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Use of Bayesian Inference to Diagnose Issues in Experimental Measurements of Mechanical Disk Resonators

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Bayesian modeling of the split-photodiode nonlinearity recovers disk-resonator decay constants up to 25% more accurately than simple exponential fitting, and fits ringdowns the old method discarded.

desk verdict Genuinely useful Bayesian framework for ringdown analysis with a real novelty in the photodiode model, but the abstract's 25% claim is unsupported and the likelihood has a documented noise-floor limitation. read the letter →

arxiv 2505.17346 v1 pith:SLGX43W3 submitted 2025-05-22 physics.ins-det

Simon C. Tait , Michael J. Williams , Joseph Bayley , Bryan W. Barr , Iain Martin This is my paper · ORCID
classification physics.ins-det
keywords mechanicallossBayesianinferencenestedsamplingringdownmeasurementsplitphotodiodecoatingBrownianthermalnoisegravitationalwavedetectorsGentleNodalSuspension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper seeks to establish that the standard analysis of mechanical ringdowns of coated disk resonators—fitting the beat between two decaying modes as a pair of simple exponentials—is systematically biased by the nonlinear response of the split-photodiode readout and by amplitude-dependent laser noise, and that a Bayesian treatment removes both problems. The authors introduce three signal models of increasing complexity, the most complete being an analytic power-series approximation (M3, with $T=7$) of the true photodiode response, paired with a likelihood whose noise variance grows with signal amplitude. Using 50 simulated ringdowns and 47 real measurements, they report that the refined model recovers the decay constants $\tau_1$ and $\tau_2$ with up to 25% better accuracy at large oscillation amplitudes, and that six real ringdowns that previously failed to fit can be analyzed without modification. This matters because these decay constants set the mechanical loss of coating materials, which controls coating Brownian thermal noise—currently the sensitivity limit of gravitational-wave detectors.

What carries the argument

The load-bearing object is the analytic signal model M3 together with the two-component noise likelihood. M3 starts from a Gaussian-beam model of a split photodiode with a finite gap, where the differential voltage is a difference of error functions; expanding that response as a Taylor series in the beam position $\mu$ to order $T$ gives $V_{S,T} = \sum_{k=0}^{T} C_k \mu^k$, with coefficients fixed by the gap and beam size. Demodulating at the mode-pair midpoint frequency and low-pass filtering casts the squared readout as a sum of beat harmonics, $s_3^2(t) = X_0 + X_1 \cos(\Delta\omega t + \Delta\varphi) + \cdots + X_T \cos(T\Delta\omega t + T\Delta\varphi)$, whose coefficients carry powers of the decaying amplitudes $A_1, A_2$ and the decay constants $\tau_1, \tau_2$. The likelihood treats each ringdown sample as Gaussian with variance $\xi_i^2 = s(t_i)^2 \xi_A^2 + \xi_S^2$, adding an amplitude-proportional relative-intensity-noise term to a stationary electronic-noise term, and nested sampling with normalizing flows is used to compute posteriors and Bayes factors for model comparison.

What would settle it

Generate 50 simulated ringdowns with the full numerical photodiode model (M2) and add a hard FFT noise floor so that the signal drops below it in the second half of each record; then run the M3, $T=7$ inference. If the posterior medians for $\tau_1$ and $\tau_2$ are systematically displaced from the injected values, or if the 3-$\sigma$ intervals exclude the truth in more than a few percent of cases, the two-component Gaussian likelihood fails for low-amplitude tails and the claimed accuracy gain does not generalise.

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Extended reading notes

Core claim

The central discovery, on the paper's own terms, is that the bias in ringdown parameter estimation comes from two effects the old analysis ignored: the Gaussian laser spot sweeping across the photodiode gap produces an amplitude-dependent, harmonic-generating response, and the readout noise is not stationary but scales with the instantaneous signal. The paper shows that modeling both effects with M3 and a two-component Gaussian likelihood recovers the true injected $\tau_1$ and $\tau_2$ in simulations where the simple model M1 is often biased—more than half of M1's posteriors exclude the true value, while M3 excludes it in only four ringdowns, all of which dip below the relative-intensity-noise floor. On real data the same framework yields tighter, more consistent mechanical-loss estimates across mode families and eliminates the spurious outliers and outright failures of the original method, including fits to six ringdowns that had previously been unanalysable.

Load-bearing premise

The load-bearing premise is that every ringdown data point is an independent Gaussian random variable whose variance is the signal squared times an amplitude-noise term plus a stationary-noise term; in reality each point is the peak of a 0.2-second FFT power spectrum, so the statistics are extreme-value rather than Gaussian and the model breaks down once the signal falls to the noise floor.

Editorial extensions

If this is right

  • Previously discarded single-decay ringdowns—six in this dataset—can be fitted without special handling, yielding posterior distributions for both $\tau_1$ and $\tau_2$ instead of failing.
  • Mechanical-loss estimates become more tightly clustered across repeated measurements of the same mode family, and the spurious outliers in $1/Q_2$ seen with the original method disappear.
  • The Bayes factor between M1 and M3 tells the experimenter when a given measurement needs the nonlinear readout model and when the simple exponential model suffices, so model complexity can be matched to oscillation amplitude.
  • Up to 25% better accuracy in $\tau_1$ and $\tau_2$ translates directly into smaller uncertainties in the inferred coating loss $\phi(f) = 1/(\pi f \tau)$, and hence in predicted coating Brownian thermal noise.
  • Residuals and posterior widths from the Bayesian fit expose amplitude-dependent distortions in the apparatus that the old analysis hid, turning the fitter into a diagnostic tool.

Reading between the lines

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

  • Because each ringdown data point is the maximum of a 0.2 s FFT power spectrum, replacing the Gaussian likelihood with an extreme-value model for that maximum is a direct way to remove the noise-floor bias the paper itself documents in Appendix B.1; this is my inference, not a claim the paper makes.
  • The same Taylor-expansion-of-the-error-function trick should transfer to any optical-lever or split-detector measurement where a Gaussian spot scans across a gap, so the framework is likely applicable beyond disk resonators to other mechanical loss measurements.
  • Switching the readout from FFT-bin maxima to heterodyne or lock-in demodulation, which the authors mention, would simplify the likelihood and reduce the computational cost of the nested-sampling analysis while sidestepping the noise-floor problem.
  • The model-comparison logic could be used diagnostically in reverse: a low-amplitude ringdown that strongly prefers M3 may signal a misaligned beam, an unexpected beam size, or an asymmetric photodiode response, giving a data-driven check on apparatus alignment.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The manuscript presents a Bayesian framework for extracting the two decay constants tau1 and tau2 from GeNS ringdown measurements of coated disk resonators used in gravitational-wave coating loss studies. Three signal models are considered: a traditional two-exponential model (M1), a numerical split-photodiode model (M2) that includes the Gaussian beam profile and photodiode gap nonlinearity, and an analytic power-series approximation (M3) that enables fast likelihood evaluation. The likelihood in Eq. (24) combines stationary and amplitude-proportional (RIN-like) Gaussian noise. The models are validated on 50 simulated ringdowns generated with M2 and applied to 47 real ringdowns, with model comparison performed via nested sampling (nessai) and Bayes factors. The paper claims up to 25% improvement in estimation accuracy over traditional methods, especially for larger oscillation amplitudes, and reports that six previously unfittable ringdowns can now be analyzed without adjustment.

Significance. The framework addresses a genuine and consequential problem: photodiode nonlinearity and RIN noise create model misspecification that biases conventional exponential fits, and the paper offers a physically motivated correction with principled model comparison. The analytic M3 construction, the two-component noise model with standardized residual diagnostics, and the public implementation (bayesbeat) with a data release are notable strengths, as is the unusually candid documentation of failure modes (four biased simulations, one tau1/tau2 swap, and prior-width sensitivity in Appendix B.1). If the accuracy claims hold after the requested quantification, this is a useful contribution to the mechanical-loss and coating thermal noise community and to the instrumentation readership of Classical and Quantum Gravity.

major comments (3)
  1. [Abstract; Section 3.2 (Figure 4); Section 5.2 (Figure 10)] The abstract's headline claim of 'improvements in estimation accuracy by up to 25%' is not supported by the accuracy metrics reported in the paper. The only 25% figure in the manuscript is in Section 3.2 (Figure 4), where it quantifies the difference between the predicted amplitudes of the T=1 and T=7 signal models, not the error in the inferred decay constants. The accuracy comparison in Figure 10 is reported qualitatively ('M3 is consistently closer to the true value'), with no percentage improvement, and the text also states that at lower amplitudes neither model is favoured. Please report a quantitative accuracy metric (e.g., median or 90th percentile of |tau_hat - tau_true|/tau_true across the 50 injections, binned by amplitude) or reword the abstract so that the 25% figure is attributed to model-output differences rather than to estimation accuracy.
  2. [Eq. (24); Eq. (2); Appendix B.1; Section 7] The likelihood in Eq. (24) treats each ringdown sample as an independent Gaussian with variance s(t_i)^2 xi_A^2 + xi_S^2, but each d(t_i) in Eq. (2) is the maximum of an FFT power spectrum over a 0.2 s window; its distribution is an extreme-value statistic with a positive noise floor that does not vanish as s -> 0. The authors themselves document the consequence: Appendix B.1 reports biased tau estimates in four of the 50 simulated ringdowns when the signal falls below the FFT/RIN noise floor, including one case with a tau1/tau2 swap, and Section 7 concedes that the model does not account for the FFT noise floor. This is precisely the regime relevant to the claim in Section 6 that six previously discarded real ringdowns, described in Appendix B.3 as low-amplitude and often single-decay signals, 'can now be reliably analysed.' As it stands, the unconditional 'superior estimation accuracy' of the abstract is not established for these low-SNR tails. I request either a quantitative assessment of the bias for the affected real ringdowns (e.g., a posterior predictive check that includes a noise-floor term, or an analysis demonstrating that those ringdowns do not enter the low-SNR regime) or a scope-limited wording of the claim.
  3. [Section 5.2; Eq. (13); Section 3.2] The simulated-data validation is largely a self-consistency check within one model family: the injections are generated with M2 (Eq. (13)) and analyzed with M3 (Section 3.2), an analytic approximation of the same photodiode model. The recovery of the injected tau values therefore demonstrates internal consistency of the signal-model family rather than the adequacy of the signal model against an independent ground truth; the comparison that matters for the paper's improvement claim is M3 versus M1, and both share the same, possibly misspecified, Gaussian likelihood of Eq. (24). A concrete strengthening would be to inject simulated data using the extreme-value statistic of Eq. (2) including the noise floor, or to validate M3 against a measured photodiode response such as the scan in Figure 2b, which would test the absolute accuracy of the inferred decay constants rather than only the relative improvement over M1.
minor comments (7)
  1. [Section 4.2, after Eq. (21)] The two noise components are mislabeled in the text: 'The first, nA is stationary Gaussian noise' and 'The second, nA is amplitude dependent noise' both refer to nA, whereas in Eq. (22) the multiplicative amplitude-dependent component is nA and the stationary component is nS; the text should be corrected to match Eq. (22).
  2. [Section 5.1 and Figure 5 caption] The condition 'xi_A < 0' appears twice ('allowing xi_A < 0 has minimal effects of the signal fit' and 'with only stationary noise versus (xi_A = 0) stationary and amplitude dependent noise (xi_A < 0)'); the intended condition is xi_A > 0, and the sign flip will confuse readers.
  3. [Eq. (17)] As typeset, the first and third terms on the right-hand side of Eq. (17) are identical and opposite in sign, so the 3*omega_1 contribution cancels exactly; presumably one sign is a typographical error. Since the full expressions for M3 are only given in the bayesbeat code, this displayed equation should be corrected so that the derivation is checkable from the paper alone.
  4. [Section 5.2, near Figure 8] The phrase 'at lower amplitudes neither model is favoured (log10 B = 1)' is inconsistent with the Jeffreys scale defined in Section 4, where log10 B > 1 is strong evidence in favor of one model; the scatter in Figure 8 suggests values near zero at low amplitude, so this should presumably read 'log10 B is close to 0'.
  5. [Section 1 and Reference [1]] The in-text claim that 'the landmark detection of GW150914_095045 in 2015 [1]' is supported by reference [1], which is the 2016 Living Reviews article by Abbott (arXiv:1304.0670), not the GW150914 discovery paper; the correct citation is Abbott et al., Phys. Rev. Lett. 116, 061102 (2016). In addition, the sentence ending '...calculated (see Figure 1). to the isotropy of the substrate...' appears to be missing the word 'Due' at the start of the second clause.
  6. [Appendix A.1] The beat-frequency prior is set using the same data that are subsequently analyzed ('This is achieved by applying a fourth-order low-pass butter filter, computing the FFT of the data, and then finding any frequencies...'), which is a double use of the data in the evidence calculation; given the enormous Bayes factors the conclusions are unlikely to change, but this data-driven prior should be acknowledged or fixed independently of the data.
  7. [Appendix C and Figure 10 caption] Two small presentation issues: in Figure 10 the caption has an unclosed parenthesis ('M3 with T = 7 (orange when analysing 50 simulated ringdowns'), and in Appendix C the text reads 'the analyses with M3 taken longer' instead of 'take longer'.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the tau estimates come from a likelihood fit, and the real-data/Vajente comparison supplies an external baseline; the M2-generated simulations make the Section 5 validation partly a self-consistency check, and the FFT noise-floor limitation (Section 7, Appendix B.1) narrows but does not circularize the accuracy claim.

full rationale

The paper's central claim is that Bayesian inference with M3 (T=7) plus a two-component noise model recovers tau1 and tau2 more accurately than simplified M1. This is not circular at the construction level: the tau values are free parameters in the likelihood (Eq. 24) and are inferred from data; no fitted parameter is relabelled as a prediction, and no equation defines the target result in terms of itself. The main soft spot is that the simulated truth is generated from M2 (Eq. 13), while M3 is derived from M2 by Taylor expansion (Eqs. 15-16), so Section 5's recovery of injected taus is partly a self-consistency test within the M2/M3 model family rather than an independent validation. The authors concede this limitation: Appendix B.1 reports four simulated ringdowns with biased tau estimates (including one tau1/tau2 swap) when the signal falls below the FFT noise floor, and Section 7 states that the model 'does not account for the noise floor that results from applying the FFT' and that 'the data will no longer be Gaussian' in that regime. This limits the scope of the claimed superior accuracy in low-amplitude tails, but it is a statistical-model misspecification, not circularity. The paper also provides external anchors: M1 is the published Vajente model, real ringdowns are compared with Vajente's original estimates, and six previously 'unfittable' ringdowns are analysed without adjustment. Self-citations (nessai, bayesbeat, and Tait et al. [9]) are to software and stated modelling choices that are code-reproduced or derived in the text; no load-bearing uniqueness theorem is imported. The abstract's 'up to 25%' improvement figure traces to Figure 4, which compares model amplitudes for T=3,5,7 versus T=1 rather than directly reporting tau-recovery statistics; this is a presentation weakness, not a circular derivation. Verdict: no circular step identified; score 2 reflects minor self-citations and the self-consistency validation caveat.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the Vajente two-mode ringdown model, a specific optical readout model, a Gaussian noise assumption, and user-chosen values such as T=7 and the beat-frequency prior width. No new physical entity is introduced.

free parameters (5)
  • Truncation order T = 7
    Chosen by hand for the M3 analyses; the claimed nonlinear correction and the reported Bayes factors depend on this choice. Higher T captures more harmonics at greater computational cost.
  • Beat-frequency prior width δ = 0.05 Hz for real data, 0.2 rad for simulated data
    User-specified and set partly from the data via an initial FFT search. Appendix B.1 notes that a tau-swapping bias can be avoided by making priors narrower, so this choice affects the results.
  • Photodiode gap xg = 0.25 mm
    Fixed from measurement; enters all M2/M3 nonlinearity coefficients. If this value is wrong, the nonlinear correction is miscalibrated.
  • Beam radius σ = 1.5 mm
    Reported as measured; determines the shape of the Gaussian-beam response in Eqs. (9)-(14) and therefore the size of the nonlinear distortion.
  • Noise parameters ξA and ξS = Inferred per ringdown
    The amplitude-dependent and stationary noise variances are jointly inferred. The claimed improvement over the stationary-noise-only model depends on these being identifiable from each ringdown.
assumptions (6)
  • domain assumption The ringdown signal is a superposition of exactly two exponentially decaying sinusoids from a degenerate mode pair (Eq. 3).
    Adopted from Vajente et al. [19]. If extra modes, frequency drift, or nonlinear damping contribute, both M1 and M3 inherit a bias.
  • domain assumption The photodiode voltage is the difference of Gaussian-beam powers integrated over two half-planes with a finite gap (Eqs. 7-12).
    Assumes a TEM00 beam, normal incidence, no clipping at the detector edges, and that the beam displacement is equal to the mechanical surface displacement.
  • domain assumption The FFT bin value equals the power of the signal components inside that bin, and taking the maximum over the spectrum selects the mode-pair beat (Eq. 13).
    Underlies both M2 and the demodulation step in M3. This fails when the signal approaches or falls below the noise floor.
  • domain assumption M3's low-pass filtering discards all terms above T(ω1-ω2), and the Taylor series truncated at order T is accurate (Eqs. 18-19).
    A computational approximation chosen for speed. Its validity depends on the amplitudes being small relative to σ and on no out-of-band power leaking into the FFT bin.
  • domain assumption Noise is stationary plus amplitude-proportional Gaussian noise, uncorrelated between samples (Eqs. 22-24).
    Each sample is actually the maximum of an FFT power spectrum, which gives extreme-value statistics, especially near the noise floor. The authors acknowledge this limitation in Section 7.
  • standard math Standard Taylor expansion and integration of the error function.
    Uncontroversial calculus used to derive Eqs. (14)-(17), not a physical assumption.

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Pith. "Pith review of Use of Bayesian Inference to Diagnose Issues in Experimental Measurements of Mechanical Disk Resonators." pith.science (2026). https://pith.science/paper/SLGX43W3

@misc{pith2026250517346,
  author       = {Pith},
  title        = {Pith review of: Use of Bayesian Inference to Diagnose Issues in Experimental Measurements of Mechanical Disk Resonators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SLGX43W3}},
  note         = {Machine review of arXiv:2505.17346}
}
abstract

Gravitational wave detectors, such as LIGO, are predominantly limited by coating Brownian thermal noise (CTN), arising from mechanical losses in the Bragg mirror coatings used on test-mass optics. Accurately characterizing and minimizing these losses is crucial for enhancing detector sensitivity. This paper introduces a general mathematical and statistical framework leveraging Bayesian inference to precisely analyse mechanical ring-down measurements of disk resonators, a standard method for quantifying mechanical loss in coating materials. Our approach presents a refined model that fully captures the non-linear behaviour of beam spot motion on split photodiode sensors, significantly improving upon traditional simplified exponential-decay methods. We achieve superior estimation accuracy for decay constants ($\tau_1$ and $\tau_2$), especially for measurements exhibiting larger oscillation amplitudes. Specifically, we observe improvements in estimation accuracy by up to 25$\%$ over traditional methods, with strong Bayesian evidence favouring our framework. Our simulations and experimental validations reveal that previously discarded measurements due to fitting inaccuracies can now be reliably analysed, maximizing the use of available data. This enhanced analytical capability not only provides more precise mechanical loss estimations but also offers deeper insights into systematic issues affecting disk resonator measurements, paving the way toward improved coating materials and ultimately, more sensitive gravitational wave detectors.

Figures

Figures reproduced from arXiv: 2505.17346 by the authors.

Figure 1
Figure 1. Example of low frequency surface deformations of mode-shapes of a cylinder calculated with finite element modelling (COMSOL) listed in ascending frequency. Regions of large displacement are denoted in red, contrasting regions with little/no displacement, are shown in blue. Mechanical loss in coating materials, a property directly related to CTN, can be [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Simple illustration of a typical photodetector setup and non-linear signal behaviour. Top: Exaggerated example of two beams (small radius and large radius) centred on a split photodetector. Middle: normalised intensity profile of two smaller and larger radius beams incident on the photodetector and gap. Bottom: Scan of the two beams across the photodetector. The larger radius beam across a configuration with no gap … view at source ↗
Figure 3
Figure 3. Simulated example of two exponential decay-pairs using the small beam spot in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Analytic signal model with varying power series terms (T) and amplitudes. Upper plots show the predicted amplitude versus time. Lower plots showing the ratio of amplitudes when including T = 3, 5, 7 terms, compared to T = 1, for the two signals. These are compared with…
Figure 5
Figure 5. Figure 5: Fit and residuals for signal 31 obtained using the simple signal model (M1) with only stationary noise versus (ξA = 0) stationary and amplitude dependent noise (ξA < 0) for a 2.5 kHz mode (see [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: True total amplitude (aT = a1+a2) vs Bayes factor comparing analyses with a simple signal model (M1) and two different noise hypotheses using simulated data; a noise model that includes stationary and amplitude dependant noise versus one that only includes stationary n…
Figure 7
Figure 7. Figure 7: Fit and residuals for signal 31 obtained using the simple signal model (M1) with only stationary noise versus (ξA = 0) stationary and amplitude dependent noise (ξA > 0) for a 2.5 kHz mode (see [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: True total amplitude (aT = a1 + a2) versus Bayes factors comparing M3 with T = 7 vs M1 for 50 simulated ringdowns. All analyses uses the noise model that include amplitude dependent noise [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Comparison of the inferred posterior distributions (indicated with a hat) and true parameters for simulated signal 31. Results are shown for analyses using model 1 with stationary noise (blue), model 1 with amplitude-dependent noise and model 3 (with T = 7) with amplit…
Figure 10
Figure 10. Figure 10: Absolute difference in inferred and true decay parameters for M1 (blue) and M3 with T = 7 (orange when analysing 50 simulated ringdowns. The difference is computed using the median of the posterior distribution. Different markers are used to indicate whether the 3-σ c…
Figure 11
Figure 11. Figure 11: Fit and residuals for ringdown 0 obtained using M1 with stationary noise (blue), M1 with amplitude-dependent and stationary noise (orange) and M3 with T = 7 amplitude-dependent and stationary noise (green). The log Bayes factors are log10 B = 4285.7 in favour of M1 wi…
Figure 12
Figure 12. Figure 12: Posterior distributions for the mechanical loss (1/Q) computed from the inferred decay constants τ1 and τ2 for ringdown 0. Results are shown for two analyses with model 1, one with only stationary noise (ξA = 0) and another with stationary and amplitude dependent nois…
Figure 13
Figure 13. Figure 13: Inferred total amplitude with stationary noise, acT = a1 + a2, versus Bayes factor comparing different models. The colour axis for each plot shows the difference between the inferred decay parameters. a correlation with the amplitude ratio ρ, see Figure B3, at lower t…
Figure 14
Figure 14. Figure 14: Posterior distributions of the mechanical loss inferred using M1 with stationary noise and the original fitting method (left), M1 both noise types and the proposed method (middle) and M3 with T = 7 and the proposed method (right) as a function of frequency. The error …

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Pith tools

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