{"id":"fb070d6f-6608-461c-a073-72d9c394529f","arxiv_id":"2606.19896","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Applies Bayesian design optimization to the marginalized posterior for non-linear parameter estimation under white noise, with examples of exponentially decaying signals.","lead":"This paper applies the Bayesian framework for optimal experimental design to the posterior distribution after marginalizing over linear parameters, for non-linear estimation in signals with additive white Gaussian noise. Smart generalists might read it for insights into efficient adaptive sampling strategies usable in NMR and relaxometry experiments with spin sensors.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption directly tracks the paper's own premise. With the central claim being an application under that premise, and no further technical detail available to expose a flaw in the marginalization or optimality argument, the UNVERDICTED verdict with LOW confidence remains appropriate.","tokens_in":1526,"tokens_out":233,"duration_ms":15631,"concrete_test":"Extract the explicit form of the marginal posterior (after integrating linear parameters) and the design criterion from the full manuscript; recompute the optimal design for one of the exponential-decay examples using that expression and verify it matches the reported result under the white-noise model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's assessment correctly isolates the additive white Gaussian noise assumption as foundational to applying the Bayesian design framework after marginalization over linear parameters. The abstract states this assumption explicitly and frames the contribution as an application plus discussion of implications for the given examples. No internal inconsistency, hidden assumption in the marginalization step, or unsupported leap from the stated premise is detectable from the provided information.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1560,"tokens_out":271,"duration_ms":19286,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":null},{"comment":"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.","section":null}],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"no","referee_comment":"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."}],"tokens_in":1023,"tokens_out":275,"duration_ms":15557,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper applies the Bayesian framework for design optimization to the posterior after marginalizing over linear parameters, under the assumption of additive white Gaussian noise. It then works through examples of exponentially decaying signals, with and without oscillations, and notes the relevance to NMR and relaxometry using solid-state spin sensors.\n\nWhat it does is take a standard approach and show how the marginalization step changes the sampling problem for the nonlinear parameters. The choice of examples is practical and directly tied to the mentioned applications, which keeps the discussion grounded.\n\nThe soft spots are that the abstract provides no derivations, no optimization details, and no validation against other methods or real data. Without those, it's impossible to judge whether the adaptive strategies actually improve estimation accuracy or remain computationally tractable. The white noise assumption is stated up front and is the usual one, but any real experiment would need to check how sensitive the designs are to that assumption.\n\nThis is for people already working in Bayesian experimental design or in parameter estimation for decaying signals. A reader in NMR relaxometry might pick up a useful concrete case, but someone outside those niches will not find a new method or surprising result.\n\nI would send it to peer review if the full paper contains the actual math and some numerical checks, because the topic is relevant to a small but active subfield and the extension looks reasonable on the surface. It is not a major advance, but it could still be worth referee time for the application details.","headline":"Applies Bayesian optimal design to the marginalized posterior for nonlinear parameters under white Gaussian noise, with examples in exponential decay signals for NMR.","tokens_in":2011,"tokens_out":367,"would_cite":false,"duration_ms":16452,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Bayesian optimal design for non-linear parameters is obtained by optimizing the posterior after marginalizing linear ones under white Gaussian noise.","keywords":["Bayesian optimal design","non-linear parameter estimation","marginalized posterior","white Gaussian noise","exponential decay","adaptive sampling","nuclear magnetic resonance"],"falsifier":"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.","tokens_in":2426,"feed_emoji":"","tokens_out":628,"duration_ms":21714,"temperature":0.7,"pith_summary":"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.","feed_headline":"Bayesian design optimizes sampling after marginalizing linear parameters","feed_subtitle":"Under white Gaussian noise the method focuses design effort on non-linear quantities, illustrated with exponential-decay examples for NMR.","key_machinery":"The marginalized posterior distribution over the non-linear parameters, which serves as the objective for the Bayesian design optimization.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Bayesian design optimizes nonlinear estimates after linear marginalization","Adaptive Bayesian sampling for nonlinear parameters post linear marginalization","Bayesian framework enables optimal sampling after marginalizing linear params","Marginalization over linear params yields Bayesian optimal nonlinear sampling"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The noise must be additive white Gaussian for the Bayesian framework to apply directly to the design optimization.","fun_headline_variants_meta":{"raw":{"variants":["Bayesian design optimizes nonlinear estimates after linear marginalization","Adaptive Bayesian sampling for nonlinear parameters post linear marginalization","Bayesian framework enables optimal sampling after marginalizing linear params","Marginalization over linear params yields Bayesian optimal nonlinear sampling"]},"model":"grok-4.3","cost_usd":0.006376,"raw_usage":{"total_tokens":2904,"prompt_tokens":492,"num_sources_used":0,"completion_tokens":63,"cost_in_usd_ticks":63762000,"prompt_tokens_details":{"text_tokens":492,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2349,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":492,"tokens_out":63,"duration_ms":20756,"temperature":1.0,"reasoning_tokens":2349,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T15:06:34.824283+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}