Bayesian Analysis

Sampling Errors in Nested Sampling Parameter Estimation

Edward Higson, Will Handley, Mike Hobson, and Anthony Lasenby

Full-text: Open access


Sampling errors in nested sampling parameter estimation differ from those in Bayesian evidence calculation, but have been little studied in the literature. This paper provides the first explanation of the two main sources of sampling errors in nested sampling parameter estimation, and presents a new diagrammatic representation for the process. We find no current method can accurately measure the parameter estimation errors of a single nested sampling run, and propose a method for doing so using a new algorithm for dividing nested sampling runs. We empirically verify our conclusions and the accuracy of our new method.

Article information

Bayesian Anal., Volume 13, Number 3 (2018), 873-896.

First available in Project Euclid: 25 October 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

nested sampling parameter estimation

Creative Commons Attribution 4.0 International License.


Higson, Edward; Handley, Will; Hobson, Mike; Lasenby, Anthony. Sampling Errors in Nested Sampling Parameter Estimation. Bayesian Anal. 13 (2018), no. 3, 873--896. doi:10.1214/17-BA1075.

Export citation


  • Bezáková, I., Stefankovic, D., Vazirani, V. V., and Vigoda, E. (2008). “Accelerating simulated annealing for the permanent and combinatorial counting problems.” SIAM, 37(5): 1429–1454.
  • Chopin, N. and Robert, C. P. (2010). “Properties of nested sampling.” Biometrika, 97(3): 741–755.
  • Del Moral, P., Doucet, A., and Jasra, A. (2006). “Sequential Monte Carlo samplers.” Journal of the Royal Statistical Society. Series B: Statistical Methodology, 68(3): 411–436.
  • Doss, C., Flegal, J., Jones, G., and Neath, R. (2015). “Markov chain monte carlo estimation of quantiles.” Electronic Journal of Statistics, 8: 2448–2478.
  • Efron, B. (1979). “Bootstrap Methods: Another Look at the Jackknife.” The Annals of Statistics, 7(1): 1–26.
  • Efron, B. and Tibshirani, R. (1986). “Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy.” Statistical Science, 1(1): 54–77.
  • Feroz, F. and Hobson, M. P. (2008). “Multimodal nested sampling: An efficient and robust alternative to Markov Chain Monte Carlo methods for astronomical data analyses.” Monthly Notices of the Royal Astronomical Society, 384(2): 449–463.
  • Feroz, F., Hobson, M. P., and Bridges, M. (2009). “MultiNest: An efficient and robust Bayesian inference tool for cosmology and particle physics.” Monthly Notices of the Royal Astronomical Society, 398(4): 1601–1614.
  • Feroz, F., Hobson, M. P., Cameron, E., and Pettitt, A. N. (2013). “Importance Nested Sampling and the MultiNest Algorithm.” arXiv preprint arXiv:1306.2144, 28.
  • Flegal, J. M., Haran, M., and Jones, G. L. (2008). “Markov Chain Monte Carlo: Can We Trust the Third Significant Figure?” Statistical Science, 23(2): 250–260.
  • Handley, W. J., Hobson, M. P., and Lasenby, A. N. (2015a). “PolyChord: Nested sampling for cosmology.” Monthly Notices of the Royal Astronomical Society: Letters, 450(1): L61–L65.
  • Handley, W. J., Hobson, M. P., and Lasenby, A. N. (2015b). “PolyChord: next-generation nested sampling.” Monthly Notices of the Royal Astronomical Society, 15: 1–15.
  • Higson, E., Handley, W., Hobson, M., and Lasenby, A. (2017). “Supplementary Material for Sampling Errors in Nested Sampling Parameter Estimation.” Bayesian Analysis.
  • Huber, M. and Schott, S. (2014). “Random construction of interpolating sets for high-dimensional integration.” Journal of Applied Probability, 51(1): 92–105.
  • Ivezić, Ż., Connolly, A., VanderPlas, J., and Gray, A. (2014). Statistics, Data Mining, and Machine Learning in Astronomy. Princeton University Press.
  • Johnson, R. W. (2001). “An Introduction to the Bootstrap.” Teaching Statistics, 23(2): 49–54.
  • Keeton, C. R. (2011). “On statistical uncertainty in nested sampling.” Monthly Notices of the Royal Astronomical Society, 414(2): 1418–1426.
  • Liu, J., Nordman, D. J., and Meeker, W. Q. (2016). “The Number of MCMC Draws Needed to Compute Bayesian Credible Bounds.” The American Statistician, 06340: 1–27.
  • Loredo, T. J. (2012). “Bayesian astrostatistics: a backward look to the future.” Astrostatistical Challenges for the New Astronomy, 15–40.
  • MacKay, D. J. C. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge University Press.
  • Planck Collaboration (2016). “Planck 2015. XX. Constraints on inflation.” Astronomy & Astrophysics, 594: A20.
  • Rubin, D. B. (1981). “The Bayesian Bootstrap.” Annals of Statistics, 9(1): 130–134.
  • Skilling, J. (2006). “Nested sampling for general Bayesian computation.” Bayesian Analysis, 1(4): 833–860.
  • Tukey, J. W. (1958). “Bias and Confidence in Not-Quite Large Samples.” The Annals of Mathematical Statistics, 29: 614.

Supplemental materials