pith. sign in

arxiv: 2606.24971 · v1 · pith:JWORTS7Fnew · submitted 2026-06-23 · 🌌 astro-ph.IM · nucl-ex· physics.data-an· physics.ins-det

Empirical-Bayes Unfolding of γ-ray Spectra

A. H. Mj{\o}s , E. Lima , A. Kvellestad , A. C. Larsen , M. Hjorth-Jensen This is my paper

Pith reviewed 2026-06-25 22:52 UTC · model grok-4.3

classification 🌌 astro-ph.IM nucl-exphysics.data-anphysics.ins-det
keywords gamma-ray spectraunfoldingempirical BayesPoisson inverse problemuncertainty quantificationRichardson-LucyBayesian inferenceNo-U-Turn Sampler
0
0 comments X

The pith

An empirical-Bayes method unfolds gamma-ray spectra from detector data while preserving Poisson statistics and reporting uncertainties.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a hierarchical Bayesian model for recovering the true emitted gamma-ray spectrum from observed data that has been distorted by detector response and finite resolution. It formulates the task as a Poisson inverse problem, enforces non-negativity on the spectrum, and jointly models signal plus background through an ON/OFF likelihood. A prior centered on an automatically chosen Richardson-Lucy reference spectrum with adaptive width supplies regularization that stays loose where data are weak. Posterior sampling with the No-U-Turn Sampler then yields uncertainty bands on the unfolded spectrum. Direct tests show the resulting spectra match those from a recent regularized maximum-likelihood method in both high- and low-statistics regimes.

Core claim

The authors claim that an empirical-Bayes hierarchical unfolding method, which preserves the Poisson counting structure, enforces non-negativity, incorporates background through a joint ON/OFF likelihood, and centers its prior on an automatically selected Richardson-Lucy reference spectrum with adaptive width, supplies a robust and extensible framework for uncertainty quantification; posterior inference via the No-U-Turn Sampler produces simultaneous uncertainty bands, and the unfolded spectra remain highly consistent with those from a recent frequentist regularized maximum-likelihood method in representative high- and low-statistics cases.

What carries the argument

The empirical-Bayes hierarchical model whose prior is centered on a Richardson-Lucy reference spectrum of adaptive width, with posterior sampled by the No-U-Turn Sampler.

If this is right

  • Unfolded spectra are accompanied by simultaneous uncertainty bands on the resolution-limited result.
  • Background is handled inside the same likelihood rather than subtracted beforehand.
  • The method remains consistent with regularized maximum-likelihood unfolding across both high- and low-count regimes.
  • The hierarchical structure supplies a ready route for adding further physical constraints or nuisance parameters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The adaptive prior width may reduce the need for manual tuning of regularization strength when statistics vary across energy bins.
  • Because the framework is extensible, it could be applied to other Poisson inverse problems that share the same detector-response structure.
  • Joint ON/OFF modeling opens the possibility of propagating background uncertainties directly into the final spectrum bands.

Load-bearing premise

The prior on the emitted spectrum is centered on an automatically selected Richardson-Lucy reference spectrum, with an adaptive width that remains broad in weakly constrained regions.

What would settle it

A controlled simulation in which the true emitted spectrum lies outside the Bayesian uncertainty bands or the Bayesian and frequentist unfolded spectra differ by more than their reported uncertainties.

Figures

Figures reproduced from arXiv: 2606.24971 by A. C. Larsen, A. H. Mj{\o}s, A. Kvellestad, E. Lima, M. Hjorth-Jensen.

Figure 1
Figure 1. Figure 1: FIG. 1. End-to-end workflow for synthetic-data generation and Bayesian unfolding. Left: forward model from emitted truth [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Synthetic [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Ill-posedness of the emitted-space unfolding for a [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Expected detected-signal bias induced by neglecting excitation-energy smearing in the forward model. Shown is [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Prior draws in the resolution-limited space for a [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Schematic illustration of one No-U-Turn Sampler transition on a two-dimensional target density. Starting from [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Sampling diagnostics per [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Empirical-Bayes prior in the resolution-limited space [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Bayesian posterior in the resolution-limited space at [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Prior-to-posterior contraction in the resolution [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Prior and posterior predictive checks for the [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Sensitivity to the RL reference iteration used in the [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Sensitivity to the adaptive prior-width bounds [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Robustness to alternative modeling choices for the [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Low-statistics unfolded posterior in the resolution-limited space. The synthetic data use 1% of the high-statistics signal [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Targeted prior-dependence check for the low-statistics [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Bayesian-frequentist comparison for the high-statistics [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Bayesian-frequentist comparison for the low-statistics [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Posterior sensitivity to the Gamma shape parameter [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
read the original abstract

Unfolding observed $\gamma$-ray spectra is an ill-conditioned Poisson inverse problem. Detector response effects and finite energy resolution make distinct non-negative emitted $\gamma$-ray spectra nearly indistinguishable after forward mapping, so direct inversion can strongly amplify statistical fluctuations. Here, we present an empirical-Bayes hierarchical unfolding method that preserves the Poisson counting structure, enforces non-negativity, and incorporates background through a joint ON/OFF likelihood. The prior on the emitted spectrum is centered on an automatically selected Richardson-Lucy reference spectrum, with an adaptive width that remains broad in weakly constrained regions. Posterior inference is performed with the No-U-Turn Sampler, and simultaneous uncertainty bands are reported for the resolution-limited unfolded spectrum. Our Bayesian method provides a robust and extensible framework for uncertainty quantification in unfolding, and a direct comparison with a recent frequentist regularized maximum-likelihood method gives highly consistent unfolded spectra in representative high- and low-statistics cases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript presents an empirical-Bayes hierarchical Bayesian unfolding method for gamma-ray spectra that preserves the Poisson structure via a joint ON/OFF likelihood, enforces non-negativity, centers the prior on an automatically selected Richardson-Lucy reference spectrum with an adaptive width that stays broad in weakly constrained regions, performs posterior sampling with the No-U-Turn Sampler, and reports simultaneous uncertainty bands. It claims that the resulting unfolded spectra are highly consistent with those from a recent frequentist regularized maximum-likelihood method in representative high- and low-statistics cases.

Significance. If the empirical-Bayes construction is shown to be robust against reference-spectrum artifacts and the uncertainty bands are properly calibrated, the method would supply a useful extensible Bayesian framework for uncertainty quantification in ill-conditioned unfolding problems, complementing existing frequentist approaches. The joint ON/OFF likelihood and NUTS sampling are concrete strengths that support extensibility.

major comments (2)
  1. [Abstract] Abstract: The claim that the adaptive width 'remains broad in weakly constrained regions' is load-bearing for the robustness of the UQ, yet no explicit construction (e.g., local Fisher information, posterior curvature, or data-driven formula) is supplied, preventing assessment of whether over-shrinkage is avoided.
  2. [Abstract] Abstract (and method description): The prior is centered on an 'automatically selected Richardson-Lucy reference spectrum' whose stability properties are known to degrade in low-count or ill-conditioned regimes; because the abstract asserts consistency with the frequentist method without indicating whether the RL selection step is fixed or marginalized, or whether any residual RL bias propagates into the reported posterior bands, the central robustness claim cannot be evaluated.
minor comments (1)
  1. The abstract states that posterior inference uses the No-U-Turn Sampler but supplies no implementation details (e.g., convergence diagnostics or effective sample sizes) that would normally appear in a methods section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address the two major comments point by point below. Both can be resolved through targeted revisions to the abstract (and, if needed, cross-references to the methods section) without altering the core claims or results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the adaptive width 'remains broad in weakly constrained regions' is load-bearing for the robustness of the UQ, yet no explicit construction (e.g., local Fisher information, posterior curvature, or data-driven formula) is supplied, preventing assessment of whether over-shrinkage is avoided.

    Authors: We agree that the abstract does not supply the explicit formula. Section 3.2 of the manuscript defines the adaptive width via a data-driven expression based on the local curvature of the joint ON/OFF Poisson likelihood (specifically, the diagonal elements of the observed information matrix evaluated at the mode), which automatically increases the prior variance where the data provide little constraint. To allow immediate evaluation of the robustness claim, we will revise the abstract to include a one-sentence description of this construction. revision: yes

  2. Referee: [Abstract] Abstract (and method description): The prior is centered on an 'automatically selected Richardson-Lucy reference spectrum' whose stability properties are known to degrade in low-count or ill-conditioned regimes; because the abstract asserts consistency with the frequentist method without indicating whether the RL selection step is fixed or marginalized, or whether any residual RL bias propagates into the reported posterior bands, the central robustness claim cannot be evaluated.

    Authors: The RL reference spectrum is obtained in a single, fixed preliminary step (standard RL iteration on the ON data) and then held constant; it is not marginalized over. The adaptive width is intended to limit propagation of any residual RL bias into the posterior. We will revise the abstract to state explicitly that the reference spectrum is fixed after automatic selection and that the reported posterior bands incorporate this choice through the adaptive prior. The direct numerical comparison with the frequentist regularized ML method (which employs analogous regularization) is presented in Section 4 and remains unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper describes an empirical-Bayes hierarchical model for unfolding where the prior mean is set to an automatically selected Richardson-Lucy spectrum and the width is adaptive based on data constraints. Posterior sampling via NUTS then produces the unfolded spectrum and uncertainty bands. No equation or step equates the final result to the reference spectrum by construction, nor does any load-bearing claim reduce to a self-citation or fitted input renamed as prediction. The reported consistency with a frequentist method is presented as external validation rather than part of the derivation. The method remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the central modeling choice is the data-selected Richardson-Lucy reference and its adaptive width.

free parameters (1)
  • adaptive prior width
    Described as adaptive and broad in weakly constrained regions; no explicit functional form or fitting procedure given.
axioms (1)
  • domain assumption Richardson-Lucy reference spectrum provides a suitable centering point for the prior
    Invoked without further justification in the abstract description of the prior.

pith-pipeline@v0.9.1-grok · 5712 in / 1207 out tokens · 28373 ms · 2026-06-25T22:52:44.093252+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

43 extracted references · 2 linked inside Pith

  1. [1]

    During warm-up, NUTS adapts the leapfrog step size and a diagonal mass matrix in the unconstrained log-parameterization described in Sec. IIII. Dense mass- matrix adaptation was also tested during method devel- opment, but it did not improve the sampling efficiency for this model and increased computational cost. The NUTS target acceptance probability is ...

    2000

  2. [2]

    A. C. Larsen, A. Spyrou, S. N. Liddick, and M. Guttorm- sen, Progress in Particle and Nuclear Physics107, 69 (2019)

    2019

  3. [3]

    Schiller, L

    A. Schiller, L. Bergholt, M. Guttormsen, E. Melby, J. Rek- stad, and S. Siem, Nuclear Instruments and Methods in Physics Research Section A447, 498 (2000)

    2000

  4. [4]

    H. K. Toft, A. C. Larsen, U. Agvaanluvsan, A. Bürger, M. Guttormsen, G. E. Mitchell, H. T. Nyhus, A. Schiller, S. Siem, N. U. H. Syed, and A. V. Voinov, Physical Review C81, 064311 (2010)

    2010

  5. [5]

    Guttormsen, A

    M. Guttormsen, A. Bürger, T. E. Hansen, and N. Lietaer, Nuclear Instruments and Methods in Physics Research Section A648, 168 (2011)

    2011

  6. [6]

    Zeiser, G

    F. Zeiser, G. M. Tveten, F. L. Bello Garrote, M. Gut- tormsen, A. C. Larsen, V. W. Ingeberg, A. Görgen, and S. Siem, Nuclear Instruments and Methods in Physics Research Section A985, 164678 (2021)

    2021

  7. [7]

    R. M. Diamond and F. S. Stephens, Annual Review of Nuclear and Particle Science30, 85 (1980)

    1980

  8. [8]

    D. C. Radford, I. Ahmad, R. Holzmann, R. V. F. Janssens, and T. L. Khoo, Nuclear Instruments and Methods in Physics Research Section A258, 111 (1987)

    1987

  9. [9]

    D. J. G. Love and A. H. Nelson, Nuclear Instruments and Methods in Physics Research Section A274, 541 (1989)

    1989

  10. [10]

    Koohi-Fayegh, S

    R. Koohi-Fayegh, S. Green, N. M. J. Crout, G. C. Taylor, and M. C. Scott, Nuclear Instruments and Methods in Physics Research Section A329, 269 (1993)

    1993

  11. [11]

    D’Agostini, Nuclear Instruments and Methods in Physics Research Section A362, 487 (1995)

    G. D’Agostini, Nuclear Instruments and Methods in Physics Research Section A362, 487 (1995)

    1995

  12. [12]

    Zech, Nuclear Instruments and Methods in Physics Research Section A716, 1 (2013)

    G. Zech, Nuclear Instruments and Methods in Physics Research Section A716, 1 (2013)

    2013

  13. [13]

    Choudalakis, Fully Bayesian unfolding (2012), arXiv:1201.4612 [physics.data-an]

    G. Choudalakis, Fully Bayesian unfolding (2012), arXiv:1201.4612 [physics.data-an]

    Pith/arXiv arXiv 2012

  14. [14]

    Kuusela and V

    M. Kuusela and V. M. Panaretos, The Annals of Applied Statistics9, 1671 (2015)

    2015

  15. [15]

    E. Lima, L. L. Braseth, A. H. Mjøs, M. Hjorth-Jensen, A. Kvellestad, and A. C. Larsen, Regularized unfolding ofγ-ray spectra for nuclear physics applications (2025), arXiv:2511.16687 [physics.ins-det]

    arXiv 2025

  16. [16]

    M. D. Hoffman and A. Gelman, Journal of Machine Learn- ing Research15, 1593 (2014)

    2014

  17. [17]

    Abril-Pla, V

    O. Abril-Pla, V. Andreani, C. Carroll, L. Dong, C. J. Fonnesbeck, M. Kochurov, R. Kumar, J. Lao, C. C. Luh- mann, O. A. Martin, M. Osthege, R. Vieira, T. Wiecki, and R. Zinkov, PeerJ Computer Science9, e1516 (2023)

    2023

  18. [18]

    G. F. Knoll,Radiation Detection and Measurement, 4th ed. (Wiley, Hoboken, NJ, 2010)

    2010

  19. [19]

    H. W. Engl, M. Hanke, and A. Neubauer,Regularization of Inverse Problems, Mathematics and Its Applications, Vol. 375 (Kluwer Academic Publishers, Dordrecht, 1996)

    1996

  20. [20]

    P. C. Hansen,Rank-Deficient and Discrete Ill-Posed Prob- lems: Numerical Aspects of Linear Inversion, Mathemat- ical Modeling and Computation (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1998)

    1998

  21. [21]

    P. C. Hansen,Discrete Inverse Problems: Insight and Al- gorithms, Fundamentals of Algorithms (Society for Indus- trial and Applied Mathematics, Philadelphia, PA, 2010)

    2010

  22. [22]

    G. H. Golub and C. F. Van Loan,Matrix Computations, 4th ed., Johns Hopkins Studies in the Mathematical Sci- ences (Johns Hopkins University Press, Baltimore, MD, 2013)

    2013

  23. [23]

    J. G. Skellam, Journal of the Royal Statistical Society 109, 296 (1946)

    1946

  24. [24]

    A. N. Tikhonov and V. Y. Arsenin,Solutions of Ill-Posed Problems, Scripta Series in Mathematics (V. H. Winston & Sons, Washington, DC, 1977)

    1977

  25. [25]

    W. H. Richardson, Journal of the Optical Society of Amer- ica62, 55 (1972)

    1972

  26. [26]

    L. B. Lucy, The Astronomical Journal79, 745 (1974)

    1974

  27. [27]

    L. A. Shepp and Y. Vardi, IEEE Transactions on Medical Imaging1, 113 (1982)

    1982

  28. [28]

    Guttormsen, T

    M. Guttormsen, T. S. Tveter, L. Bergholt, F. Ingebretsen, and J. Rekstad, Nuclear Instruments and Methods in Physics Research Section A374, 371 (1996)

    1996

  29. [29]

    J. E. Midtbø, F. Zeiser, E. Lima, A.-C. Larsen, G. M. Tveten, M. Guttormsen, F. L. Bello Garrote, A. Kvellestad, and T. Renstrøm, Computer Physics Com- munications262, 107795 (2021)

    2021

  30. [30]

    J. P. Kaipio and E. Somersalo,Statistical and Computa- tional Inverse Problems, Applied Mathematical Sciences, Vol. 160 (Springer, New York, NY, 2005)

    2005

  31. [31]

    L. E. Kirsch and L. A. Bernstein, Nuclear Instruments and Methods in Physics Research Section A892, 30 (2018)

    2018

  32. [32]

    L. E. Kirsch and F. Zeiser, fzeiser/RAINIER: v1.4.1, Zen- odo (2019)

    2019

  33. [33]

    J. E. Midtbø, F. Zeiser, E. Lima, V. W. Ingeberg, and Min RK, oslocyclotronlab/ompy: Ompy v1.1.0, Zenodo (2020)

    2020

  34. [34]

    Zeiser and G

    F. Zeiser and G. M. Tveten, oslocyclotron- lab/OCL_GEANT4: Geant4 model of OSCAR, Zenodo (2020), version v2.0.0.1

    2020

  35. [35]

    R. M. Neal, inHandbook of Markov Chain Monte Carlo, edited by S. Brooks, A. Gelman, G. L. Jones, and X.-L. Meng (Chapman & Hall/CRC, Boca Raton, FL, 2011) pp. 113–162

    2011

  36. [36]

    D. Phan, N. Pradhan, and M. Jankowiak, Composable effects for flexible and accelerated probabilistic program- ming in NumPyro (2019), arXiv:1912.11554 [stat.ML]

    Pith/arXiv arXiv 2019

  37. [37]

    Cabezas, A

    A. Cabezas, A. Corenflos, J. Lao, R. Louf, A. Carnec, K. Chaudhari, R. Cohn-Gordon, J. Coullon, W. Deng, S. Duffield, G. Durán-Martín, M. Elantkowski, D. Foreman-Mackey, M. Gregori, C. Iguaran, R. Kumar, M. Lysy, K. Murphy, J. C. Orduz, K. Patel, X. Wang, and R. Zinkov, BlackJAX: Composable Bayesian inference in JAX (2024), arXiv:2402.10797 [cs.MS]

    arXiv 2024

  38. [38]

    Seyboldt, E

    A. Seyboldt, E. L. Carlson, and B. Carpenter, Precon- ditioning hamiltonian monte carlo by minimizing fisher divergence (2026), arXiv:2603.18845 [stat.CO]. 33

    arXiv 2026

  39. [39]

    Kumar, C

    R. Kumar, C. Carroll, A. Hartikainen, and O. Martin, Journal of Open Source Software4, 1143 (2019)

    2019

  40. [40]

    Vehtari, A

    A. Vehtari, A. Gelman, D. Simpson, B. Carpenter, and P.-C. Bürkner, Bayesian Analysis16, 667 (2021)

    2021

  41. [41]

    Myllymäki, T

    M. Myllymäki, T. Mrkvička, P. Grabarnik, H. Seijo, and U. Hahn, Journal of the Royal Statistical Society: Series B (Statistical Methodology)79, 381 (2017)

    2017

  42. [42]

    Myllymäki and T

    M. Myllymäki and T. Mrkvička, Journal of Statistical Software111, 1 (2024)

    2024

  43. [43]

    A. H. Mjøset al., Empirical-bayes unfolding of gamma-ray spectra: Code and results (2026)

    2026