Published December 2, 2024 | Published
Journal Article Open

Bayesian evidence estimation from posterior samples with normalizing flows

  • 1. ROR icon International School for Advanced Studies
  • 2. ROR icon Institute for Fundamental Physics of the Universe
  • 3. ROR icon California Institute of Technology

Abstract

We propose a novel method (floz), based on normalizing flows, to estimate the Bayesian evidence (and its numerical uncertainty) from a preexisting set of samples drawn from the unnormalized posterior distribution. We validate it on distributions whose evidence is known analytically, up to 15 parameter space dimensions, and compare with two state-of-the-art techniques for estimating the evidence: nested sampling (which computes the evidence as its main target) and a đ‘˜-nearest-neighbors technique that produces evidence estimates from posterior samples. Provided representative samples from the target posterior are available, our method is more robust to posterior distributions with sharp features, especially in higher dimensions. For a simple multivariate Gaussian, we demonstrate its accuracy for up to 200 dimensions with 105 posterior samples. floz has wide applicability, e.g., to estimate evidence from variational inference, Markov chain Monte Carlo samples, or any other method that delivers samples and their likelihood from the unnormalized posterior density. As a physical application, we use floz to compute the Bayes factor for the presence of the first overtone in the ringdown signal of the gravitational wave data of GW150914, finding good agreement with nested sampling.

Copyright and License

© 2024 American Physical Society.

Acknowledgement

We thank Alan Heavens for insightful discussions and Uroš Seljak for comments on an earlier draft. E. B., R. S. and M. B. acknowledge support from the European Union’s H2020 ERC Consolidator Grant “GRavity from Astrophysical to Microscopic Scales” (Grant No. GRAMS-815673), the PRIN 2022 grant “GUVIRP—Gravity tests in the UltraViolet and InfraRed with Pulsar timing,” and the EU Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 101007855. M. C. is funded by the European Union under the Horizon Europe’s Marie Sklodowska-Curie Project No. 101065440. R. T. acknowledges co-funding from Next Generation EU, in the context of the National Recovery and Resilience Plan, Investment PE1—Project FAIR “Future Artificial Intelligence Research.” This resource was co-financed by the Next Generation EU [DM 1555 del 11.10.22]. R. T. is partially supported by the Fondazione ICSC, Spoke 3 “Astrophysics and Cosmos Observations”, Piano Nazionale di Ripresa e Resilienza Project ID CN00000013 “Italian Research Center on High-Performance Computing, Big Data and Quantum Computing” funded by MUR Missione 4 Componente 2 Investimento 1.4: Potenziamento strutture di ricerca e creazione di “campioni nazionali di R&S (M4C2-19)”—Next Generation EU (NGEU).

Data Availability

The floz code to reproduce the results in this paper, and to estimate the evidence for any set of posterior samples, likelihood, and prior densities is available at R. Srinivasan, Github Repository, https://github.com/Rahul-Srinivasan/floZ.

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Additional details

Created:
February 14, 2025
Modified:
February 14, 2025