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arxiv: 2606.30504 · v1 · pith:W5NNLHO7new · submitted 2026-06-29 · 🌌 astro-ph.CO · astro-ph.IM· gr-qc

Rapid Hubble constant inference from GW170817 using GPU-accelerated nested sampling: prior sensitivity and the limits of post-hoc reweighting

Ming Han Yang , Metha Prathaban , David Yallup , Will Handley This is my paper

Pith reviewed 2026-06-30 04:43 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.IMgr-qc
keywords GW170817Hubble constantluminosity distance priornested samplingpost-hoc reweightingbright sirenprior sensitivitybimodality
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The pith

Switching the luminosity-distance prior inside the sampler shifts the GW170817 high-H0 tail from 1.7 percent to 15.9 percent probability, while post-hoc reweighting recovers only a fraction of that change.

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

The paper shows that the Hubble-constant posterior from the bright-siren event GW170817 depends on whether the luminosity-distance prior is imposed during nested sampling or applied afterward by reweighting existing samples. Direct sampling under a uniform-in-dL prior moves both the high tail and the median of H0, effects driven by a bimodality in the (dL, iota) plane that the volumetric prior largely misses. The data give comparable evidence for the different priors, indicating that the observed shifts are prior properties rather than new information extracted from the waveform. Because GPU acceleration makes full reruns fast, the authors conclude that prior-sensitivity checks should become routine by resampling rather than by reweighting.

Core claim

When the volumetric distance prior is replaced by a uniform-in-dL prior inside the nested sampler, P(H0 > 120 km/s/Mpc) rises from 0.017 to 0.159 and the weighted-median H0 moves from 77.6 to 87.6 km/s/Mpc; post-hoc reweighting of the baseline samples recovers only P = 0.041 in the tail. Targeted runs reveal a (dL, iota) bimodality whose high-H0 branch receives negligible weight under the volumetric prior, explaining the reweighting deficit. The three priors that allow independent evidence calculations agree to within Delta ln Z less than or equal to 1.8, confirming that the data do not strongly prefer one distance prior over another.

What carries the argument

The (dL, iota) bimodality in the likelihood surface under the IMRPhenomXAS_NRTidalv3 waveform, which causes the volumetric prior to assign negligible mass to the high-H0 branch.

If this is right

  • The binned MAP value of H0 stays at 70.5 km/s/Mpc even while the tail and median move.
  • The reweighted posterior has a lower effective sample size than the baseline samples.
  • The three distance priors that carry independent evidence agree to within Delta ln Z less than or equal to 1.8.
  • GPU-native heterodyned nested sampling completes each n_live=5000 run in roughly 13 minutes on one A100.

Where Pith is reading between the lines

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

  • Future bright-siren analyses should treat full prior-sensitivity reruns as the default robustness check rather than relying on reweighting.
  • The same bimodality mechanism may affect inclination or distance inferences for other compact-binary events.
  • Rapid GPU sampling lowers the computational cost of testing multiple distance priors to the point where such checks can be performed routinely.

Load-bearing premise

The bimodality between luminosity distance and inclination is a real feature of the likelihood surface rather than an artifact of the sampler or waveform approximation.

What would settle it

Repeating the full analysis with an independent waveform family or a different nested-sampling code and finding that the same (dL, iota) bimodality and prior-induced tail shift either disappear or remain.

Figures

Figures reproduced from arXiv: 2606.30504 by David Yallup, Metha Prathaban, Ming Han Yang, Will Handley.

Figure 1
Figure 1. Figure 1: GW150914 validation: corner overlay of the LVK GWTC-2.1 IMR￾PhenomXPHM posterior (Abbott et al. 2024; LIGO Scientific Collaboration & Virgo Collaboration 2022) and our heterodyned IMRPhenomXPHM run at 𝑛live = 8000, 𝑛mcmc = 160. The two posteriors overlap throughout the recovered support; our component-mass prior [5, 100] 𝑀⊙ encompasses the LVK [10, 80] 𝑀⊙ range without adding posterior mass at the boundari… view at source ↗
Figure 3
Figure 3. Figure 3: Prior-sensitivity comparison for IMRPhenomXAS_NRTidalv3. Left panel (a): kernel-density estimates of the 𝐻0 posterior under the four distance-prior variants of Section 2.4 (volumetric baseline, direct uniform-in-𝑑𝐿, reweighted uniform-in-𝑑𝐿, and 𝜎𝑣𝑝 = 250 km s−1 ). The directly sampled uniform-in-𝑑𝐿 posterior places substantially more mass at high 𝐻0 than the reweighted estimate, despite both targeting the… view at source ↗
Figure 2
Figure 2. Figure 2: Heterodyned vs full-resolution consistency for GW170817 under IMRPhenomD_NRTidalv2 at 𝑛live = 1500. Source-parameter corner over￾lay of the heterodyned (relative-binning) and full-resolution (full 259 201- bin) posteriors, with the public GWTC-1 IMRPhenomPv2_NRTidal pos￾terior (Abbott et al. 2019b; LIGO Scientific Collaboration & Virgo Col￾laboration 2018) shown for reference. The heterodyned and full-reso… view at source ↗
Figure 4
Figure 4. Figure 4: Cross-waveform check on the (𝑑𝐿 , 𝜄) bimodality: weighted joint posteriors from the unrestricted direct uniform-in-𝑑𝐿 runs at IMR￾PhenomD_NRTidalv2 (left; Mode-B-anchored heterodyne reference) and IMRPhenomXAS_NRTidalv3 (right; default GWTC-1 reference). Both show the same two-peak structure with the Mode-B region 𝑑𝐿 < 30 Mpc carrying 0.428 (IMRPhenomD_NRTidalv2) and 0.325 (IMRPhe￾nomXAS_NRTidalv3) of the … view at source ↗
Figure 5
Figure 5. Figure 5: The 𝑑𝐿–𝜄 bimodality in the GW170817 uniform-in-𝑑𝐿 posterior. Left: joint (𝑑𝐿 , 𝜄) posterior from the unrestricted 𝑑𝐿 ∈ [10, 75] Mpc run (Mode￾B-anchored heterodyne reference); the prior-restricted Mode-A and Mode-B contours from the two separate restricted-prior runs are overlaid in distinct colours, with the 𝑑𝐿 = 30 Mpc prior boundary marked. Right: per-mode 𝐻0 marginals as Silverman-bandwidth kernel-dens… view at source ↗
Figure 6
Figure 6. Figure 6: shows the GW170817 𝐻0 posterior under the LVK-matched prior (Section 2.4) for IMRPhenomXAS_NRTidalv3 and TaylorF2, together with the Planck and SH0ES reference bands. The plot￾ted curves are Silverman-bandwidth kernel-density estimates; the sample-derived HPDs (computed from the weighted samples, no KDE) are reported in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: GW170817 source-parameter corner overlay for IMRPhenomXAS_NRTidalv3 and TaylorF2 on the LVK-matched prior, with the public GWTC-1 IMRPhenomPv2_NRTidal posterior (Abbott et al. 2019b; LIGO Scientific Collaboration & Virgo Collaboration 2018) included as a literature reference. Two￾dimensional contours and one-dimensional marginals are KDE-smoothed [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

The bright-siren measurement of the Hubble constant from GW170817 (Abbott et al. 2017) assumes that switching from a volumetric to a uniform-in-$d_L$ luminosity-distance prior can be implemented by post-hoc reweighting of the baseline samples, rather than by re-running the inference under the target prior. Using a GPU-native heterodyned nested sampling pipeline that completes the full $n_{\rm live}=5000$ analysis in about 13 min on a single A100, we recompute the GW170817 $H_0$ posterior under four prior variants for the modern aligned-spin tidal waveform IMRPhenomXAS_NRTidalv3. Switching from the volumetric to a uniform-in-$d_L$ distance prior raises the high-tail probability $P(H_0>120\,\mathrm{km/s/Mpc})$ from 0.017 to 0.159 when imposed during sampling and shifts the weighted-median $H_0$ from 77.6 to 87.6 km/s/Mpc, while the binned MAP stays at 70.5 km/s/Mpc: both the tail and the bulk move under a change of prior that leaves the mode in place. Post-hoc reweighting of the baseline samples to the same target prior recovers only $P=0.041$ in the tail, approximately 17% of the directly sampled shift. The three prior variants that carry an independent nested sampling evidence agree to $\Delta\ln Z\lesssim 1.8$, so the data show at most a weak preference among the distance priors; the tail and bulk shifts are therefore properties of the prior, not a data update. Targeted mode-isolated runs reveal a $(d_L,\iota)$ bimodality whose high-$H_0$, low-$d_L$ branch (Mode B; $|\ln\mathcal{B}_{\rm B/A}|<1$) the volumetric prior assigns negligible mass: this is the mechanism behind the reweighting deficit. The reweighted posterior has a lower effective sample size than the baseline, independently flagging the coverage failure. The runtime budget makes full-sample prior-sensitivity reruns the default robustness tool for bright-siren cosmology, replacing post-hoc reweighting.

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

1 major / 2 minor

Summary. The manuscript claims that for the GW170817 bright-siren H0 inference with IMRPhenomXAS_NRTidalv3, direct GPU-accelerated nested sampling under a uniform-in-d_L prior yields P(H0>120 km/s/Mpc)=0.159 and median H0=87.6 km/s/Mpc versus 0.017 and 77.6 km/s/Mpc under the volumetric prior, while post-hoc reweighting recovers only P=0.041 (17% of the shift); a (d_L, ι) bimodality is identified as the mechanism, with evidence values agreeing to ΔlnZ≲1.8 across priors and lower ESS flagging the reweighting coverage failure. Full prior-sensitivity reruns are recommended over reweighting.

Significance. If the results hold, the work supplies a concrete, quantitatively documented case that post-hoc reweighting can substantially under-recover prior-induced shifts in multimodal bright-siren posteriors, while GPU-native nested sampling makes direct reruns feasible (∼13 min on A100). The independent evidence agreement and ESS diagnostic are explicit strengths that support the central comparison without circularity.

major comments (1)
  1. [Targeted mode-isolated runs] Targeted mode-isolated runs (abstract): the claim that the (d_L, ι) bimodality with |ln B_B/A|<1 is the mechanism for the reweighting deficit would be strengthened by explicit checks that the bimodality persists when n_live, proposal kernel, or heterodyning settings are varied; without those, the quantitative 17% recovery figure remains valid from the direct runs but the mechanistic interpretation carries a residual risk of sampling artifact.
minor comments (2)
  1. The abstract states that the binned MAP remains at 70.5 km/s/Mpc under the uniform-in-d_L prior; a brief description of the binning procedure or reference to the relevant figure would aid reproducibility.
  2. Consider adding a compact table that collects the four priors, their lnZ values, P(H0>120), medians, and ESS for direct visual comparison.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the work, and recommendation for minor revision. We respond to the single major comment below.

read point-by-point responses

  1. Referee: Targeted mode-isolated runs (abstract): the claim that the (d_L, ι) bimodality with |ln B_B/A|<1 is the mechanism for the reweighting deficit would be strengthened by explicit checks that the bimodality persists when n_live, proposal kernel, or heterodyning settings are varied; without those, the quantitative 17% recovery figure remains valid from the direct runs but the mechanistic interpretation carries a residual risk of sampling artifact.

    Authors: We agree that systematic variation of n_live, proposal kernel, and heterodyning settings in the mode-isolated runs would provide additional reassurance against sampling artifacts and would strengthen the mechanistic claim. The manuscript reports the (d_L, ι) bimodality as revealed by the targeted mode-isolated runs performed with the pipeline's standard configuration; the primary quantitative result (the 17% recovery of the tail shift under post-hoc reweighting) is obtained from the direct full-prior nested-sampling comparisons and is independent of the mode-isolation procedure. The agreement of the evidence values across priors (ΔlnZ ≲ 1.8) and the ESS diagnostic supply separate support for the reweighting deficit. We will revise the text to state explicitly the settings employed for the targeted runs and to note the residual risk of sampling artifact in the mechanistic interpretation, thereby clarifying the scope of the claim without altering the central numerical findings. revision: yes

Circularity Check

0 steps flagged

No circularity: independent nested-sampling runs under each prior

full rationale

The paper's central results (tail probabilities P(H0>120), median shifts, and evidence ratios) are obtained by executing separate GPU-native nested-sampling runs under each distance prior variant and then directly comparing those outputs to the post-hoc reweighted baseline; none of the reported quantities are obtained by algebraic rearrangement or re-labeling of quantities already fitted inside the same run. The (dL,ι) bimodality is presented as an empirical observation from the targeted mode-isolated runs rather than an input assumption that is later recovered by construction. No self-citations are invoked to establish uniqueness theorems or to smuggle in ansatzes, and the cited Abbott et al. (2017) result is an external benchmark. The derivation chain therefore remains self-contained against the computational evidence the authors themselves produce.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard correctness of nested sampling for multimodal posteriors and on the IMRPhenomXAS_NRTidalv3 waveform model; no new free parameters, axioms beyond standard Bayesian inference, or invented entities are introduced.

axioms (1)
  • standard math Nested sampling with n_live=5000 produces accurate posterior samples and evidence estimates for the given likelihood and prior.
    Invoked implicitly when the paper treats the four independent runs and their ΔlnZ values as reliable.

pith-pipeline@v0.9.1-grok · 5979 in / 1481 out tokens · 59677 ms · 2026-06-30T04:43:21.245760+00:00 · methodology

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Works this paper leans on

77 extracted references · 46 canonical work pages · 2 internal anchors

  1. [1]

    Physical Review Letters , year =

  2. [2]

    Physical Review X , year =

  3. [3]

    Astronomy and Astrophysics , year =

  4. [4]

    The Astrophysical Journal , year =

  5. [5]

    The Astrophysical Journal Letters , year =

  6. [6]

    The Astrophysical Journal Supplement Series , year =

  7. [9]

    Monthly Notices of the Royal Astronomical Society , year =

  8. [10]

    Bayesian Analysis , year =

  9. [11]

    2018 , howpublished =

    2018

  10. [18]

    Physical Review D , year =

  11. [19]

    2017 , howpublished =

    2017

  12. [20]

    2022 , howpublished =

    2022

  13. [21]

    Nature Astronomy , year =

  14. [22]

    Phys. Rev. D , year =

  15. [23]

    Living Reviews in Relativity , year =

  16. [25]

    1507.02646 , archivePrefix =

    Journal of Machine Learning Research , year =. 1507.02646 , archivePrefix =

    arXiv

  17. [26]

    2013 , howpublished =

    2013

  18. [28]

    2026 , eprint =

    Transactions on Machine Learning Research , issn =. 2026 , eprint =

    2026

  19. [36]

    Abac A., Dietrich T., Buonanno A., Steinhoff J., Ujevic M., 2024, @doi [Physical Review D] 10.1103/PhysRevD.109.024062 , 109, 024062

    work page doi:10.1103/physrevd.109.024062 2024

  20. [37]

    P., et al., 2016, @doi [Physical Review Letters] 10.1103/PhysRevLett.116.061102 , 116, 061102

    Abbott B. P., et al., 2016, @doi [Physical Review Letters] 10.1103/PhysRevLett.116.061102 , 116, 061102

    work page doi:10.1103/physrevlett.116.061102 2016

  21. [38]

    P., et al., 2017a, @doi [Physical Review Letters] 10.1103/PhysRevLett.119.161101 , 119, 161101

    Abbott B. P., et al., 2017a, @doi [Physical Review Letters] 10.1103/PhysRevLett.119.161101 , 119, 161101

    work page doi:10.1103/physrevlett.119.161101

  22. [39]

    P., et al., 2017b, @doi [Nature] 10.1038/nature24471 , 551, 85

    Abbott B. P., et al., 2017b, @doi [Nature] 10.1038/nature24471 , 551, 85

    work page doi:10.1038/nature24471

  23. [40]

    P., et al., 2019a, @doi [Physical Review X] 10.1103/PhysRevX.9.011001 , 9, 011001

    Abbott B. P., et al., 2019a, @doi [Physical Review X] 10.1103/PhysRevX.9.011001 , 9, 011001

    work page doi:10.1103/physrevx.9.011001

  24. [41]

    P., et al., 2019b, @doi [Physical Review X] 10.1103/PhysRevX.9.031040 , 9, 031040

    Abbott B. P., et al., 2019b, @doi [Physical Review X] 10.1103/PhysRevX.9.031040 , 9, 031040

    work page doi:10.1103/physrevx.9.031040

  25. [42]

    P., et al., 2020, @doi [Living Reviews in Relativity] 10.1007/s41114-020-00026-9 , 23, 3

    Abbott B. P., et al., 2020, @doi [Living Reviews in Relativity] 10.1007/s41114-020-00026-9 , 23, 3

    work page doi:10.1007/s41114-020-00026-9 2020

  26. [43]

    Abbott R., et al., 2024, @doi [Physical Review D] 10.1103/PhysRevD.109.022001 , 109, 022001

    work page doi:10.1103/physrevd.109.022001 2024

  27. [44]

    Ashton G., 2026, @doi [RAS Techniques and Instruments] 10.1093/rasti/rzag012 , 5, rzag012

    work page doi:10.1093/rasti/rzag012 2026

  28. [45]

    Ashton G., et al., 2019, @doi [The Astrophysical Journal Supplement Series] 10.3847/1538-4365/ab06fc , 241, 27

    work page doi:10.3847/1538-4365/ab06fc 2019

  29. [46]

    Bradbury J., et al., 2018, JAX: composable transformations of Python+NumPy programs , https://github.com/jax-ml/jax

    2018

  30. [47]

    Branchesi M., et al., 2023, @doi [Journal of Cosmology and Astroparticle Physics] 10.1088/1475-7516/2023/07/068 , 2023, 068

    work page doi:10.1088/1475-7516/2023/07/068 2023

  31. [48]

    Cabezas A., Corenflos A., Lao J., Louf R., et al., 2024, BlackJAX: Composable Bayesian inference in JAX ( @eprint arXiv 2402.10797 )

    arXiv 2024

  32. [49]

    Chan R., Narola H., Ng T. C. K., Wouters T., Wong I. C. F., Gittins F., Janquart J., Van Den Broeck C., 2026, in prep

    2026

  33. [50]

    E., 2018, @doi [Nature] 10.1038/s41586-018-0606-0 , 562, 545

    Chen H.-Y., Fishbach M., Holz D. E., 2018, @doi [Nature] 10.1038/s41586-018-0606-0 , 562, 545

    work page doi:10.1038/s41586-018-0606-0 2018

  34. [51]

    J., 2010, Fast Fisher Matrices and Lazy Likelihoods ( @eprint arXiv 1007.4820 )

    Cornish N. J., 2010, Fast Fisher Matrices and Lazy Likelihoods ( @eprint arXiv 1007.4820 )

    Pith/arXiv arXiv 2010

  35. [52]

    Dax M., et al., 2025, @doi [Nature] 10.1038/s41586-025-08593-z , 639, 49

    work page doi:10.1038/s41586-025-08593-z 2025

  36. [53]

    Di Valentino E., et al., 2021, @doi [Classical and Quantum Gravity] 10.1088/1361-6382/ac086d , 38, 153001

    work page doi:10.1088/1361-6382/ac086d 2021

  37. [54]

    Edwards T. D. P., Wong K. W. K., Lam K. K. H., Coogan A., Foreman-Mackey D., Isi M., Zimmerman A., 2024, @doi [Physical Review D] 10.1103/PhysRevD.110.064028 , 110, 064028

    work page doi:10.1103/physrevd.110.064028 2024

  38. [55]

    S., Chernoff D

    Finn L. S., Chernoff D. F., 1993, @doi [Phys. Rev. D] 10.1103/PhysRevD.47.2198 , 47, 2198

    work page doi:10.1103/physrevd.47.2198 1993

  39. [56]

    E., Hughes S

    Holz D. E., Hughes S. A., 2005, @doi [The Astrophysical Journal] 10.1086/431341 , 629, 15

    work page doi:10.1086/431341 2005

  40. [57]

    P., Deller A

    Hotokezaka K., Nakar E., Gottlieb O., Nissanke S., Masuda K., Hallinan G., Mooley K. P., Deller A. T., 2019, @doi [Nature Astronomy] 10.1038/s41550-019-0820-1 , 3, 940

    work page doi:10.1038/s41550-019-0820-1 2019

  41. [58]

    M., 2020, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/staa049 , 492, 3803

    Howlett C., Davis T. M., 2020, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/staa049 , 492, 3803

    work page doi:10.1093/mnras/staa049 2020

  42. [59]

    Hu Q., Veitch J., 2025, @doi [Physical Review D] 10.1103/dj7k-tk37 , 112, 084039

    work page doi:10.1103/dj7k-tk37 2025

  43. [60]

    John Wiley & Sons, New York

    Kish L., 1965, Survey Sampling . John Wiley & Sons, New York

    1965

  44. [61]

    N., Williams N., Zimmerman A., 2023, Accelerated parameter estimation in Bilby with relative binning ( @eprint arXiv 2312.06009 )

    Krishna K., Vijaykumar A., Ganguly A., Talbot C., Biscoveanu S., George R. N., Williams N., Zimmerman A., 2023, Accelerated parameter estimation in Bilby with relative binning ( @eprint arXiv 2312.06009 )

    arXiv 2023

  45. [62]

    LIGO Scientific Collaboration Virgo Collaboration 2017, A gravitational-wave standard siren measurement of the Hubble constant -- Data Release , LIGO Document P1700296; https://dcc.ligo.org/LIGO-P1700296/public

    2017

  46. [63]

    LIGO Scientific Collaboration Virgo Collaboration 2018, Properties of the binary neutron star merger GW170817 -- Data Release , LIGO Document P1800061; https://dcc.ligo.org/LIGO-P1800061/public

    2018

  47. [64]

    LIGO Scientific Collaboration Virgo Collaboration 2022, GWTC-2.1: Deep Extended Catalog of Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run -- Parameter Estimation Data Release , Zenodo, https://doi.org/10.5281/zenodo.6513631, @doi 10.5281/zenodo.6513631

    work page doi:10.5281/zenodo.6513631 2022

  48. [65]

    LIGO Scientific Collaboration Virgo Collaboration KAGRA Collaboration 2026, GWTC-5.0: Constraints on the Cosmic Expansion Rate and Modified Gravitational-wave Propagation ( @eprint arXiv 2605.27227 ), @doi 10.48550/arXiv.2605.27227

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.27227 2026

  49. [66]

    P., et al., 2018, @doi [Nature] 10.1038/s41586-018-0486-3 , 561, 355

    Mooley K. P., et al., 2018, @doi [Nature] 10.1038/s41586-018-0486-3 , 561, 355

    work page doi:10.1038/s41586-018-0486-3 2018

  50. [67]

    Morisaki S., 2021, @doi [Physical Review D] 10.1103/PhysRevD.104.044062 , 104, 044062

    work page doi:10.1103/physrevd.104.044062 2021

  51. [68]

    R., Jasche J., Wandelt B

    Mukherjee S., Lavaux G., Bouchet F. R., Jasche J., Wandelt B. D., Nissanke S., Leclercq F., Hotokezaka K., 2021, @doi [Astronomy and Astrophysics] 10.1051/0004-6361/201936724 , 646, A65

    work page doi:10.1051/0004-6361/201936724 2021

  52. [69]

    Nicolaou C., Lahav O., Lemos P., Hartley W., Braden J., 2020, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/staa1120 , 495, 90

    work page doi:10.1093/mnras/staa1120 2020

  53. [70]

    B., 2013, Monte Carlo Theory, Methods and Examples , https://artowen.su.domains/mc/

    Owen A. B., 2013, Monte Carlo Theory, Methods and Examples , https://artowen.su.domains/mc/

    2013

  54. [71]

    Palmese A., Mastrogiovanni S., 2025, Gravitational Wave Cosmology ( @eprint arXiv 2502.00239 )

    arXiv 2025

  55. [72]

    Palmese A., Kaur R., Hajela A., Margutti R., McDowell A., MacFadyen A., 2024, @doi [Physical Review D] 10.1103/PhysRevD.109.063508 , 109, 063508

    work page doi:10.1103/physrevd.109.063508 2024

  56. [73]

    Payne E., Talbot C., Thrane E., 2019, @doi [Physical Review D] 10.1103/PhysRevD.100.123017 , 100, 123017

    work page doi:10.1103/physrevd.100.123017 2019

  57. [74]

    Planck Collaboration 2020, @doi [Astronomy and Astrophysics] 10.1051/0004-6361/201833910 , 641, A6

    work page doi:10.1051/0004-6361/201833910 2020

  58. [75]

    H., Templeton W., Handley W., 2026, @doi [RAS Techniques and Instruments] 10.1093/rasti/rzag034 , 5, rzag034

    Prathaban M., Yallup D., Alvey J., Yang M. H., Templeton W., Handley W., 2026, @doi [RAS Techniques and Instruments] 10.1093/rasti/rzag034 , 5, rzag034

    work page doi:10.1093/rasti/rzag034 2026

  59. [76]

    Pratten G., Husa S., Garc\'ia-Quir\'os C., Colleoni M., Ramos-Buades A., Estell\'es H., Jaume R., 2020, @doi [Physical Review D] 10.1103/PhysRevD.102.064001 , 102, 064001

    work page doi:10.1103/physrevd.102.064001 2020

  60. [77]

    Pratten G., et al., 2021, @doi [Physical Review D] 10.1103/PhysRevD.103.104056 , 103, 104056

    work page doi:10.1103/physrevd.103.104056 2021

  61. [78]

    G., et al., 2016, @doi [The Astrophysical Journal] 10.3847/0004-637X/826/1/56 , 826, 56

    Riess A. G., et al., 2016, @doi [The Astrophysical Journal] 10.3847/0004-637X/826/1/56 , 826, 56

    work page doi:10.3847/0004-637x/826/1/56 2016

  62. [79]

    G., et al., 2022, @doi [The Astrophysical Journal Letters] 10.3847/2041-8213/ac5c5b , 934, L7

    Riess A. G., et al., 2022, @doi [The Astrophysical Journal Letters] 10.3847/2041-8213/ac5c5b , 934, L7

    work page doi:10.3847/2041-8213/ac5c5b 2022

  63. [80]

    Salvarese A., Chen H.-Y., 2024, @doi [The Astrophysical Journal Letters] 10.3847/2041-8213/ad7bbc , 974, L16

    work page doi:10.3847/2041-8213/ad7bbc 2024

  64. [81]

    F., 1986, @doi [Nature] 10.1038/323310a0 , 323, 310

    Schutz B. F., 1986, @doi [Nature] 10.1038/323310a0 , 323, 310

    work page doi:10.1038/323310a0 1986

  65. [82]

    Skilling J., 2006, @doi [Bayesian Analysis] 10.1214/06-BA127 , 1, 833

    work page doi:10.1214/06-ba127 2006

  66. [83]

    E., Blackburn K., Haster C.-J., P\"urrer M., Raymond V., Schmidt P., 2016, @doi [Physical Review D] 10.1103/PhysRevD.94.044031 , 94, 044031

    Smith R., Field S. E., Blackburn K., Haster C.-J., P\"urrer M., Raymond V., Schmidt P., 2016, @doi [Physical Review D] 10.1103/PhysRevD.94.044031 , 94, 044031

    work page doi:10.1103/physrevd.94.044031 2016

  67. [84]

    S., 2020, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/staa278 , 493, 3132

    Speagle J. S., 2020, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/staa278 , 493, 3132

    work page doi:10.1093/mnras/staa278 2020

  68. [85]

    Vehtari A., Simpson D., Gelman A., Yao Y., Gabry J., 2024, Journal of Machine Learning Research, 25, 1

    2024

  69. [86]

    Vinciguerra S., Veitch J., Mandel I., 2017, @doi [Classical and Quantum Gravity] 10.1088/1361-6382/aa6d44 , 34, 115006

    work page doi:10.1088/1361-6382/aa6d44 2017

  70. [87]

    J., Veitch J., Messenger C., 2021, @doi [Physical Review D] 10.1103/PhysRevD.103.103006 , 103, 103006

    Williams M. J., Veitch J., Messenger C., 2021, @doi [Physical Review D] 10.1103/PhysRevD.103.103006 , 103, 103006

    work page doi:10.1103/physrevd.103.103006 2021

  71. [88]

    J., Karamanis M., Luo Y., Seljak U., 2025, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/staf1458 , 543, 1479

    Williams M. J., Karamanis M., Luo Y., Seljak U., 2025, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/staf1458 , 543, 1479

    work page doi:10.1093/mnras/staf1458 2025

  72. [89]

    Wong K. W. K., Isi M., Edwards T. D. P., 2023, @doi [The Astrophysical Journal] 10.3847/1538-4357/acf5cd , 958, 129

    work page doi:10.3847/1538-4357/acf5cd 2023

  73. [90]

    Wouters T., Pang P. T. H., Dietrich T., Van Den Broeck C., 2024, @doi [Physical Review D] 10.1103/PhysRevD.110.083033 , 110, 083033

    work page doi:10.1103/physrevd.110.083033 2024

  74. [91]

    Yallup D., Prathaban M., Alvey J., Handley W., 2025, Parallel Nested Slice Sampling for Gravitational Wave Parameter Estimation ( @eprint arXiv 2509.24949 )

    arXiv 2025

  75. [92]

    Yallup D., Kroupa N., Handley W., 2026, @doi [Transactions on Machine Learning Research] 10.48550/arXiv.2601.23252

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2601.23252 2026

  76. [93]

    Yang M. H., Prathaban M., Yallup D., Handley W., 2026, GW170817 bright-siren H_0 : data and analysis release , https://github.com/ming-256/GW170817-bright-siren-H0, @doi 10.5281/zenodo.21038511

    work page doi:10.5281/zenodo.21038511 2026

  77. [94]

    Zackay B., Dai L., Venumadhav T., 2018, Relative binning and fast likelihood evaluation for gravitational-wave parameter estimation ( @eprint arXiv 1806.08792 )

    Pith/arXiv arXiv 2018