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Nested sampling for physical scientists

Nature Reviews Methods Primers volume 2, Article number: 39 (2022) Cite this article

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Abstract

This Primer examines Skilling’s nested sampling algorithm for Bayesian inference and, more broadly, multidimensional integration. The principles of nested sampling are summarized and recent developments using efficient nested sampling algorithms in high dimensions surveyed, including methods for sampling from the constrained prior. Different ways of applying nested sampling are outlined, with detailed examples from three scientific fields: cosmology, gravitational-wave astronomy and materials science. Finally, the Primer includes recommendations for best practices and a discussion of potential limitations and optimizations of nested sampling.

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Fig. 1: Illustrations of nested sampling algorithm.
Fig. 2: Experimentation in nested sampling.
Fig. 3: Cosmological applications of nested sampling.
Fig. 4: Applications of nested sampling in gravitational-wave astronomy.
Fig. 5: Illustrations of materials science applications.

Change history

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Acknowledgements

The authors thank J. Skilling for his wonderful algorithm. The success of nested sampling may be that simple beats clever; but the beauty of nested sampling is that it is both simple and clever. They thank K. Barbary for discussions. A.F. was supported by a National Natural Science Foundation of China (NSFC) Research Fund for International Young Scientists (grant 11950410509). L.B.P. acknowledges support from the Engineering and Physical Sciences Research Council (EPSRC) through an Early Career Fellowship (EP/T000163/1). M. Habeck acknowledges support from the Carl Zeiss Foundation. N.B. was funded by the US Naval Research Laboratory’s base 6.1 research program, and CPU time from the US Department of Defence (DoD) High Performance Computing Modernization Program Office (HPCMPO) at the Air Force Research Laboratory (AFRL) and Army Research Laboratory (ARL) DoD Supercomputing Research Centers (DSRCs). M.P. acknowledges support from the Science and Technology Facilities Council (STFC) (ST/V001213/1 and ST/V005707/1). W.H. was supported by a Royal Society University Research Fellowship.

Author information

Authors and Affiliations

  1. School of Physics and Astronomy, Monash University, Melbourne, Victoria, Australia

    Greg Ashton

  2. Department of Physics, Royal Holloway, University of London, London, UK

    Greg Ashton

  3. Center for Materials Physics and Technology, US Naval Research Laboratory, Washington, DC, USA

    Noam Bernstein

  4. Max Planck Institute for Extraterrestrial Physics, Garching, Germany

    Johannes Buchner

  5. Department of Computer Science, University of Bath, Bath, UK

    Xi Chen

  6. Engineering Laboratory, University of Cambridge, Cambridge, UK

    Gábor Csányi

  7. Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, China

    Andrew Fowlie

  8. London, UK

    Farhan Feroz

  9. Concr Ltd, Cambridge, UK

    Matthew Griffiths

  10. Astrophysics Group, Cavendish Laboratory, University of Cambridge, Cambridge, UK

    Will Handley, Anthony Lasenby & David Yallup

  11. Kavli Institute for Cosmology, University of Cambridge, Cambridge, UK

    Will Handley, Michael Hobson, Anthony Lasenby & David Yallup

  12. Microscopic Image Analysis Group, Jena University Hospital, Jena, Germany

    Michael Habeck

  13. The D. E. Shaw group, New York, NY, USA

    Edward Higson

  14. Center for Theoretical Astrophysics, Korea Astronomy and Space Science Institute, Daejeon, Korea

    David Parkinson

  15. Department of Chemistry, University of Warwick, Coventry, UK

    Livia B. Pártay

  16. Department of Physics, Lancaster University, Lancaster, UK

    Matthew Pitkin

  17. Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany

    Doris Schneider & Philipp Wacker

  18. Dunlap Institute for Astronomy and Astrophysics, University of Toronto, Toronto, Ontario, Canada

    Joshua S. Speagle

  19. David A. Dunlap Department of Astronomy & Astrophysics, University of Toronto, Toronto, Ontario, Canada

    Joshua S. Speagle

  20. Department of Statistical Sciences, University of Toronto, Toronto, Ontario, Canada

    Joshua S. Speagle

  21. School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia

    Leah South

  22. School of Physics and Astronomy, University of Glasgow, Glasgow, UK

    John Veitch

  23. Yusuf Hamied Department of Chemistry, University of Cambridge, Cambridge, UK

    David J. Wales

Authors
  1. Greg Ashton
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  2. Noam Bernstein
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  3. Johannes Buchner
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  4. Xi Chen
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  5. Gábor Csányi
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  6. Andrew Fowlie
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  7. Farhan Feroz
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  8. Matthew Griffiths
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  9. Will Handley
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  10. Michael Habeck
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  11. Edward Higson
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  12. Michael Hobson
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  13. Anthony Lasenby
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  14. David Parkinson
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  15. Livia B. Pártay
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  16. Matthew Pitkin
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  17. Doris Schneider
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  18. Joshua S. Speagle
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  19. Leah South
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  20. John Veitch
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  21. Philipp Wacker
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  22. David J. Wales
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  23. David Yallup
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Contributions

Introduction (A.F., M. Habeck, D.S., L.S. and P.W.); Experimentation (J.B., E.H. and J.S.S.); Results (A.F.); Applications (G.A., N.B., X.C., G.C., F.a.F., M.G., M. Habeck, M. Hobson, W.H., A.L., L.B.P., M.P., D.P., J. V., D.J.W. and D.Y.); Reproducibility and data deposition (A.F.); Limitations and optimizations (A.F.); Outlook (A.F.); Overview of the Primer (A.F.).

Corresponding author

Correspondence to Andrew Fowlie.

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Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Reviews Methods Primers thanks Roberto Trotta and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Related links

anesthetic: https://github.com/williamjameshandley/anesthetic

dynesty: https://github.com/joshspeagle/dynesty

dyPolyChord: https://github.com/ejhigson/dyPolyChord

MultiNest: https://github.com/farhanferoz/MultiNest

nestcheck: https://github.com/ejhigson/nestcheck

nessai: https://github.com/mj-will/nessai

PolyChord: https://github.com/PolyChord/PolyChordLite

Supplementary information

Glossary

Markov chain Monte Carlo

A class of algorithms for drawing a correlated sequence of samples from a distribution using the states of a Markov chain.

Multimodal

Refers to problems in which the integrand contains more than one mode. In inference problems, modes may correspond to distinct ways in which the model could predict the observed data.

Integrands

Functions that are being integrated.

Prior

A probability density that is not yet conditioned on observed data. In nested sampling, it is a measure in our integral.

Likelihood

The probability of the observed data as a function of a model’s parameters.

Evidence

A factor in Bayes’ theorem that may be written as an integral and that plays an important role in Bayesian model comparison. In nested sampling, this is the multidimensional integral we wish to compute.

Posterior

A probability density conditioned on the observed data found by updating a prior using Bayes’ theorem. In nested sampling, it describes the shape of our integrand.

Bayesian model comparison

A method for comparing models based on computing the change in their relative plausibility in light of data using Bayes’ theorem.

Curse of dimensionality

The phenomenon that the difficulty of a problem often increases dramatically with dimension.

Unimodal

Problems in which the integrand contains only one mode.

Constrained prior

The prior for the parameters restricted to the region in which the likelihood exceeds a threshold.

Bulk

A region with size of order e−H, where H is the Kullback–Leibler divergence, that contains the overwhelming majority of the posterior mass. Closely related to typical sets. Usually, the bulk will not lie near the mode of the posterior, especially in high dimensions.

Tractable

An adjective used to describe problems that are feasible to solve under computational, monetary or time constraints.

Markov chain

A sequence of random states for which the probability of a state depends only on the previous state.

Measure

A probability measure is a function assigning probabilities to events.

Kullback–Leibler divergence

(H). A measure of difference between two distributions that may be interpreted as the information gained by switching from one to the other.

Modes

Peaks in a probability distribution.

Parameter domain

The set of a priori possible parameters, usually the reals Rn or a subset thereof.

Overloaded

A function with a definition that depends on the type of its argument. In nested sampling, the likelihood L is an overloaded function as we consider separate functions L(θ) and L(X).

Survival function

A function F(x) associated with a distribution that returns the probability of obtaining a sample greater than x.

Super-level sets

A λ-super-level set of any function contains all points for which the function value exceeds λ.

Push-forward measure

(Also known as image measure). The distribution of a random variable under a probability measure.

Transition kernels

Functions that describe the likely steps of a Markov chain.

Iso-likelihood contour

The set of points for which the likelihood is equal to a particular constant; in two dimensions, this set forms a contour line.

Bootstrapping

Refers to techniques that estimate statistical variation by repeatedly drawing samples from the true data set with replacement.

Indicator function

A function that takes the value one if a condition holds, and zero otherwise.

Partition function

A normalizing constant that fully characterizes a physical system, because many important thermodynamic variables can be derived from it.

Insertion indexes

The indexes at which the elements of a list must be inserted into an ordered list to maintain ordering.

Simulation-based calibration

Techniques that use simulations from the model to check the correctness of Bayesian computation.

Microstate

The state of all degrees of freedom in a physical system; for example, the microstate of a multi-particle system includes the positions and momenta of all particles.

Microcanonical ensemble

Assigns equal probability to states Θ with E(Θ) = ε and zero probability otherwise, such that the energy level ε rather than the inverse temperature β characterizes a thermodynamic state.

Pseudo-importance sampling

Using algorithms in which an importance sampling density is defined a posteriori.

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Ashton, G., Bernstein, N., Buchner, J. et al. Nested sampling for physical scientists. Nat Rev Methods Primers 2, 39 (2022). https://doi.org/10.1038/s43586-022-00121-x

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