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arxiv: 2606.19896 · v1 · pith:5RYJ2TITnew · submitted 2026-06-18 · ⚛️ physics.data-an

Optimal and Adaptive Bayesian Sampling for Non-Linear Parameter Estimation under White Noise

Lennart H. Bosch , Martin B. Plenio This is my paper

Pith reviewed 2026-06-26 15:06 UTC · model grok-4.3

classification ⚛️ physics.data-an
keywords Bayesian optimal designnon-linear parameter estimationmarginalized posteriorwhite Gaussian noiseexponential decayadaptive samplingnuclear magnetic resonance
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The pith

Bayesian optimal design for non-linear parameters is obtained by optimizing the posterior after marginalizing linear ones under white Gaussian noise.

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

The paper applies Bayesian experimental design to the problem of choosing sampling times that minimize uncertainty in non-linear parameters. It does so by first integrating out any linear parameters from the posterior, leaving an effective distribution over the quantities of interest. This is done under the assumption of additive white Gaussian noise. Concrete examples are given for exponentially decaying signals, both with and without oscillations, and the approach is connected to nuclear magnetic resonance and relaxometry measurements. A reader would care because the method supplies a systematic way to decide when to take data in order to estimate decay rates or frequencies more efficiently.

Core claim

The central claim is that the Bayesian framework for design optimization, when applied to the posterior distribution obtained after marginalization over linear parameters, yields optimal and adaptive sampling strategies for estimating non-linear parameters in the presence of additive white Gaussian noise, as demonstrated through examples of exponentially decaying signals.

What carries the argument

The marginalized posterior distribution over the non-linear parameters, which serves as the objective for the Bayesian design optimization.

If this is right

  • Optimal sampling times can be computed for exponentially decaying signals by maximizing the expected information gain in the marginalized posterior.
  • The same procedure extends directly to oscillating exponential decays.
  • Adaptive sampling becomes possible by updating the marginalized posterior after each measurement and re-optimizing the next design point.
  • The resulting designs improve efficiency for parameter estimation tasks such as those arising in nuclear magnetic resonance and relaxometry with solid-state spin sensors.

Where Pith is reading between the lines

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

  • The marginalization step may reduce computational cost when the number of linear parameters is large relative to the non-linear ones.
  • If the white-noise assumption is relaxed, the same marginalization idea could be tested with colored noise models to see whether the optimality properties survive.
  • The approach suggests a general template for separating linear and non-linear contributions in other inverse problems that admit closed-form marginalization.

Load-bearing premise

The noise must be additive white Gaussian for the Bayesian framework to apply directly to the design optimization.

What would settle it

An experiment that compares parameter-estimation errors obtained from the paper's recommended sampling times against those from uniform or heuristic sampling, under controlled additive white Gaussian noise; if the recommended times do not reduce error, the claim is falsified.

Figures

Figures reproduced from arXiv: 2606.19896 by Lennart H. Bosch, Martin B. Plenio.

Figure 1
Figure 1. Figure 1: (a) Example plot of an oscillating signal with a single frequency subject to Lorentzian decay over time in units of its lifetime τˆ. The gray shaded signal correspond to a re-initialization at two times the lifetime. (b) Plot of the posterior distributions logarithm together with its quadratic approximation and the correction by linear expansion of maximum and curvature in the noise. (c) Plot of the full p… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Expected information entropy over the latest sam￾pling time as obtained via nested integration and the leading-order term of the Laplace approximation while keeping all previous fixed for the simple exponential model discussed in Example A of sec￾tion 3. The next sample is taken each at location of the inverted tri￾angles. (b) Ratio of the expected information entropy from Laplace approximation and int… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Analytical uncertainty from eq. (42) and as obtained by numerical fit of synthetic data with artifical noise for κˆ = 1. The shaded area is drawn to indicate the remaining signal amplitude. (b) Analytic uncertainty multiplied by √ t (normalized) to reveal the optimal time spent per repetition of the measurement and the loss in precision upon detuning from the optimal reinitialization rate. (c) Estimate… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Frequency estimate over the number of samples for iterative sampling with the shaded area indicating estimated uncertainty. Blue lines indicate the expected uncertainty for uniform sampling on t ∈ [0, 2/κˆ], uncertainty obtained from batch optimization is indicated by the gray dashed lines. Divergence in blue curves occur when the sampling frequency is close to the signal frequency and the signal appea… view at source ↗
Figure 5
Figure 5. Figure 5: Histograms of parameter estimates for the exponentially decaying signal in (a) and for the oscillating signal in (b)-(c) af￾ter 100 iteratively obtained samples. Under- and overflow bins make up approximately 7.1 % and 9.1 % in histograms (b) and (c), respectively. Vertical lines indicate true values of the respective pa￾rameters. imum of ϕ with ϕ({tj}) = S2 − S 2 1/S0 (C.11) where Sl = ∑ n j=1 t l j e −2κ… view at source ↗
Figure 6
Figure 6. Figure 6: Schematic plot of ϕ({tj}) after fixing all tj but one, illustrating that ϕ always takes higher values at t∗ + compared to t ∗ −. Inserting the values for t ∗ − and t ∗ + derived from eqs. (C.16) and (C.17), respectively, simplifies the dif￾ference to S˜ 0(1 + κδ) 2 κ 2(1 + S˜ 0 exp(2 + 2κ(δ + S˜ 1/S˜ 0))) . (C.19) Above expression is evidently positive, indicating that ϕ(t ∗ +) > ϕ(t ∗ −) upon fixing all r… view at source ↗
read the original abstract

The question of optimal experimental design has been addressed in a vast variety of contexts and answered using manifold approaches. Assuming additive white Gaussian noise, this work applies the Bayesian framework for design optimization to the posterior distribution after marginalization over linear parameters and discusses the implications. Examples of exponentially decaying signals with and without oscillations complement the discussion. Application of the examples considered include but are not limited to nuclear magnetic resonance and relaxometry experiments using solid-state spins sensors.

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 applies the Bayesian framework for optimal experimental design to the posterior distribution after marginalization over linear parameters, under the assumption of additive white Gaussian noise. It discusses implications of this approach and complements the discussion with examples of exponentially decaying signals, both with and without oscillations, relevant to nuclear magnetic resonance and relaxometry experiments using solid-state spin sensors.

Significance. Marginalization over linear parameters before optimizing designs for non-linear ones is a standard technique that reduces the dimensionality of the optimization problem in Bayesian experimental design. The examples illustrate potential practical implications for adaptive sampling in physical measurement contexts. The work is an application of an established framework rather than a derivation of new theory.

minor comments (2)
  1. The abstract states the core contribution but provides no indication of the specific design criterion (e.g., expected information gain) or the form of the marginalized posterior used; adding one sentence on this would improve clarity for readers.
  2. The title emphasizes both 'Optimal and Adaptive' sampling, yet the abstract focuses on design optimization; ensure the full text explicitly distinguishes or connects the adaptive aspect to the optimal design results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review of our manuscript. We appreciate the recognition that our examples have potential practical implications for adaptive sampling in physical measurement contexts such as NMR and relaxometry.

read point-by-point responses

  1. Referee: Marginalization over linear parameters before optimizing designs for non-linear ones is a standard technique that reduces the dimensionality of the optimization problem in Bayesian experimental design. The examples illustrate potential practical implications for adaptive sampling in physical measurement contexts. The work is an application of an established framework rather than a derivation of new theory.

    Authors: We agree that marginalization over linear parameters is a standard technique that reduces the dimensionality of the design optimization. Our manuscript applies this established approach specifically to the marginalized posterior for non-linear parameter estimation under additive white Gaussian noise. The contribution consists of a focused discussion of the implications of this choice together with concrete examples of exponentially decaying signals (with and without oscillations) that are directly relevant to nuclear magnetic resonance and relaxometry experiments using solid-state spin sensors. While the underlying Bayesian framework is not new, the specific application to this setting and the accompanying analysis of adaptive sampling strategies provide practical guidance that, to our knowledge, has not been presented in this form for these experimental contexts. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract frames the work as an application of the Bayesian design optimization framework to the marginalized posterior under the explicit additive white Gaussian noise assumption, with examples of exponentially decaying signals. No equations, derivations, self-citations, or load-bearing steps are present in the provided text that reduce a claimed prediction or result to its own inputs by construction. The central contribution is presented as an application plus discussion of implications, with no evidence of self-definitional fits, renamed known results, or uniqueness theorems imported from the authors' prior work. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no details provided on free parameters, axioms, or invented entities.

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