- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Tool for accurate model selection using Bayes factors

Ankur Barsode*

  • *Contact author: ankur.barsode@icts.res.in
PDF

Phys. Rev. D 114, 064045 – Published 9 September, 2026

DOI: https://doi.org/10.1103/wv11-h53b

Abstract

A common task in physics and astronomy is studying which of the competing hypotheses the data prefer. This is usually done by computing the Bayes factor between the two hypotheses and either interpreting it in terms of the posterior odds or as a ranking statistic for a frequentist p-value test. Here, we describe a relationship between the Bayes factor and its distributions under the two competing hypotheses, called the Bayes factor–Bayes factor (B-B) relationship, expressed as a diagnostic plot. Using examples from gravitational wave (GW) astronomy, we demonstrate how the B-B plot can validate the accuracy of Bayes factor calculations. The B-B relationship may also be useful for estimating background distributions of the Bayes factor at low computational cost, even analytically in some cases. We apply this technique in the context of wave-optics lensing of GWs, extrapolating the background distribution from GWTC4 to put a rough bound of ≲4.1σ on the statistical significance of the lensing hypothesis of GW231123.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (64)

  1. B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari et al., Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).
  2. C. Collaboration et al., Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716, 30 (2012).
  3. B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya et al., GW170817: Observation of gravitational waves from a binary neutron star inspiral, Phys. Rev. Lett. 119, 161101 (2017).
  4. W.-W. Yu and B. Allen, Stochastic gravitational-wave background search using data from five pulsar timing arrays, arXiv:2512.08666.
  5. A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116, 1009 (1998).
  6. A. Adame, J. Aguilar, S. Ahlen, S. Alam, D. Alexander, M. Alvarez, O. Alves, A. Anand, U. Andrade, E. Armengaud et al., DESI 2024 VI: Cosmological constraints from the measurements of baryon acoustic oscillations, J. Cosmol. Astropart. Phys. 02 (2025) 021.
  7. H. Jeffreys, Theory of Probability (Oxford University Press, Oxford, 1961).
  8. J. Neyman and E. S. Pearson, Ix. on the problem of the most efficient tests of statistical hypotheses, Phil. Trans. R. Soc. A 231, 289 (1933).
  9. A. Fowlie, Neyman-pearson lemma for bayes factors, Commun. Stat.-Theor. Methods 52, 5379 (2023).
  10. J. Skilling, Nested sampling for general bayesian computation, Bayesian Analysis 1, 833 (2006).
  11. D. Sivia and J. Skilling, Data Analysis: A Bayesian Tutorial (OUP, Oxford, 2006).
  12. A. Fowlie, W. Handley, and L. Su, Nested sampling cross-checks using order statistics, Mon. Not. R. Astron. Soc. 497, 5256 (2020).
  13. I. J. Good, A list of properties of bayes-turing factors, NSA Tech. J. 10, 1 (1965), https://www.nsa.gov/portals/75/documents/news-features/declassified-documents/tech-journals/list-of-properties.pdf.
  14. N. Sekulovski, M. Marsman, and E.-J. Wagenmakers, A good check on the Bayes factor, Behav. Res. Meth. Instrum. Comput. 56, 8552 (2024).
  15. K. Okada, Optimality of the half-order exponent in the turing-good identities for Bayes factors, arXiv:2602.19838.
  16. M. Modrák, A. H. Moon, S. Kim, P. Bürkner, N. Huurre, K. Faltejsková, A. Gelman, and A. Vehtari, Simulation-based calibration checking for Bayesian computation: The choice of test quantities shapes sensitivity, Bayesian Anal. 20, 461 (2023).
  17. M. Modrák, S. Stroppel, and P.-C. Bürkner, Simulation-based validation of Bayes factor computation, arXiv:2508.11814.
  18. A. Barsode, K. Maity, and P. Ajith, Lensing, not luck! detection prospects of strongly lensed gravitational waves, Astrophys. J. 998, 294 (2026).
  19. A. Abac, I. Abouelfettouh, F. Acernese, K. Ackley, C. Adamcewicz, S. Adhicary, D. Adhikari, N. Adhikari, R. Adhikari, V. Adkins et al., GW231123: A binary black hole merger with total mass 190–265 msun, Astrophys. J. Lett. 993, L25 (2025).
  20. N. Cornish, L. Sampson, N. Yunes, and F. Pretorius, Gravitational wave tests of general relativity with the parameterized post-Einsteinian framework, Phys. Rev. D 84, 062003 (2011).
  21. J. Lawrence, Focusing of gravitational radiation by interior gravitational fields, Il Nuovo Cimento B (1971–1996) 6, 225 (1971).
  22. R. Takahashi and T. Nakamura, Wave effects in the gravitational lensing of gravitational waves from chirping binaries, Astrophys. J. 595, 1039 (2003).
  23. L. Dai and T. Venumadhav, On the waveforms of gravitationally lensed gravitational waves, arXiv:1702.04724.
  24. J. U. Lange, nautilus: Boosting Bayesian importance nested sampling with deep learning, Mon. Not. R. Astron. Soc. 525, 3181 (2023).
  25. C. Cutler and E. E. Flanagan, Gravitational waves from merging compact binaries: How accurately can one extract the binary’s parameters from the inspiral waveform?, Phys. Rev. D 49, 2658 (1994).
  26. J. Roulet, S. Olsen, J. Mushkin, T. Islam, T. Venumadhav, B. Zackay, and M. Zaldarriaga, Removing degeneracy and multimodality in gravitational wave source parameters, Phys. Rev. D 106, 123015 (2022).
  27. K. Haris, A. K. Mehta, S. Kumar, T. Venumadhav, and P. Ajith, Identifying strongly lensed gravitational wave signals from binary black hole mergers, arXiv:1807.07062.
  28. C. R. Harris et al., Array programming with NumPy, Nature (London) 585, 357 (2020).
  29. J. Janquart, O. A. Hannuksela, K. Haris, and C. Van Den Broeck, A fast and precise methodology to search for and analyse strongly lensed gravitational-wave events, Mon. Not. R. Astron. Soc. 506, 5430 (2021).
  30. R. K. Lo and I. Magaña Hernandez, Bayesian statistical framework for identifying strongly lensed gravitational-wave signals, Phys. Rev. D 107, 123015 (2023).
  31. D. H. Cheung, S. Rinaldi, M. Toscani, and O. A. Hannuksela, Mitigating the effect of population model uncertainty on strong lensing bayes factor using nonparametric methods, Phys. Rev. D 111, 063012 (2025).
  32. J. Janquart, K. Haris, O. A. Hannuksela, and C. Van Den Broeck, The return of golum: Improving distributed joint parameter estimation for strongly lensed gravitational waves, Mon. Not. R. Astron. Soc. 526, 3088 (2023).
  33. A. Barsode, S. Goyal, and P. Ajith, Fast and efficient bayesian method to search for strongly lensed gravitational waves, Astrophys. J. 980, 258 (2025).
  34. O. A. Hannuksela, K. Haris, J. Janquart, H. Narola, H. Phurailatpam, J. D. Creighton, and C. V. den Broeck, Strong gravitational-wave lensing posterior odds, Astrophys. J. 1002, 1 (2026).
  35. R. K. Lo, denmarf: A Python package for density estimation using masked autoregressive flow, arXiv:2305.14379.
  36. M. Wright et al., Lensingflow: An automated workflow for gravitational wave lensing analyses, RAS Tech. Instrum. 5, rzag003 (2026).
  37. LIGO-Virgo-KAGRA Collaboration, Gravitational-wave candidate event database (gracedb) (2024), https://gracedb.ligo.org/.
  38. A. Abac, I. Abouelfettouh, F. Acernese, K. Ackley, C. Adamcewicz, S. Adhicary, D. Adhikari, N. Adhikari, R. Adhikari, V. Adkins et al., GWTC-4.0: Searches for gravitational-wave lensing signatures, arXiv:2512.16347.
  39. Q. Hu, H. Narola, J. Heynen, M. Wright, J. Veitch, J. Janquart, and C. V. D. Broeck, GW231123: Overlapping gravitational wave signals?, Astrophys. J. 1003, 187 (2026).
  40. V. De Luca, G. Franciolini, and A. Riotto, GW231123: A possible primordial black hole origin, Phys. Rev. Lett. 136, 201401 (2026).
  41. C. Yuan, Z.-C. Chen, and L. Liu, GW231123 mass gap event and the primordial black hole scenario, Phys. Rev. D 112, L081306 (2025).
  42. S. Goyal, H. Villarrubia-Rojo, and M. Zumalacarregui, Across the universe: GW231123 as a magnified and diffracted black hole merger, arXiv:2512.17631.
  43. J. C. Chan, J. M. Ezquiaga, R. K. Lo, J. Bowman, L. M. Zertuche, and L. Vujeva, Discovering gravitational waveform distortions from lensing: A deep dive into GW231123, Phys. Rev. D 114, L021502 (2026).
  44. A. Chakraborty and S. Mukherjee, The first model-independent upper bound on micro-lensing signature of the highest mass binary black hole event GW231123, Astrophys. J. 1003, 20 (2026).
  45. X. Shan, H. Yang, and S. Mao, GW231123: A case for binary microlensing in a strong lensing field, arXiv:2512.19118.
  46. B. Wang and T. Yang, Gw231123: False massive graviton signatures from unmodeled point-mass lensing, arXiv:2604.08179.
  47. M. Vallisneri, Testing general relativity with gravitational waves: A reality check, Phys. Rev. D 86, 082001 (2012).
  48. W. D. Pozzo, K. Grover, I. Mandel, and A. Vecchio, Testing general relativity with compact coalescing binaries: Comparing exact and predictive methods to compute the bayes factor, Classical Quantum Gravity 31, 205006 (2014).
  49. S. Bini, K. Król, K. Chatziioannou, and M. Isi, Impact of waveform systematics and Gaussian noise on the interpretation of GW231123, Phys. Rev. D 113, 083036 (2026).
  50. C. Chatterjee, K. McGowan, S. Deshmukh, N. Tyler-Howard, and K. Jani, Machine learning confirms GW231123 is a “lite” intermediate mass black hole merger, Astrophys. J. Lett. 995, L6 (2025).
  51. A. Ray, S. Banagiri, E. Thrane, and P. D. Lasky, GW231123: Extreme spins or microglitches?, arXiv:2510.07228.
  52. A. Sasli, N. Karnesis, and N. Stergioulas, Heavy-tailed likelihoods for robustness against data outliers: Applications to the analysis of gravitational wave data, Phys. Rev. D 108, 103005 (2023).
  53. J. C. Chan, E. Seo, A. K. Li, H. Fong, and J. M. Ezquiaga, Detectability of lensed gravitational waves in matched-filtering searches, Phys. Rev. D 111, 084019 (2025).
  54. T. Islam, J. Roulet, and T. Venumadhav, Factorized parameter estimation for real-time gravitational wave inference, arXiv:2210.16278.
  55. P. Virtanen et al. (scipy 1.0 Contributors), scipy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  56. J. C. Chan, L. M. Zertuche, J. M. Ezquiaga, R. K. Lo, L. Vujeva, and J. Bowman, Identification and characterization of distorted gravitational waves by lensing using deep learning, Phys. Rev. D 113, 024041 (2026).
  57. A. Barsode, The code repository for “B-B Plot: A tool for accurate model selection using Bayes factors”, https://github.com/anbarsode/bb_plot (2026).
  58. A. Abac, I. Abouelfettouh, F. Acernese, K. Ackley, C. Adamcewicz, S. Adhicary, D. Adhikari, N. Adhikari, R. Adhikari, V. Adkins et al., GWTC-4.0: Population properties of merging compact binaries, Astrophys. J. Lett. 1005, L51 (2026).
  59. P. Madau and M. Dickinson, Cosmic star-formation history, Annu. Rev. Astron. Astrophys. 52, 415 (2014).
  60. T. E. Collett, The population of galaxy–galaxy strong lenses in forthcoming optical imaging surveys, Astrophys. J. 811, 20 (2015).
  61. R. Kormann, P. Schneider, and M. Bartelmann, Isothermal elliptical gravitational lens models, Astron. Astrophys. 284, 285 (1994), https://ui.adsabs.harvard.edu/abs/1994A%26A...284..285K/abstract.
  62. M. Fukugita and E. L. Turner, Gravitational lensing frequencies-galaxy cross-sections and selection effects, Mon. Not. R. Astron. Soc. 253, 99 (1991).
  63. LIGO-Virgo Collaboration, Noise curves used for simulations in the update of the observing scenarios paper (2020), LIGO Document T2000012-v1.
  64. G. Pratten, C. García-Quirós, M. Colleoni, A. Ramos-Buades, H. Estellés, M. Mateu-Lucena, R. Jaume, M. Haney, D. Keitel, J. E. Thompson et al., Computationally efficient models for the dominant and subdominant harmonic modes of precessing binary black holes, Phys. Rev. D 103, 104056 (2021).

Outline

Information

Phys. Rev. D 114, 064045– Published 9 September, 2026
  • Received 14 May 2026
  • Accepted 14 August 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation