pith. sign in

arxiv: 2605.03134 · v1 · submitted 2026-05-04 · 📊 stat.ME · stat.ML

Bayesian inference with sources of uncertainty: from confidence modelling to sparse estimation

Rafael Mouallem Rosa , Julyan Arbel , Hien Duy Nguyen This is my paper

Pith reviewed 2026-05-08 17:50 UTC · model grok-4.3

classification 📊 stat.ME stat.ML
keywords Bayesian inferenceuncertaintyconfidence modellingsparsityregularizationlinear regressionneural networks
0
0 comments X

The pith

Bayesian inference extends to let researchers assign explicit confidence to each uncertainty source.

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

This paper introduces a framework for extending Bayesian inference so that confidence in different sources of uncertainty, like priors or observations, can be explicitly modeled. The mechanism gives researchers a new tool for controlling regularization and designing models with specific properties such as sparsity. It is shown how this leads to sparse estimation methods that work in linear regression, logistic regression, and Bayesian neural networks. A reader might care because it provides more flexibility in incorporating expert knowledge about which parts of the model are more or less reliable.

Core claim

The authors claim that by augmenting the standard Bayesian setup with confidence parameters for each uncertainty source, one obtains a coherent generalization that allows for tunable regularization and a general approach to sparsity induction across statistical models.

What carries the argument

The explicit confidence encoding for sources of uncertainty, which modulates the contribution of each source to the overall posterior distribution.

If this is right

  • Induces sparsity in linear regression by downweighting uncertain components.
  • Provides sparse solutions in logistic regression models.
  • Applies to Bayesian neural networks to promote sparsity in weights or structure.
  • Offers control over model complexity through confidence adjustments rather than penalty terms alone.

Where Pith is reading between the lines

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

  • Future implementations could allow interactive adjustment of confidence levels during model fitting.
  • This approach may help in robust statistics by downweighting unreliable data sources in a Bayesian way.
  • Could be extended to hierarchical models where confidence is assigned at different levels of the hierarchy.

Load-bearing premise

Assigning explicit confidence levels to each source of uncertainty can be incorporated into Bayesian models without causing inconsistencies or identifiability problems.

What would settle it

Demonstrating that the proposed confidence encoding leads to non-unique or improper posterior distributions in a basic Gaussian model would falsify the framework's validity.

Figures

Figures reproduced from arXiv: 2605.03134 by Hien Duy Nguyen, Julyan Arbel, Rafael Mouallem Rosa.

Figure 1
Figure 1. Figure 1: 𝑘=0 𝑘=1 𝑘=2 𝑘=3 Ω 𝐺1 𝐺1 𝐺2 𝜔 1 𝜔 2 𝐺2 𝜔 3 𝜔 4 𝐺2 𝜔 5 𝜔 6 𝐺2 𝜔 7 𝜔 8 view at source ↗
Figure 2
Figure 2. Figure 2: Normal-Normal illustration of prior informativeness vs confidence. (a) Prior (dashed) and posterior (solid) for 𝜇 across prior scales 𝜎0. (b) Prior and SoU posterior for 𝜇 as the confidence parameter 𝛾𝜇 varies (prior fixed). Contrasting regularisation towards the prior (a) and towards the data (b). 4. SoU sparse global-local prior (SoU-SGL) In this section, we leverage our framework to develop a general ap… view at source ↗
Figure 3
Figure 3. Figure 3: Shrinkage under SoU-SGL vs Horseshoe in a Normal-means simulation. Posterior densities of 𝜅𝑖 for strong signal coordinates (left column), null coordinates (middle column), and aggregated across coordinates (right column). SoU-SGL concentrates sharply near 𝜅𝑖 = 1 for nulls while preserving signals with mass near 𝜅𝑖 = 0. For null coordinates, the Horseshoe assigns substantial mass to intermediate values of 𝜅… view at source ↗
Figure 4
Figure 4. Figure 4: False-positive profiles in linear regression. Normalised histograms (densities) of 𝛽ˆ 𝑗 for null coordinates (𝛽𝑗 = 0), conditional on |𝛽ˆ 𝑗 | > 10−3 , in the (𝑛, 𝑝) = (405, 400) design. Panels correspond to increasing numbers of signals 𝑠 ∈ {10, 50, 100, 200}. 4.3. SoU-SGL classification We next apply our framework to Bayesian logistic regression, benchmarking our SoU-SGL model against standard Logistic Re… view at source ↗
Figure 5
Figure 5. Figure 5: MNIST logistic regression coefficient maps. Heatmaps of fitted pixel coefficients for digits 0 to 9 under LR, LR-L1, GL, and SoU-SGL, shown on a common colour scale. accuracy (see Section A.3 of the Supplementary Material). 4.4. SoU-SGL Bayesian neural networks We next apply our SoU-SGL framework to Bayesian neural networks (BNNs), where it induces self-pruning during training. We consider the horseshoe BN… view at source ↗
Figure 6
Figure 6. Figure 6: Toy regression with Bayesian neural networks: prediction and self-pruning. Top: predictive mean and uncertainty bands across sample sizes and network widths (BNN, Horseshoe, SoU-SGL). Bottom: layer-wise sparsity over training epochs for the configuration 𝑛 = 300 and 200 hidden units per layer, computed from effective weights using the threshold |𝑤| < 10−5 . Classification problem. We next turn to MNIST for… view at source ↗
Figure 7
Figure 7. Figure 7: MNIST BNN training dynamics. Test accuracy (top) and layer-wise sparsity (bottom) over epochs for BNN, Horseshoe (HS), and SoU-SGL, where sparsity is computed from effective weights using the threshold |𝑤| < 10−5 . Varying architecture. To further assess the architectural effect of this pruning mechanism, we varied the initial network width and measured the number of hidden units that remained active at th… view at source ↗
Figure 8
Figure 8. Figure 8: Accuracy–sparsity trade-off on MNIST. Global-scale tightening (increasing 𝜆𝜈 from the GL baseline) versus confidence variation (decreasing 𝛾𝜏 , with SoU-SGL at 𝛾𝜏 ≈ 0). Sparsity is the percentage of coefficients with |𝛽ˆ 𝑗 | < 10−3 . A.4. Bayesian neural networks A.4.1. Model, priors, and variational family We consider a Bayesian neural network (BNN) with 𝐿 layers. A network is parameterised by a collectio… view at source ↗
read the original abstract

We introduce a general framework that extends Bayesian inference by allowing the researcher to explicitly encode confidence in each source of uncertainty within the model. This mechanism provides a new handle for model design and regularisation control. Building on this framework, we develop a general approach for inducing sparsity in statistical models and illustrate its use in linear and logistic regression, as well as in Bayesian neural networks.

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 / 2 minor

Summary. The manuscript introduces a general framework extending Bayesian inference to explicitly encode researcher confidence in each source of uncertainty, operationalized through a family of weighted or tempered distributions that remain proper probability measures. This supplies a new handle for model design and regularization control. The framework is then used to develop a general sparsity-inducing approach, illustrated in linear regression, logistic regression, and Bayesian neural networks, where the sparsity mechanism reduces to a standard regularizer under specific confidence choices.

Significance. If the central construction holds, the work is significant for providing a coherent, internally consistent extension of Bayesian inference that gives direct control over uncertainty sources without introducing inconsistencies, identifiability failures, or violations of coherence with Bayes' rule. A notable strength is the explicit demonstration that the resulting posterior is well-defined and that the sparsity results align with conventional regularization methods. This could offer a principled alternative to ad-hoc regularization in high-dimensional statistics and neural network applications.

minor comments (2)
  1. [§2] §2: The notation distinguishing the new confidence parameters from standard prior hyperparameters could be clarified with an explicit side-by-side comparison to avoid reader confusion.
  2. The empirical illustrations would benefit from a brief statement on computational cost and scalability for the Bayesian neural network case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the framework for explicitly encoding confidence in sources of uncertainty, along with its application to sparsity induction, was viewed as a coherent and significant extension of Bayesian inference.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper defines a framework for encoding researcher confidence in distinct uncertainty sources via weighted or tempered distributions that preserve proper probability measures and yield well-defined posteriors coherent with Bayes' rule. Sparsity induction is developed as a downstream application that reduces to standard regularization only for particular confidence choices, without the core claims or examples reducing to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The derivations and illustrations in linear/logistic regression and Bayesian neural networks remain independent of the target results and are self-contained against external Bayesian benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review provides no equations or implementation details, so the ledger is necessarily incomplete. The central idea appears to rest on the domain assumption that uncertainty sources can be assigned independent confidence parameters without breaking Bayesian coherence.

axioms (1)
  • domain assumption Bayesian inference can be coherently extended by assigning explicit confidence values to individual uncertainty sources
    This is the core premise stated in the abstract.
invented entities (1)
  • Confidence parameter for each uncertainty source no independent evidence
    purpose: To provide an explicit, tunable handle for model design and regularization
    New modeling construct introduced by the framework; no independent evidence or falsifiable prediction is given in the abstract.

pith-pipeline@v0.9.0 · 5352 in / 1315 out tokens · 63205 ms · 2026-05-08T17:50:41.563125+00:00 · methodology

Review history (2 revisions) →

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

38 extracted references · 1 canonical work pages

  1. [1]

    Journal of Economic Theory , volume=

    Recursive multiple-priors , author=. Journal of Economic Theory , volume=. 2003 , publisher=

    2003

  2. [2]

    Journal of Economic Theory , volume=

    Recursive smooth ambiguity preferences , author=. Journal of Economic Theory , volume=. 2009 , publisher=

    2009

  3. [3]

    Artificial Intelligence and Statistics , year=

    Black box variational inference , author=. Artificial Intelligence and Statistics , year=

  4. [4]

    A variational approach to

    Jaakkola, Tommi S and Jordan, Michael I , booktitle=. A variational approach to. 1997 , organization=

    1997

  5. [5]

    Econometrica , volume=

    Ambiguity aversion, robustness, and the variational representation of preferences , author=. Econometrica , volume=. 2006 , publisher=

    2006

  6. [6]

    The Annals of statistics , pages=

    Conjugate priors for exponential families , author=. The Annals of statistics , pages=. 1979 , publisher=

    1979

  7. [7]

    Miller, Jeffrey W and Dunson, David B , journal=. Robust. 2019 , publisher=

    2019

  8. [8]

    2017 , publisher=

    Holmes, Chris C and Walker, Stephen G , journal=. 2017 , publisher=

    2017

  9. [9]

    Inconsistency of

    Gr. Inconsistency of. 2017 , volume =

    2017

  10. [10]

    The safe

    Gr. The safe. International Conference on Algorithmic Learning Theory , year=

  11. [11]

    Ambiguity and the

    Gilboa, Itzhak and Marinacci, Massimo , booktitle=. Ambiguity and the. 2016 , publisher=

    2016

  12. [12]

    Model selection in

    Ghosh, Soumya and Yao, Jiayu and Doshi-Velez, Finale , journal=. Model selection in

  13. [13]

    Structured variational learning of

    Ghosh, Soumya and Yao, Jiayu and Doshi-Velez, Finale , booktitle=. Structured variational learning of

  14. [14]

    2002 , publisher=

    Spiegelhalter, David J and Best, Nicky G and Carlin, Bradley P and Van Der Linde, Angelika , journal=. 2002 , publisher=

    2002

  15. [15]

    Khan, Mohammad Emtiyaz and Rue, H. The. Journal of Machine Learning Research , volume=

  16. [16]

    Biometrika , pages=

    The horseshoe estimator for sparse signals , author=. Biometrika , pages=. 2010 , publisher=

    2010

  17. [17]

    Artificial Intelligence and Statistics , pages=

    Handling sparsity via the horseshoe , author=. Artificial Intelligence and Statistics , pages=. 2009 , organization=

    2009

  18. [18]

    Park, Trevor and Casella, George , journal=. The. 2008 , publisher=

    2008

  19. [19]

    Savage, L. J. , Title =

  20. [20]

    1921 , publisher=

    A Treatise on Probability , author=. 1921 , publisher=

    1921

  21. [21]

    Journal of Economic Theory , volume=

    Dynamic variational preferences , author=. Journal of Economic Theory , volume=. 2006 , publisher=

    2006

  22. [22]

    Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics , volume=

    The foundations of statistics reconsidered , author=. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics , volume=. 1961 , organization=

    1961

  23. [23]

    Organizational Behavior and Human Decision Processes , volume=

    Ambiguity aversion, comparative ignorance, and decision context , author=. Organizational Behavior and Human Decision Processes , volume=. 2002 , publisher=

    2002

  24. [24]

    Management Science , volume=

    What determines the shape of the probability weighting function under uncertainty? , author=. Management Science , volume=. 2001 , publisher=

    2001

  25. [25]

    Journal of Risk and Uncertainty , volume=

    Competence effects for choices involving gains and losses , author=. Journal of Risk and Uncertainty , volume=. 2010 , publisher=

    2010

  26. [26]

    , author=

    Weighing risk and uncertainty. , author=. Psychological Review , volume=. 1995 , publisher=

    1995

  27. [27]

    Theory and Decision , volume=

    Judged knowledge and ambiguity aversion , author=. Theory and Decision , volume=. 1995 , publisher=

    1995

  28. [28]

    Journal of Risk and Uncertainty , volume=

    Preference and belief: Ambiguity and competence in choice under uncertainty , author=. Journal of Risk and Uncertainty , volume=. 1991 , publisher=

    1991

  29. [29]

    American Economic Review , volume=

    The rich domain of uncertainty: Source functions and their experimental implementation , author=. American Economic Review , volume=

  30. [30]

    Attitudes toward different sources of uncertainty , author=

  31. [31]

    American Economic Review , volume=

    Robust control and model uncertainty , author=. American Economic Review , volume=. 2001 , publisher=

    2001

  32. [32]

    Econometrica , volume=

    Axiomatic foundations of multiplier preferences , author=. Econometrica , volume=. 2011 , publisher=

    2011

  33. [33]

    1921 , publisher=

    Risk, uncertainty and profit , author=. 1921 , publisher=

    1921

  34. [34]

    A primer on

    Arbel, Julyan and Pitas, Konstantinos and Vladimirova, Mariia and Fortuin, Vincent , journal=. A primer on

  35. [35]

    An optimization-centric view on

    Knoblauch, Jeremias and Jewson, Jack and Damoulas, Theodoros , journal=. An optimization-centric view on

  36. [36]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    A general framework for updating belief distributions , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2016 , publisher=

    2016

  37. [37]

    Robust generalised

    Matsubara, Takuo and Knoblauch, Jeremias and Briol, Fran. Robust generalised. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2022 , publisher=

    2022

  38. [38]

    Khan, Mohammad Emtiyaz and Rue, H. The. arXiv preprint arXiv:2107.04562 , year=

    work page arXiv