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arxiv: 2606.27090 · v1 · pith:AX4C6WSUnew · submitted 2026-06-25 · 📊 stat.ML · cs.AI· cs.LG

Beyond Global Divergences: A Local-Mass Perspective on Bayesian Inference

Hanli Xu , Fengxiang He , Sarat Moka This is my paper

Pith reviewed 2026-06-26 02:27 UTC · model grok-4.3

classification 📊 stat.ML cs.AIcs.LG
keywords Bayesian inferencelocal massKL divergenceMass IndexRE-KLsmall-ball probabilitiesdistributional discrepancy
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The pith

Bayesian updating changes local small-ball mass through explicit likelihood factors and support adjustments in ways global divergences miss.

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

The paper develops a local-mass account of Bayesian inference by introducing two tools that track behaviour inside small regions rather than across entire distributions. A Mass Index records the polynomial and logarithmic scales at which local mass decays, while regularised extended KL provides a way to compare masses inside chosen sets even when distributions have singular parts. These let the authors show how power-log likelihood terms shift mass directly and how parameter-dependent supports alter the remaining local scale. The result is a set of inequalities that relate local masses under the two directions of KL, giving a finer description of what updating actually does at the scale that often determines practical performance.

Core claim

Mass Indices characterise how Bayesian updating changes local mass through power-log likelihood factors that shift it explicitly and through parameter-dependent supports that change the local scale by the amount of mass remaining near the parameter value. Using local RE-KL, absolute, relative, and directional inequalities are proved for comparing local small-ball masses under the two KL directions.

What carries the argument

Mass Index, which records the polynomial and logarithmic decay scales of local mass, together with regularised extended KL (RE-KL), a set-localised divergence that can be defined even with singular components.

If this is right

  • Power-log likelihood factors shift local mass explicitly.
  • Parameter-dependent supports or their smooth softenings change the local scale through the amount of mass that remains near the parameter value.
  • Local RE-KL yields absolute, relative, and directional inequalities for comparing local small-ball masses under the two KL directions.
  • Experiments supply controlled illustrations of how these local changes appear in practice.

Where Pith is reading between the lines

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

  • The same local-mass lens could be used to compare variational approximations that match globally but diverge inside particular high-density regions.
  • Mass Index values might serve as a diagnostic for when a model’s support choices are likely to distort inference on subsets of the data.
  • Extending the inequalities to other local divergences could link the framework to existing analyses of concentration and tail behaviour.

Load-bearing premise

The local-mass behaviour not captured by global objectives is both measurable with the introduced Mass Index and RE-KL and relevant to the performance of Bayesian inference procedures.

What would settle it

A controlled example in which local mass scales change measurably under updating yet produce no detectable difference in posterior accuracy or predictive behaviour on held-out points near the parameter of interest.

Figures

Figures reproduced from arXiv: 2606.27090 by Fengxiang He, Hanli Xu, Sarat Moka.

Figure 3
Figure 3. Figure 3: illustrates that local RE-KL control is direc￾tional. On the same shrinking neighbourhoods, the p∥q direction stays bounded while the q∥p direction diverges. Thus the order of the two arguments deter￾mines which local mismatch is detected. The explicit construction is given in Appendix C.3. 7 Conclusion This paper studied local small-ball mass p(Br(θ)) as a measure-level quantity in Bayesian inference. The… view at source ↗
Figure 2
Figure 2. Figure 2: UCI prior/posterior small-ball masses and [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

Global objectives, such as KL divergence and ELBO, are widely used in Bayesian inference for measuring distributional discrepancy. This paper studies their local-mass behaviour that is not directly captured by such objectives. We introduce and use two mathematical tools: (1) Mass Index for recording the polynomial and logarithmic decay scales of local mass, and (2) regularised extended KL (RE-KL), a set-localised divergence that can be formulated in the presence of singular components. Mass Indices help characterise how Bayesian updating changes local mass: (1) power-log likelihood factors shift it explicitly, and (2) parameter-dependent supports, or their smooth softenings, may change the local scale through the amount of mass that remains near the parameter value. Using local RE-KL, we prove absolute, relative, and directional inequalities for comparing local small-ball masses under the two KL directions. Together, these results provide a local theoretical account of local mass behaviour. Experiments provide controlled illustrations of the local behaviour. Code is available at https://github.com/Forsythia0604/Local-Mass-Framework.

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

0 major / 3 minor

Summary. The manuscript introduces the Mass Index to record polynomial and logarithmic decay scales of local mass and the regularised extended KL (RE-KL) divergence as a set-localised measure that accommodates singular components. It shows how Bayesian updating alters local mass via power-log likelihood factors and parameter-dependent supports (or their smoothings), then uses local RE-KL to prove absolute, relative, and directional inequalities comparing local small-ball masses under the two KL directions. Controlled experiments illustrate the local behaviour, with code released.

Significance. If the inequalities hold, the work supplies a local theoretical account of mass behaviour that is not directly captured by global objectives such as KL or ELBO. The explicit handling of singular components and the provision of reproducible code are strengths that support verifiability.

minor comments (3)
  1. [Abstract] Abstract: the sentence on 'power-log likelihood factors' and 'parameter-dependent supports' could be expanded with a brief parenthetical example to clarify the two mechanisms before the inequalities are stated.
  2. [Experiments] The experiments section states that the illustrations are 'controlled'; adding a short table or paragraph mapping each figure to the corresponding inequality (absolute/relative/directional) would improve traceability.
  3. [§2] Notation: the definition of the Mass Index (presumably in §2) uses both polynomial and logarithmic scales; a single displayed equation collecting the two cases would reduce ambiguity when the index is later invoked in the proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its contributions regarding local mass behaviour, handling of singular components, and the release of reproducible code. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation introduces Mass Index and RE-KL as new constructs, then proves absolute/relative/directional inequalities relating local small-ball masses. No quoted step reduces a claimed result to a fitted parameter, self-definition, or self-citation chain; the inequalities are presented as following from the definitions of the new tools. The framework is mathematically self-contained against external benchmarks and does not rename known results or smuggle ansatzes via citation. This is the normal case of an independent theoretical contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Abstract introduces two new mathematical objects (Mass Index, RE-KL) whose definitions and supporting lemmas are not supplied; no free parameters or invented physical entities are mentioned.

axioms (1)
  • standard math Standard properties of probability measures and KL divergence hold for the local small-ball setting.
    Invoked implicitly when defining local RE-KL and proving inequalities.
invented entities (2)
  • Mass Index no independent evidence
    purpose: Record polynomial and logarithmic decay scales of local mass.
    New tool introduced to characterise local-mass behaviour under Bayesian updating.
  • RE-KL divergence no independent evidence
    purpose: Set-localised divergence that handles singular components.
    New divergence defined to enable local inequalities.

pith-pipeline@v0.9.1-grok · 5724 in / 1253 out tokens · 33870 ms · 2026-06-26T02:27:14.002140+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

30 extracted references · 1 linked inside Pith

  1. [1]

    On lo- cal divergences between two probability measures

    G Avlogiaris, A Micheas, and K Zografos. On lo- cal divergences between two probability measures. Metrika, 79(3):303–333, 2016

    2016

  2. [2]

    Minimum hellinger distance es- timates for parametric models.The annals of Statistics, pages 445–463, 1977

    Rudolf Beran. Minimum hellinger distance es- timates for parametric models.The annals of Statistics, pages 445–463, 1977

    1977

  3. [3]

    John Wiley & Sons, 2013

    PatrickBillingsley.Convergence of probability mea- sures. John Wiley & Sons, 2013

    2013

  4. [4]

    Cambridge university press, 1989

    Nicholas H Bingham, Charles M Goldie, and Jef L Teugels.Regular variation, volume 27. Cambridge university press, 1989

    1989

  5. [5]

    Variational inference: A review for statisticians.Journal of the American statistical Association, 112(518):859–877, 2017

    David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians.Journal of the American statistical Association, 112(518):859–877, 2017

    2017

  6. [6]

    Bayesian linear regression with Hanli Xu, F engxiang He, Sarat Moka sparse priors.The Annals of Statistics, 43(5):1986– 2018, 2015

    Ismaël Castillo, Johannes Schmidt-Hieber, and Aad van der Vaart. Bayesian linear regression with Hanli Xu, F engxiang He, Sarat Moka sparse priors.The Annals of Statistics, 43(5):1986– 2018, 2015

    1986

  7. [7]

    Wiley Chichester, 1997

    Kenneth J Falconer and KJ Falconer.Techniques in fractal geometry, volume 3. Wiley Chichester, 1997

    1997

  8. [8]

    John Wiley & Sons, 1999

    Gerald B Folland.Real analysis: modern tech- niques and their applications. John Wiley & Sons, 1999

    1999

  9. [9]

    Ober, Florian Wenzel, Gunnar Ratsch, Richard E Turner, Mark van der Wilk, and Lau- rence Aitchison

    Vincent Fortuin, Adrià Garriga-Alonso, Sebas- tian W. Ober, Florian Wenzel, Gunnar Ratsch, Richard E Turner, Mark van der Wilk, and Lau- rence Aitchison. Bayesian neural network priors revisited. InInternational Conference on Learning Representations, 2022

    2022

  10. [10]

    Convergence rates of posterior distributions.Annals of Statistics, pages 500–531, 2000

    Subhashis Ghosal, Jayanta K Ghosh, and Aad W Van Der Vaart. Convergence rates of posterior distributions.Annals of Statistics, pages 500–531, 2000

    2000

  11. [11]

    Model selection in bayesian neural networks via horseshoe priors.Journal of Machine Learning Research, 20(182):1–46, 2019

    Soumya Ghosh, Jiayu Yao, and Finale Doshi-Velez. Model selection in bayesian neural networks via horseshoe priors.Journal of Machine Learning Research, 20(182):1–46, 2019

    2019

  12. [12]

    Dimension of measures: the probabilistic approach.Publicacions Matemà- tiques, 51(2):243–290, 2007

    Yanick Heurteaux. Dimension of measures: the probabilistic approach.Publicacions Matemà- tiques, 51(2):243–290, 2007

    2007

  13. [13]

    Asymp- totic efficiency in parametric structural models with parameter-dependent support.Econometrica, 71(5):1307–1338, 2003

    Keisuke Hirano and Jack R Porter. Asymp- totic efficiency in parametric structural models with parameter-dependent support.Econometrica, 71(5):1307–1338, 2003

    2003

  14. [14]

    Sur un mode de croissance régulière des fonctions.Mathematica (Cluj), 4:38– 53, 1930

    Jovan Karamata. Sur un mode de croissance régulière des fonctions.Mathematica (Cluj), 4:38– 53, 1930

    1930

  15. [15]

    Sur un mode de croissance régulière

    Jovan Karamata. Sur un mode de croissance régulière. théorèmes fondamentaux.Bulletin de la Société Mathématique de France, 61:55–62, 1933

    1933

  16. [16]

    Auto- encoding variational bayes.arXiv preprint arXiv:1312.6114, 2013

    Diederik P Kingma and Max Welling. Auto- encoding variational bayes.arXiv preprint arXiv:1312.6114, 2013

    Pith/arXiv arXiv 2013

  17. [17]

    On information and sufficiency.The annals of mathe- matical statistics, 22(1):79–86, 1951

    Solomon Kullback and Richard A Leibler. On information and sufficiency.The annals of mathe- matical statistics, 22(1):79–86, 1951

    1951

  18. [18]

    Rényi Diver- gence Variational Inference

    Yingzhen Li and Richard Turner. Rényi Diver- gence Variational Inference. InAdvances in Neural Information Processing Systems, volume 29. Cur- ran Associates, Inc., 2016

    2016

  19. [19]

    Pacgan: The power of two samples in generative adversarial networks.Advances in neural information processing systems, 31, 2018

    Zinan Lin, Ashish Khetan, Giulia Fanti, and Se- woong Oh. Pacgan: The power of two samples in generative adversarial networks.Advances in neural information processing systems, 31, 2018

    2018

  20. [20]

    Bayesian compression for deep learning

    Christos Louizos, Karen Ullrich, and Max Welling. Bayesian compression for deep learning. In I. Guyon, U. Von Luxburg, S. Bengio, H. Wal- lach, R. Fergus, S. Vishwanathan, and R. Garnett, editors,Advances in Neural Information Process- ing Systems, volume 30. Curran Associates, Inc., 2017

    2017

  21. [21]

    Cambridge University Press, Cambridge, 1999

    Pertti Mattila.Geometry of sets and measures in Euclidean spaces: fractals and rectifiability, vol- ume 44 ofCambridge Studies in Advanced Mathe- matics. Cambridge University Press, Cambridge, 1999

    1999

  22. [22]

    Sequential Monte Carlo for Inclusive KL Minimization in Amortized Variational Inference

    Declan McNamara, Jackson Loper, and Jeffrey Regier. Sequential Monte Carlo for Inclusive KL Minimization in Amortized Variational Inference. InProceedings of The 27th International Confer- ence on Artificial Intelligence and Statistics, pages 4312–4320. PMLR, April 2024

    2024

  23. [23]

    Thomas P. Minka. Divergence measures and mes- sage passing. Technical Report MSR-TR-2005-173, Microsoft Research, 2005

    2005

  24. [24]

    Markovian score climbing: Variational infer- ence with kl (p|| q).Advances in Neural Informa- tion Processing Systems, 33:15499–15510, 2020

    Christian Naesseth, Fredrik Lindsten, and David Blei. Markovian score climbing: Variational infer- ence with kl (p|| q).Advances in Neural Informa- tion Processing Systems, 33:15499–15510, 2020

    2020

  25. [25]

    On the es- timation ofα-divergences

    Barnabás Póczos and Jeff Schneider. On the es- timation ofα-divergences. InProceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, pages 609–617. JMLR Workshop and Conference Proceedings, 2011

    2011

  26. [26]

    Poste- rior Concentration for Sparse Deep Learning

    Nicholas G Polson and Veronika Ročková. Poste- rior Concentration for Sparse Deep Learning. In Advances in Neural Information Processing Sys- tems, volume 31. Curran Associates, Inc., 2018

    2018

  27. [27]

    van der Vaart and J

    Aad W. van der Vaart and J. Harry van Zan- ten. Rates of contraction of posterior distributions based on gaussian process priors.The Annals of Statistics, 36(3):1435–1463, 2008

    2008

  28. [28]

    F-divergence variational inference.Advances in neural information processing systems, 33:17370– 17379, 2020

    Neng Wan, Dapeng Li, and Naira Hovakimyan. F-divergence variational inference.Advances in neural information processing systems, 33:17370– 17379, 2020

    2020

  29. [29]

    Variational f-divergence Minimization, December 2024

    Mingtian Zhang, Thomas Bird, Raza Habib, Tian- lin Xu, and David Barber. Variational f-divergence Minimization, December 2024. arXiv:1907.11891 [stat.ML]. Beyond Global Divergences

    arXiv 2024

  30. [30]

    Rethinking generative mode coverage: A pointwise guaranteed approach

    Peilin Zhong, Yuchen Mo, Chang Xiao, Pengyu Chen, and Changxi Zheng. Rethinking generative mode coverage: A pointwise guaranteed approach. Advances in Neural Information Processing Sys- tems, 32, 2019. Hanli Xu, F engxiang He, Sarat Moka A Examples and Counterexamples Example A.1(Oscillating small-ball order).This example shows why the upper and lower Pow...

    2019