Bayesian evidence estimation from posterior samples with normalizing flows
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
- European Research Council
- H2020 ERC Consolidator Grant âGRavity from Astrophysical to Microscopic Scalesâ GRAMS-815673
- Ministero dell'Istruzione e del Merito
- PRIN âGUVIRPâGravity tests in the UltraViolet and InfraRed with Pulsar timingâ -
- European Commission
- Marie Sklodowska-Curie Grant Agreement 101007855
- European Commission
- Marie Sklodowska-Curie Project 101065440
- European Union
- Next Generation EU, National Recovery and Resilience Plan, Investment PE1âProject FAIR âFuture Artificial Intelligence Research.â DM 1555 del 11.10.22
- Fondazione ICSC Centro Nazionale di Ricerca in High Performance Computing, Big Data e Quantum Computing
- Spoke 3 âAstrophysics and Cosmos Observationsâ, Piano Nazionale di Ripresa e Resilienza Project CN00000013
- Accepted
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2024-10-04Accepted
- Available
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2024-12-02Published online
- Caltech groups
- TAPIR
- Publication Status
- Published