Nested sampling algorithm

The nested sampling algorithm is a computational approach to the problem of comparing models in Bayesian statistics, developed in 2004 by physicist John Skilling.^[1]

Background

Bayes' theorem can be applied to a pair of competing models M1 and M2 for data D, one of which may be true (though which one is not known) but which both cannot simultaneously be true, as follows:

$\begin{align} P(M1|D) & {} = \frac{P(D|M1) P(M1)}{P(D)} \\ & {} = \frac{P(D|M1) P(M1)}{P(D|M1) P(M1) + P(D|M2) P(M2)} \\ & {} = \frac{1}{1 + \frac{P(D|M2)}{P(D|M1)} \frac{P(M2)}{P(M1)} } \end{align}$

Given no a priori information in favor of M1 or M2, it is reasonable to assign prior probabilities P(M1) = P(M2) = 1 / 2, so that P(M2) / P(M1) = 1. The remaining ratio P(D | M2) / P(D | M1) is not so easy to evaluate since in general it requires marginalization of nuisance parameters. Generally, M1 has a collection of parameters that can be lumped together and called θ, and M2 has its own vector of parameters that may be of different dimensionality but is still referred to as θ. The marginalization for M1 is

$P(D|M1) = \int d \theta P(D|\theta,M1) P(\theta|M1)$

and likewise for M2. 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 P(θ | D,M1).^[2] It is an alternative to methods from the Bayesian literature^[3] 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 Z = P(D | M) where M is M1 or M2:

  Start with N points θ1,...,θN sampled from prior.
  for i = 1 to j do        % The number of iterations j is chosen by guesswork.
      Li: = min(current likelihood values of the points);
      Xi: = exp( − i / N);
      wi: = Xi − 1 − Xi
      Z: = Z + Li * wi;
      Save the point with least likelihood as a sample point with weight wi.
      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 Li.
  end
  return Z;

At each iteration, X_i is an estimate of the amount of prior mass covered by the hypervolume in parameter space of all points with likelihood greater than θ_i. The weight factor w_i is an estimate of the amount of prior mass that lies between two nested hypersurfaces {θ | P(D | θ,M) = P(D | θ_{i − 1},M)} and {θ | P(D | θ,M) = P(D | θ_i,M)}. The update step Z: = Z + L_i * w_i computes the sum over i of L_i * w_i to numerically approximate the integral

$\begin{array}{lcl} P(D|M) &=& \int P(D|\theta,M) P(\theta|M) d \theta \\ &=& \int P(D|\theta,M) dP(\theta|M)\\ \end{array}$

The idea is to chop up the range of f(θ) = P(D | θ,M) and estimate, for each interval [f(θ_{i − 1}),f(θ_i)], how likely it is a priori that a randomly chosen θ would map to this interval. This can be thought of as a Bayesian's way to numerically implement Lebesgue integration.

Simple example code written in C, R, or Python demonstrating this algorithm can be downloaded from John Skilling's website. There is also a Haskell port on Hackage.

Applications

Since nested sampling was proposed in 2004, it has been used in multiple settings within 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."^[4] A refinement of the nested sampling algorithm to handle multimodal posteriors has also been suggested as a means of detecting astronomical objects in existing datasets.^[5]

See also

• Bayesian model comparison

References

1. ^ Skilling, John (2004). "Nested Sampling". AIP Conference Proceedings 735: 395–405. doi:10.1063/1.1835238. 
2. ^ Skilling, John (2006). "Nested Sampling for General Bayesian Computation". Bayesian Analysis 1 (4): 833–860. doi:10.1214/06-BA127. 
3. ^ Chen, Ming-Hui, Shao, Qi-Man, and Ibrahim, Joseph George (2000). Monte Carlo methods in Bayesian computation. Springer. ISBN 9780387989358. http://books.google.com/?id=R3GeFfshc7wC. 
4. ^ Mukherjee, P., Parkinson, D., and Liddle, A.R. (2006). "A Nested Sampling Algorithm for Cosmological Model Selection". Astrophysical Journal 638 (2): 51–54. Bibcode 2005astro.ph..8461M. doi:10.1086/501068. 
5. ^ Feroz, F., Hobson, M.P. (2008). "Multimodal nested sampling: an efficient and robust alternative to Markov Chain Monte Carlo methods for astronomical data analyses". MNRAS 384 (2): 449–463. Bibcode 2008MNRAS.384..449F. doi:10.1111/j.1365-2966.2007.12353.x. http://adsabs.harvard.edu/cgi-bin/bib_query?arXiv:0704.3704.