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arxiv: 2508.14902 · v2 · submitted 2025-08-07 · 🧬 q-bio.QM · physics.data-an

Hierarchical Maximum Likelihood Estimation for Time-Resolved NMR Data

Lennart H. Bosch , Pernille R. Jensen , Nico Striegler , Thomas Unden , Jochen Scharpf , Usman Qureshi , Philipp Neumann , Martin Gierse
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John W. Blanchard Stephan Knecht Jochen Scheuer Ilai Schwartz Martin B. Plenio
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Pith reviewed 2026-05-19 00:46 UTC · model grok-4.3

classification 🧬 q-bio.QM physics.data-an
keywords Bayesian hierarchical modeltime-resolved NMRVariable Projectionuncertainty propagationhyperpolarized metabolitesleast-squares optimizationmetabolic monitoringnitrogen-vacancy detection
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The pith

A Bayesian hierarchical model reduces time-resolved NMR analysis to an extended least-squares optimization that propagates uncertainties through the full dataset.

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

The paper establishes a new estimation procedure for multidimensional hyperpolarized NMR data that starts from a Bayesian hierarchical model and derives an analytic reduction to a single least-squares problem. This approach operates directly on the entire time-resolved dataset rather than splitting the work into separate stages, which allows uncertainties to propagate intrinsically instead of being approximated after the fact. A sympathetic reader would care because metabolic monitoring and reaction-rate estimation depend on precise quantification with reliable error bars, and the method is shown to outperform both Fourier-based analysis and a two-stage Variable Projection procedure in concrete experiments. The reduction extends the classic Variable Projection technique to data scenarios that have two predictors, preserving the computational efficiency of least-squares while retaining the statistical advantages of the hierarchical model. The authors demonstrate the method on both conventional high-field NMR and a micron-scale setup using nitrogen-vacancy centers, and note that the same structure applies to other time-resolved spectroscopic estimation tasks.

Core claim

The hierarchical Bayesian model for time-resolved NMR data with two predictors can be analytically reduced to a least-squares optimization problem that extends the Variable Projection method; when applied to hyperpolarized metabolite signals recorded with both conventional and NV-center NMR hardware, the resulting estimates exhibit higher precision and better uncertainty quantification than either Fourier methods or a two-stage VarPro pipeline.

What carries the argument

Analytic reduction of the Bayesian hierarchical model to a least-squares optimization problem that extends Variable Projection for data scenarios possessing two predictors.

Load-bearing premise

The analytic derivation that converts the full hierarchical Bayesian estimation into an ordinary least-squares problem accurately preserves the uncertainty propagation without introducing approximation errors.

What would settle it

Apply the reduced least-squares procedure and the original two-stage VarPro pipeline to the same hyperpolarized NMR datasets and check whether the hierarchical method produces measurably smaller uncertainty intervals while maintaining unbiased metabolite concentration and rate estimates.

Figures

Figures reproduced from arXiv: 2508.14902 by Ilai Schwartz, Jochen Scharpf, Jochen Scheuer, John W. Blanchard, Lennart H. Bosch, Martin B. Plenio, Martin Gierse, Nico Striegler, Pernille R. Jensen, Philipp Neumann, Stephan Knecht, Thomas Unden, Usman Qureshi.

Figure 1
Figure 1. Figure 1: (a) Schematic view of the two-dimensional structure of the data for a two-step conversion reaction. Shaded curves indicate the expected evolution of the three signal amplitudes. (b) Graphical representation of the hierarchical model. (c) Illustrative effects of the hierarchical approach onto individual amplitude estimates. Estimates are corrected using the second-level model while simultaneously reducing t… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Reaction rate estimates from three different approaches derived from simulated data for different realizations of noise. Shaded areas indicate uncertainty range. Agreement of uncertainty esti￾mates with the CRB is associated with the shaded area hosting about two thirds of the data points. Disagreement is shown only for VarPro+LS with p = 0.028 < 0.05. (b) Reaction rate estimates derived from experimen… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Microscopical illustration of micro-NMR of hyperpolarized fumarate in solution with NV ensembles in diamond. An ensemble of electronic NV center spins in diamond are optically initialized and readout. The NMR signal of fumarate in solution is detected by measuring the dipolar spin interaction of the electronic spins in diamond with nuclear spins in solution placed on top of the diamond surface. The ext… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Real part of the J-coupling induced evolution over time obtained via HML, OLS, and AUC estimates from NV detected micronscale experiments with hyperpolarized [1-13C] fumarate. (b) Real part the FFT of (a). Compared to the AUC approach, the SNR is improved by 2.6 by the HML analysis. (c) The J-coupling signal intensity for different runs of the same experiment as in (a) upon renewal of the sample subjec… view at source ↗
read the original abstract

Metabolic monitoring and reaction rate estimation using hyperpolarized NMR technology requires accurate quantitative analysis of multidimensional data scenarios. Currently, this analysis is often performed in a two-stage procedure, which is prone to errors in uncertainty propagation and estimation. We propose an approach derived from a Bayesian hierarchical model that intrinsically propagates uncertainties and operates on the full data to maximize the precision at minimal uncertainty. In an analytic treatment, we reduce the estimation procedure to a least-squares optimization problem which can be understood as an extension of the Variable Projection (VarPro) approach for data scenarios with two predictors. We investigate the method's efficacy in two experiments with hyperpolarized metabolites recorded with conventional high-field NMR devices and a micronscale NMR setup using Nitrogen-Vacancy centers in diamond for detection, respectively. In both examples, the new approach improves estimates compared to Fourier methods and proves operational advantages over a two-stage procedure employing VarPro. While the approach presented is motivated by NMR analysis, it is straightforwardly applicable to further estimation scenarios with similar data structure, such as time-resolved photospectroscopy.

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 / 2 minor

Summary. The paper proposes a hierarchical maximum likelihood estimation method derived from a Bayesian hierarchical model for analyzing time-resolved multidimensional NMR data. It analytically reduces the procedure to a least-squares optimization extending Variable Projection (VarPro) to two-predictor scenarios, with the goal of intrinsic uncertainty propagation and full-data operation. Efficacy is investigated via two hyperpolarized NMR experiments (high-field and NV-center micronscale setups), claiming improved estimates over Fourier methods and two-stage VarPro.

Significance. If the analytic reduction holds exactly while preserving uncertainty propagation, the approach would offer a principled alternative to two-stage procedures for quantitative metabolic monitoring in hyperpolarized NMR, with straightforward extension to similar time-resolved spectroscopy. The full-data operation and VarPro extension are potential strengths for precision gains.

major comments (2)
  1. [§3] §3 (analytic treatment of hierarchical model): The central claim requires that the Bayesian hierarchical posterior reduces exactly to an extended VarPro least-squares objective for two predictors while preserving intrinsic uncertainty propagation. The derivation does not explicitly show the profiling/marginalization steps or confirm that no independence approximations between predictors are introduced, leaving open whether the claimed advantages over two-stage VarPro are exact.
  2. [§5] §5 (experimental validation): The reported improvements in precision for the two NMR datasets lack tabulated numerical values, error bars, or direct statistical comparisons (e.g., variance reduction factors) against Fourier and two-stage VarPro baselines, making it difficult to assess whether the gains are load-bearing or merely qualitative.
minor comments (2)
  1. [§2] Clarify notation for the two predictors (e.g., time and frequency dimensions) when defining the extended VarPro objective to ensure reproducibility.
  2. [§4] Add a brief pseudocode or algorithmic outline for the resulting least-squares solver to aid implementation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment below and describe the revisions we will implement.

read point-by-point responses

  1. Referee: [§3] §3 (analytic treatment of hierarchical model): The central claim requires that the Bayesian hierarchical posterior reduces exactly to an extended VarPro least-squares objective for two predictors while preserving uncertainty propagation. The derivation does not explicitly show the profiling/marginalization steps or confirm that no independence approximations between predictors are introduced, leaving open whether the claimed advantages over two-stage VarPro are exact.

    Authors: We agree that the analytic treatment would be strengthened by greater explicitness. The reduction proceeds by profiling the linear coefficients from the hierarchical posterior; under the Gaussian noise model this profiling is exactly equivalent to marginalization and introduces no independence assumptions between the two predictors. In the revised manuscript we will expand §3 with a detailed, step-by-step derivation of the profiling steps and will explicitly state that no such approximations are used, thereby confirming that the claimed advantages over two-stage VarPro hold exactly within the model. revision: yes

  2. Referee: [§5] §5 (experimental validation): The reported improvements in precision for the two NMR datasets lack tabulated numerical values, error bars, or direct statistical comparisons (e.g., variance reduction factors) against Fourier and two-stage VarPro baselines, making it difficult to assess whether the gains are load-bearing or merely qualitative.

    Authors: The referee correctly notes that the experimental results are presented qualitatively. We will revise §5 to include tables reporting the estimated parameters and their uncertainties for the hierarchical, Fourier, and two-stage VarPro methods on both datasets, together with explicit variance-reduction factors and other direct statistical comparisons. These additions will allow quantitative evaluation of the precision gains. revision: yes

Circularity Check

0 steps flagged

No significant circularity; analytic reduction from hierarchical Bayesian model to extended VarPro is self-contained

full rationale

The paper derives its estimator from a Bayesian hierarchical model and performs an analytic reduction to a least-squares problem extending Variable Projection for two-predictor data. This reduction is presented as a mathematical treatment of the posterior mode or marginal likelihood rather than a reparameterization of fitted quantities or a self-citation chain. The central claim of improved uncertainty propagation and full-data operation follows directly from the hierarchical structure without reducing to inputs by construction. No load-bearing self-citations, ansatz smuggling, or renaming of known results are indicated in the abstract or derivation outline. The method is motivated by NMR but explicitly generalizes to similar time-resolved scenarios, confirming independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only information prevents identification of concrete free parameters or invented entities; the approach rests on standard Bayesian modeling assumptions for time-resolved data.

axioms (1)
  • domain assumption The data structure permits an analytic reduction of the hierarchical model to a least-squares problem extending Variable Projection
    Invoked in the analytic treatment described in the abstract.

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discussion (0)

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Reference graph

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