Importance nested sampling with normalising flows

Michael J. Williams*, John Veitch, Chris Messenger

*Corresponding author for this work

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We present an improved version of the nested sampling algorithm nessai in which the core algorithm is modified to use importance weights. In the modified algorithm, samples are drawn from a mixture of normalising flows and the requirement for samples to be independently and identically distributed (i.i.d.) according to the prior is relaxed. Furthermore, it allows for samples to be added in any order, independently of a likelihood constraint, and for the evidence to be updated with batches of samples. We call the modified algorithm i-nessai. We first validate i-nessai using analytic likelihoods with known Bayesian evidences and show that the evidence estimates are unbiased in up to 32 dimensions. We compare i-nessai to standard nessai for the analytic likelihoods and the Rosenbrock likelihood, the results show that i-nessai is consistent with nessai whilst producing more precise evidence estimates. We then test i-nessai on 64 simulated gravitational-wave signals from binary black hole coalescence and show that it produces unbiased estimates of the parameters. We compare our results to those obtained using standard nessai and dynesty and find that i-nessai requires 2.68 and 13.3 times fewer likelihood evaluations to converge, respectively. We also test i-nessai of an 80 s simulated binary neutron star signal using a reduced-order-quadrature basis and find that, on average, it converges in 24 min, whilst only requiring 1.01 × 10 6 likelihood evaluations compared to 1.42 × 10 6 for nessai and 4.30 × 10 7 for dynesty. These results demonstrate that i-nessai is consistent with nessai and dynesty whilst also being more efficient.

Original languageEnglish
Article number035011
Number of pages27
JournalMachine Learning: Science and Technology
Issue number3
Early online date25 Jul 2023
Publication statusPublished - 1 Sept 2023


  • Bayesian inference
  • gravitational waves
  • machine learning
  • nested sampling
  • normalising flows
  • UKRI
  • STFC
  • ST/V005634/1
  • ST/I006285/1

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