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arxiv: 2507.16913 · v2 · submitted 2025-07-22 · ✦ hep-ph · hep-ex

A linear PDF model for Bayesian inference

Mark N. Costantini , Luca Mantani , James M. Moore , Maria Ubiali This is my paper

Pith reviewed 2026-05-19 03:10 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords parton distribution functionsBayesian inferencelinear modelsneural network reductiondeep inelastic scatteringglobal fitsuncertainty quantificationmodel selection
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The pith

A linear model using reduced neural-network bases supports fast Bayesian inference for parton distribution functions.

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

The paper introduces a linear representation for PDFs to make Bayesian methods feasible for global fits. PDFs are expressed as vectors in a functional space spanned by bases obtained through dimensional reduction of a neural network functional space, yielding a compact yet flexible form. Low dimensionality speeds up sampling while the adjustable basis size lets users balance under- and over-fitting and perform principled model selection inside Bayesian workflows. The approach is tested on synthetic deep-inelastic-scattering data through multi-closure tests to establish readiness for full global analyses.

Core claim

PDFs are represented as linear combinations of specially chosen basis functions derived from dimensional reduction of a neural-network functional space, producing low-dimensional models that enable computationally efficient Bayesian inference with transparent control over model complexity.

What carries the argument

Linear PDF model whose functional space is spanned by bases obtained from dimensional reduction of a neural network functional space, which carries the representation for fast sampling and adjustable expressivity.

If this is right

  • Inference becomes fast enough for routine use because the preferred models remain low-dimensional.
  • Basis size can be increased or decreased systematically to trade off underfitting against overfitting.
  • Bayesian model selection becomes straightforward once the basis dimension is treated as a discrete hyperparameter.
  • The same framework can be scaled to global PDF fits once validated on synthetic data.

Where Pith is reading between the lines

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

  • The reduced bases may let analysts compare the impact of different prior choices more directly than with full neural-network models.
  • Low-dimensional linear models could be combined with existing PDF codes to produce uncertainty bands that are both faster and easier to interpret.
  • If the bases prove stable under addition of new data sets, the method would lower the barrier to frequent re-fits during the High-Luminosity LHC era.

Load-bearing premise

Bases obtained by reducing a neural-network functional space still span every feature of PDFs that matters for accurate global fits.

What would settle it

A multi-closure test in which the linear model recovers input PDFs from synthetic data with systematically larger uncertainties or biased central values compared with a full neural-network parametrization in the same kinematic range.

read the original abstract

A robust uncertainty estimate in global analyses of Parton Distribution Functions (PDFs) is essential at the Large Hadron Collider (LHC), especially in view of the high-precision data anticipated by experimentalists in the High-Luminosity phase of the LHC. A Bayesian framework to determine PDFs provides a rigorous treatment of uncertainties and full control on the prior, though its practical implementation can be computationally demanding. To address these challenges, we introduce a novel approach to PDF determination tailored for Bayesian inference, based on the use of linear models. Unlike traditional parametrisations, our method represents PDFs as vectors in a functional space spanned by specially chosen bases, derived from the dimensional reduction of a neural network functional space, providing a compact yet versatile representation of PDFs. The low-dimensionality of the preferred models allows for particularly fast inference. The size of the bases can be systematically adjusted, allowing for transparent control over underfitting and overfitting, and facilitating principled model selection through Bayesian workflows. In this work, the methodology is applied to a fit of Deep Inelastic Scattering synthetic data, and thoroughly tested via multi-closure tests, thus paving the way to its application to global PDF fits.

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

3 major / 2 minor

Summary. The paper introduces a linear parametrization of PDFs as vectors in a low-dimensional functional space obtained via dimensional reduction of a neural network ensemble. This basis is intended to enable computationally efficient Bayesian inference while allowing systematic adjustment of basis size to control under- and overfitting. The approach is demonstrated on synthetic DIS data with multi-closure tests.

Significance. If the retained basis truly spans all relevant PDF shapes without material loss of expressivity, the method would offer a practical route to Bayesian global PDF fits with transparent model selection and reduced computational cost. The adjustable dimensionality and emphasis on Bayesian workflows are positive features that could complement existing NN-based approaches.

major comments (3)
  1. [Methodology / basis construction] The central claim that the NN-derived linear basis provides a compact yet versatile representation spanning all relevant PDF features (abstract and methodology) rests on the untested assumption that variations orthogonal to the retained principal directions are negligible. No explicit reproduction of independent parametrizations (NNPDF or CT18 replicas) is reported, nor is preservation of sum rules and positivity quantified beyond the DIS training ensemble.
  2. [Results / multi-closure tests] Multi-closure tests are performed exclusively on synthetic DIS data (results section). This setup cannot detect systematic bias when the same fixed basis is applied to global fits that include Drell-Yan, jet, or top data, because any PDF feature outside the span of the original NN ensemble is projected out by construction.
  3. [Results] The paper does not provide quantitative metrics (e.g., bias, variance, or uncertainty coverage) from the closure tests that would allow assessment of whether the linear model recovers the input PDFs to within the target precision while remaining parameter-free in the sense claimed.
minor comments (2)
  1. [Methodology] Notation for the basis vectors and the projection operator should be defined explicitly with an equation number to avoid ambiguity when discussing linear combinations.
  2. [Figures] Figure captions for the closure-test results should include the number of replicas, the chosen basis dimension, and a direct comparison to a standard NN fit on the same data.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed report. We address each major comment below, indicating where revisions will be made to strengthen the manuscript while clarifying the scope of the current work, which focuses on a proof-of-concept application to synthetic DIS data.

read point-by-point responses

  1. Referee: [Methodology / basis construction] The central claim that the NN-derived linear basis provides a compact yet versatile representation spanning all relevant PDF features (abstract and methodology) rests on the untested assumption that variations orthogonal to the retained principal directions are negligible. No explicit reproduction of independent parametrizations (NNPDF or CT18 replicas) is reported, nor is preservation of sum rules and positivity quantified beyond the DIS training ensemble.

    Authors: We agree that explicit quantification of the retained variance and constraints would strengthen the presentation. In the revised manuscript we will add a supplementary figure showing the cumulative explained variance ratio from the principal component analysis of the NN ensemble, confirming that the chosen basis dimensionality captures the large majority of functional variation. We will also report numerical measures of sum-rule violation and the incidence of positivity violations across the closure-test replicas. Direct reproduction of NNPDF or CT18 replicas lies outside the present scope, which is restricted to synthetic DIS data; we will add a clarifying sentence in the methodology section noting that the basis construction procedure is general and can be applied to broader ensembles in future global analyses. revision: partial

  2. Referee: [Results / multi-closure tests] Multi-closure tests are performed exclusively on synthetic DIS data (results section). This setup cannot detect systematic bias when the same fixed basis is applied to global fits that include Drell-Yan, jet, or top data, because any PDF feature outside the span of the original NN ensemble is projected out by construction.

    Authors: The referee correctly identifies the controlled nature of the test. The manuscript explicitly frames the study as a validation of the linear-model methodology on synthetic DIS data, with the explicit goal of paving the way for global fits. We will revise the discussion and conclusions to state this limitation more prominently and to outline how a future basis can be constructed from an NN ensemble trained on a wider set of processes. Within the DIS setting the multi-closure tests already demonstrate faithful recovery, and the adjustable dimensionality of the linear model provides a direct mechanism to enlarge the span when additional data are included. revision: partial

  3. Referee: [Results] The paper does not provide quantitative metrics (e.g., bias, variance, or uncertainty coverage) from the closure tests that would allow assessment of whether the linear model recovers the input PDFs to within the target precision while remaining parameter-free in the sense claimed.

    Authors: We thank the referee for this observation. Although the results section presents visual comparisons and qualitative statements of recovery, we agree that explicit numerical metrics would improve clarity. In the revised version we will add a table (or inline values) reporting the mean bias, root-mean-square deviation, and the fraction of input PDF points lying inside the 68 % and 95 % credible intervals, averaged over the full set of closure-test replicas. These quantities will be computed both for the central values and for the uncertainty bands, directly addressing the assessment of precision and coverage. revision: yes

Circularity Check

0 steps flagged

Basis construction presented as independent upstream step; no reduction to fitted inputs by construction

full rationale

The paper describes representing PDFs via linear combinations of bases obtained from dimensional reduction of a neural network functional space, then performing Bayesian inference on the resulting low-dimensional coefficients. This basis choice is framed as a prior methodological step separate from the subsequent fit to synthetic DIS data. No equations or claims in the abstract equate a derived quantity (such as a predicted PDF shape or uncertainty) directly to a fitted parameter or self-citation by construction. Multi-closure tests are described as validation on held-out synthetic data rather than tautological recovery of the same inputs. The derivation chain therefore remains self-contained against external benchmarks and does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only information limits the ledger to the core modeling assumption stated in the text; no explicit free parameters or invented entities are quantified.

axioms (1)
  • domain assumption The functional space of PDFs can be effectively spanned by a reduced basis obtained from dimensional reduction of a neural network functional space.
    This premise underpins the entire linear model construction and is invoked to justify compactness and versatility.
invented entities (1)
  • Linear PDF model in reduced NN-derived basis no independent evidence
    purpose: To enable fast Bayesian inference while retaining control over underfitting and overfitting
    New representation introduced to address computational demands of traditional Bayesian PDF fits.

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Forward citations

Cited by 2 Pith papers

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  2. Precision QCD with the Electron-Ion Collider

    hep-ph 2026-04 unverdicted novelty 1.0

    A workshop summary report outlines discussion topics in perturbative QCD, nuclear structure, and related techniques for the upcoming Electron-Ion Collider.

Reference graph

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