithm is used to choose an optimal finite element model, and this was applied to structural dynamics. This sampling method has also been used in the field of materials modeling. It can be used to learn the partition function from statistical mechanics and derive thermodynamic properties. 
Dynamic nested sampling is a generalisation of the nested sampling algorithm in which the number of samples taken in different regions of the parameter space is dynamically adjusted to maximise calculation accuracy. This can lead to large improvements in accuracy and computational efficiency when compared to the original nested sampling algorithm, in which the allocation of samples cannot be changed and often many samples are taken in regions which have little effect on calculation accuracy.
Publicly available dynamic nested sampling software packages include:
Dynamic nested sampling has been applied to a variety of scientific problems, including analysis of gravitational waves, mapping distances in space and exoplanet detection.
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