Global Analysis of Nuclear Parton Distribution Functions using Markov Chain Monte Carlo

Nasim Derakhshanian

REVIEW

Global Analysis of Nuclear Parton Distribution Functions using Markov Chain Monte Carlo

Nasim Derakhshanian

High Energy Physics - Phenomenology · Submitted 27 Oct 2026 · 271 references

Pith's verdict: First genuine MCMC-based nPDF global fit with a clever Pb-only single-nucleus setup, but the central multimodality claim rests on chain ergodicity that the thesis itself has not established.

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CitationNasim Derakhshanian. "Global Analysis of Nuclear Parton Distribution Functions using Markov Chain Monte Carlo." arXiv:2610.37254 (2026). https://pith.science/paper/DGVGDPOB

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Editor's note

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Quick take: this is the first real attempt to sample the nPDF posterior directly with MCMC, and the Pb-only single-nucleus fit is a genuinely new diagnostic. Both deserve referee time. But the headline claim—that the valence-sector posterior is genuinely multimodal and that Hessian bands are therefore misleading—is not yet established. The thesis's own trace plots show drift in auv3 (Fig. 6.1) and the Multi-nuclei uv trace is captioned as showing “potential remnants of multi-modal structure” (Fig. 7.21). The stress-test concern lands: adaptive Metropolis–Hastings chains that spend long blocks in one mode can pass trace, cumulative-mean, and Gamma-method checks while still failing to mix between modes. The Gamma method in particular can underestimate autocorrelation when between-mode transitions are rarer than the observation window.

What the paper does well: the Bayesian pipeline is coherent and the statistical diagnostics are the right ones—trace plots, cumulative means, Gamma-method autocorrelation times, thinning, cross-chain consistency checks, and explicit minima tests. The independent chi-squared scan along auv3 (Fig. 6.2) and the Hessian comparisons (Tables 7.3–7.4) are genuine external checks, not circular. The paper also states its standard QCD assumptions plainly and gives a fair account of the limitations of both Hessian and Monte Carlo replica methods. The self-citation to the group's published proton MCMC paper is appropriate, not a red flag.

Soft spots, in proportion: the ergodicity premise is load-bearing and unresolved. If the chains are not practically ergodic, the secondary minima could be artifacts of metastable regions rather than features of the likelihood, and the MCMC-versus-Hessian comparison would be comparing a biased sample against a quadratic approximation. The thesis is honest about the “potential remnants” language, but the multimodality claim needs stronger support than that: parallel tempering, multiple well-separated initializations, or quantitative mode-hopping diagnostics. No code or data artifacts ship, so the analysis is not independently reproducible as it stands, and the main companion paper is under review. The qualitative results are also conditional on the 10-parameter CTEQ-style ansatz; multimodality inside that ansatz is not necessarily multimodality of the physical nPDFs, though that is a normal and acceptable limitation.

Who this is for: nPDF fitters and anyone working on Bayesian uncertainty quantification for PDFs. They get a useful template and a clear warning about Hessian limitations. My recommendation: send it to peer review, and ask for code/data release plus serious mixing diagnostics before the multimodality conclusion is accepted.

Referee report

Significance

If the multimodality claim survives scrutiny, this is a substantial methodological result: it would demonstrate that the standard nPDF uncertainty pipeline—linearizing around one minimum and enlarging bands by a tolerance Δχ²—misses a qualitative feature of the posterior in the valence sector, the region most relevant for the EMC effect, and that MCMC sampling is a required check rather than an optional alternative in nPDF fits with strong parameter correlations. Credit is due where the manuscript earns it: the convergence and autocorrelation diagnostics are concrete and reproducible (Γ-method, thinning, cross-chain and minima tests, Secs. 6.3 and 7.1), the central claims are checked against external benchmarks (Hessian fits, the independent χ² scan of Fig. 6.2), and the authors explicitly flag the unresolved spots in their own results (the “potential remnants of multi-modal structure” caption of Fig. 7.21). The main risk is that the secondary-minima claim is only as strong as the between-mode ergodicity of the adaptive-MH sampler; the current diagnostics, thorough within a basin, are not by themselves sufficient to establish mixing between modes.

Major objections (4)

[Sec. 7.1.2, Figs. 6.1–6.2, 7.10–7.12, 7.21] Sec. 7.1.2 (with Figs. 6.1–6.2, 7.10–7.12, 7.21): the central claim of genuine bimodality in the valence parameters (Table 7.2) requires that the adaptive-MH chains mix between the two basins, and the diagnostics of Sec. 6.3 (trace plots, cumulative means, Γ-method integrated autocorrelation) are within-basin diagnostics: a chain that spends long blocks in one metastable mode can pass all of them, and the Γ-method can underestimate the autocorrelation when between-mode transitions are rarer than the observation window. The manuscript itself documents the two signs that make this more than a formal concern: the a_uv3 trace drifts (Fig. 6.1), and the Multi-nuclei uv trace is captioned as showing “potential remnants of multi-modal structure” (Fig. 7.21). I request that the authors report: (i) the number of between-mode transitions in each independent chain and in the combined Pb-only sample; (ii) the posterior mass assigned to each mode, with its sampling uncertainty; and (iii) the integrated autocorrelation time of the mode-indicator variable itself, as a cross-check on the Γ-method values in Table 7.1. If the combined sample contains only one or two transitions, the relative weights of the two minima are effectively unconstrained, and the posterior-averaged PDFs in Figs. 7.15–7.16 inherit a normalization uncertainty that the current error bands do not display.
[Table 7.2, Figs. 7.15–7.16] Table 7.2 and Figs. 7.15–7.16: under the flat-prior Gaussian likelihood of Eq. (4.50), the posterior weight of a secondary minimum relative to the global minimum is approximately exp(−Δχ²/2), up to the local volume factors of the two basins. The physical impact of the reported secondary minima is therefore set by their Δχ² separation from the main minimum, and the paper should quote this separation for the a_dv3 and a_uv3 secondary minima together with the implied relative posterior weight. If Δχ² ≈ 2–5, the secondary mode carries a sizable fraction of the posterior mass and the multimodality claim is directly relevant to the credible bands; if Δχ² ≳ 8–10, the secondary minima contribute little to the 68% bands and the Hessian-versus-MCMC conclusion should be re-anchored to non-Gaussian tails and asymmetry rather than to multiple minima. As it stands, Table 7.2 lists the minima but the separation that determines which of these two readings is correct is not stated in the text available to the reader.
[Sec. 7.1.3, Figs. 7.14, 7.22, Tables 7.3–7.4] Sec. 7.1.3, Figs. 7.14 and 7.22, Tables 7.3–7.4: the Hessian-versus-MCMC comparison of uncertainty bands should state explicitly the Δχ² tolerance used to generate the Hessian error sets and should be presented at a coverage matched to the MCMC credible intervals. The tolerance is a free input, ranging from Δχ² = 1 (the 68% single-parameter interval under Gaussianity) to Δχ² = 35 in nCTEQ15 (Sec. 3.4.4), and wider tolerances make Hessian bands broader by construction. I recommend adding a coverage check: the fraction of the MCMC posterior mass enclosed by the Hessian 68% and 90% bands, evaluated as a function of x for the valence distributions of Fig. 7.14. This single number would convert the qualitative statement that the Hessian bands “misrepresent” the uncertainty into a quantitative one and would separate the effect of tolerance choice from the effect of genuine non-Gaussianity. The authors acknowledge this: “In practice, this assumption can fail, espe- cially in poorly constrained directions of parameter space, leading to underestimated or inaccurate uncertainty bands. • Dependence on Tolerance Choice: The uncertainty estimates are sensitive to the chosen tolerance ∆χ 2, which is not fixed by first principles and must be specified empirically.”
[Sec. 7.3, Figs. 7.23–7.24] Sec. 7.3 and Figs. 7.23–7.24: the headline comparison that including lighter nuclei “significantly constrains quark distributions and changes the shape of the extracted lead PDFs, while the gluon distribution remains comparatively stable” needs a quantitative criterion. Without a stated metric (for example, the shift of the central value relative to the combined uncertainty, or the x-dependent overlap of the 68% credible bands), the reader cannot distinguish a genuinely stable gluon from a gluon that is simply unconstrained in both fits, particularly at small x. Please define the comparison statistic and report it per flavor; the numerical stability of the MCMC bands across thinning rates (Figs. 7.25–7.26) suggests this statistic can be computed without new samples.

Minor objections (5)

[Sec. 4.3.2] The log-likelihood derivation refers to “the multivariate Gaussian likelihood in Eq. (??)”; the equation number is unresolved, which breaks the chain of references leading to Eq. (4.39).
[Sec. 3.4.4] The sentence “A detailed study in this paper demonstrates that the replica ensemble reproduces the correct posterior only in the special case of linear models with Gaussian errors” should cite Ref. [52] explicitly or say “in the present thesis”; as written, “this paper” is ambiguous in a chapter that reviews the literature.
[Chapters 2–4] Chapters 2 through 4 contain roughly eighty pages of textbook material (Standard Model, QCD, DIS, Bayesian inference, MCMC theory); for a journal version this should be condensed to the definitions and equations actually used, with the original contribution starting at Chapter 6.
[Table 7.1] Table 7.1 reports the thinning rates and autocorrelation times for “a representative chain”; reporting the maximum over chains and over parameters would be more informative, since the valence parameters are the least constrained and the slowest mixing.
[Figs. 7.25–7.26] Figures 7.25 and 7.26 have nearly identical captions (“at different thinning rates” versus “at different thinning rate”); the captions should state the two thinned intervals explicitly.

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The paper

The paper claims

By sampling the nuclear-PDF posterior directly with MCMC instead of expanding the fit statistic around a single minimum, this thesis finds non-Gaussian, multimodal parameter structure in the valence sector that Hessian uncertainty bands…

The reading

Global fits of nuclear parton distributions have long estimated parameter uncertainties with the Hessian method, which expands the fit statistic quadratically around one best-fit point and thereby assumes a Gaussian posterior. This thesis argues that for nuclear PDFs, where data are sparse and parameter correlations are strong, that assumption breaks down, and that direct sampling of the posterior with Markov Chain Monte Carlo (MCMC) reveals structure the Hessian cannot see: skewed margins, flat directions, and secondary minima, most visibly in the valence-quark parameters. To make the case it performs the first global single-nucleus fit of lead PDFs from lead data alone, with the standard mass-number dependence frozen at $A=208$, and a complementary multi-nuclei fit that lets lighter nuclei constrain the lead distributions. The practical consequence is that Hessian-based nuclear-PDF error bands inherit the Gaussian assumption; if the posterior is genuinely multimodal, those bands misstate the uncertainty in exactly the region where new nuclear data are arriving.

What carries the argument

The load-bearing machinery is the adaptive Metropolis-Hastings sampler applied to a ten-parameter nuclear-PDF ansatz (an nCTEQ-style functional form with standard analytic $A$-dependence), with the posterior proportional to $\exp(-\chi^2/2)$ built from a covariance-matrix likelihood. The algorithm adapts its proposal covariance during burn-in and then runs fixed-kernel chains; convergence is assessed through trace plots, cumulative means, and $\Gamma$-method autocorrelation times, with thinning chosen from those times. Two data setups carry the scientific comparison: a Pb-only fit ($A$ frozen at 208, lead data only) that isolates the intrinsic posterior structure, and a Multi-nuclei fit in which the $A$-dependence links lead to lighter nuclei. The observed object that distinguishes the Bayesian picture from the Hessian picture is the appearance of secondary minima in the valence sector, which a single quadratic minimum cannot represent.

Extended reading notes

Core claim

Within the nCTEQ analysis framework, the thesis constructs a Bayesian posterior $P(\theta|D)\propto\exp(-\chi^2(\theta)/2)$ over the ten parameters of its nuclear-PDF ansatz and samples it with an adaptive Metropolis-Hastings algorithm. The posterior of the Pb-only fit is reported as markedly non-Gaussian, with the valence parameters $a_{dv3}$ and $a_{uv3}$ showing secondary minima beside the main minimum, so that a Hessian expansion around the single best point misrepresents the uncertainty. Comparing fits with and without lighter-nucleus data, the thesis finds that multi-nuclei constraints shift and narrow the quark distributions while the gluon stays comparatively stable. The central scientific claim is that the valence-sector posterior is genuinely multimodal and strongly correlated, making MCMC-based sampling, rather than the Hessian approximation, the appropriate uncertainty tool for nuclear PDF fits of this kind.

Load-bearing premise

The results assume the MCMC chains have converged and visited every relevant mode of the posterior; convergence is checked only empirically, and the paper's own trace plots show parameter drift and leftover hints of multimodality, so the reported secondary minima could in principle be sampling artifacts rather than true features of the likelihood.

Editorial extensions

If this is right

If the posterior is genuinely multimodal, Hessian-built 68% uncertainty bands understate the spread of valence-quark nuclear PDFs, and every observable computed from those bands inherits the error.
MCMC sampling becomes a necessary diagnostic, not an optional cross-check, for nPDF fits whose parameter spaces are strongly correlated and weakly constrained.
The Pb-only fit establishes a controlled single-nucleus benchmark: the first lead-PDF extraction constrained by lead data alone, against which future single-nucleus fits can be compared.
Including lighter nuclei changes the extracted quark distributions but leaves the gluon stable, implying that current datasets constrain nuclear gluons only weakly in the fitted kinematic range.
The comparison maps where the Hessian quadratic approximation is adequate (approximately Gaussian, single-mode regions) and where it is not, giving future analyses a picture of the unreliable regions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same multimodality check should be applied to proton global fits, whose parameter spaces are similarly small and correlated; the companion proton MCMC study cited in the thesis is the natural place to look for valence-sector secondary minima there.
A nested-sampling or thermodynamic-integration run on the same likelihood would supply the Bayesian evidence and the relative posterior weight of each minimum, upgrading 'secondary minima exist' into a quantitative statement about how much posterior mass each mode carries.
If the gluon's stability under added $A$-dependence survives new data, it suggests the quark-sector shifts reflect genuinely improved quark constraints, while gluon insensitivity is a statement about current data coverage (mainly DIS and Drell-Yan) rather than about nuclear gluon physics itself.

Original abstract

Global analyses of nuclear parton distribution functions (nPDFs) have traditionally used the Hessian method for uncertainty estimation, relying on a local quadratic approximation of the chi-squared function. This Gaussian approximation can be inadequate when experimental constraints are limited and parameter correlations are strong, as it cannot fully capture asymmetric uncertainties, flat directions, or multiple minima. This thesis develops a Bayesian framework for nPDF determination using Markov Chain Monte Carlo (MCMC) techniques within the nCTEQ analysis framework. An adaptive Metropolis-Hastings algorithm enables direct sampling of the posterior distribution, revealing pronounced non-Gaussian behavior and secondary minima, particularly in the valence distributions. The analysis presents the first global single-nucleus determination of lead (Pb) PDFs based exclusively on lead data, complemented by a multi-nuclei analysis incorporating different nuclear targets through a standard analytic dependence on the mass number. Including lighter nuclei significantly constrains quark distributions and changes the extracted lead PDFs, while the gluon distribution remains comparatively stable. A comparison with Hessian-based results highlights the limitations of the Gaussian approximation in regions with strong nonlinearity or multimodality. Overall, this work establishes Bayesian MCMC inference as a flexible framework for nPDF determination, providing a more comprehensive characterization of parameter uncertainties and the underlying parameter space.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.89 · states limitation 0.28 · human participants 0.01 · controlled comparison 0.47 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.97 · mentions code or data availability 0.03.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Global analyses of nuclear parton distribution functions (nPDFs) have traditionally used the Hessian method for uncertainty estimation, relying on a local quadratic approximation of the chi-squared function. This Gaussian approximation can be inadequate when experimental constraints are limited and parameter correlations are strong, as it cannot fully capture asymmetric uncertainties, flat directions, or multiple minima. This thesis develops a Bayesian framework for nPDF determination using Markov Chain Monte Carlo (MCMC) techniques within the nCTEQ analysis framework. An adaptive Metropolis-Hastings algorithm enables direct sampling of the posterior distribution, revealing pronounced non-Gaussian behavior and secondary minima, particularly in the valence distributions. The analysis presents the first global single-nucleus determination of lead (Pb) PDFs based exclusively on lead data, complemented by a multi-nuclei analysis incorporating different nuclear targets through a standard analytic dependence on the mass number. Including lighter nuclei significantly constrains quark distributions and changes the extracted lead PDFs, while the gluon distribution remains comparatively stable. A comparison with Hessian-based results highlights the limitations of the Gaussian approximation in regions with strong nonlinearity or multimodality. Overall, this work establishes Bayesian MCMC inference as a flexible framework for nPDF determination, providing a more comprehensive characterization of parameter uncertainties and the underlying parameter space.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	Global analyses of nuclear parton distribution functions (nPDFs) have traditionally used the Hessian method for uncertainty estimation, relying on a local quadratic approximation of the chi-squared function.	background 1.00	0.05	0.02	asserted 1.00
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s2	This thesis develops a Bayesian framework for nPDF determination using Markov Chain Monte Carlo (MCMC) techniques within the nCTEQ analysis framework.	contribution 0.88	0.92	0.08	asserted 1.00
s3	An adaptive Metropolis-Hastings algorithm enables direct sampling of the posterior distribution, revealing pronounced non-Gaussian behavior and secondary minima, particularly in the valence distributions.	result 0.71	0.95	0.11	asserted 1.00
s4	The analysis presents the first global single-nucleus determination of lead (Pb) PDFs based exclusively on lead data, complemented by a multi-nuclei analysis incorporating different nuclear targets through a standard analytic dependence on the mass number.	contribution 0.89	0.95	0.08	asserted 1.00
s5	Including lighter nuclei significantly constrains quark distributions and changes the extracted lead PDFs, while the gluon distribution remains comparatively stable.	result 0.99	0.93	0.07	asserted 1.00
s6	A comparison with Hessian-based results highlights the limitations of the Gaussian approximation in regions with strong nonlinearity or multimodality.	interpretation 0.58	0.84	0.09	asserted 0.99
s7	Overall, this work establishes Bayesian MCMC inference as a flexible framework for nPDF determination, providing a more comprehensive characterization of parameter uncertainties and the underlying parameter space.	interpretation 0.80	0.87	0.89	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.92 · evidence from analysis of existing data 0.75.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 2.** 1: Classification of elementary particles according to the Standard Model of particle

**Figure 3.** 1: Nuclear correction ratio F Fe 2 /F D 2 , the figure taken from [126]. 1. Shadowing (x ≲ 0.1): At very low x, the nuclear structure function is suppressed relative to that of free nucleons, yielding RA(x) < 1. This suppression, known as shadowing [29, 30, 31], is believed to arise from coherent multiple scattering and the overlap of partons from different nucleons. In this regime, gluon recombination and coherence effects reduce the effective parton densities, especially for gluons and sea 4This expectation holds exactly only for isoscalar nuclei, where the number of protons equals

**Figure 6.** 2: Scan of the χ 2 function along the parameter a uv 3 in the Pb-only analysis. The parameter a uv 3 is varied while all other parameters are fixed at their values corresponding to the MCMC global minimum of the full fit. The profile exhibits a nearly constant χ 2 over a broad interval, indicating weak sensitivity of the likelihood to this parameter, together with a sharp, localized increase around a uv 3 = 0 that strongly disfavors this region. volume of the parameter space, the Markov chain can efficiently traverse it and explore the extended, nearly degenerate likelihood regi

**Figure 7.** 13: Pb-only fit: Comparison of 68% uncertainty bands for the Pb valence (uv, dv), light-sea (u¯+d¯), and gluon (g) distributions obtained using three MCMC-based estimators: cumulative-χ 2 (red), Percentile (blue), and MCSE (orange). For each flavor, the upper panel shows the absolute distributions, while the lower panel displays ratios to the cumulative-χ 2 best-fit prediction. The differences are more pronounced in the uncertainty bands. While the cumulative-χ 2 method consistently yields the widest (most conservative) uncertainties, the Percentile and MCSE bands overlap very closely

Showing 4 of the paper's 7 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.37254 by the authors.

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