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REVIEW 3 major objections 4 minor 88 references

A neural population model recovers intrinsic merger distributions and absolute rates from sparse, selection-biased gravitational-wave catalogs without presupposing a functional form.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 06:25 UTC pith:KFXWEG73

load-bearing objection A plausible but not yet reliable ML population-inference framework: the new compound architecture and LISA toy are worth engaging, but the per-event likelihood double-counts selection and the GWTC-3 mass-ratio result is partly imposed. the 3 major comments →

arxiv 2602.00195 v2 pith:KFXWEG73 submitted 2026-01-30 astro-ph.IM astro-ph.HEgr-qc

Model-Agnostic Population Inference for Gravitational-Wave Astronomy: From LVK to LISA

Yi-kun Li , Cheng Cheng , Lang Cui , Yun Fang This is my paper
classification astro-ph.IM astro-ph.HEgr-qc
keywords gravitational-wave astronomypopulation inferencehierarchical Bayesian inferencedeep generative modelsnormalizing flowsGaussian copulaselection effectsLISA
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to replace rigid functional forms in gravitational-wave population inference with a neural network that learns the intrinsic distribution of merger parameters directly from data. It builds a flexible population model—a weighted mixture of flexible marginal distributions joined by a Gaussian copula—and trains it with amortized variational inference, correcting selection effects and measurement uncertainties in the same step. On simulated LISA-like catalogs of supermassive black hole mergers, the method recovers the joint mass–redshift–mass-ratio distribution and the absolute merger rate from as few as a dozen observed events. On 69 real events from the GWTC-3 catalog, it reproduces the expected low-mass peak, high-mass cutoff, redshift turnover near cosmic noon, and a preference for equal-mass binaries without assuming those features in advance. The contribution is a model-agnostic, scalable alternative to parametric hierarchical Bayesian inference for current and future observatories.

Core claim

The central claim is that a correlated compound mixture density network—mixture weights for multimodality, Gaussian copulas for correlations, flexible marginals for shape—trained with normalizing-flow-guided variational inference, can approximate the hyperparameter posterior of a gravitational-wave population and compute selection corrections from the learned density itself. The absolute merger rate follows in a second stage from a Gamma posterior conditioned on the detection efficiency. In simulated LISA scenarios with 17 to 412 events, the learned distributions pass marginal and joint statistical tests against ground truth and recover the intrinsic event count; on a misspecified parametric

What carries the argument

The central object is the Correlated Compound Mixture Density Network: a weighted sum of K components, each combining a library of flexible one-dimensional marginals (Kumaraswamy, truncated Pareto, truncated normal) with a Gaussian copula that encodes parameter correlations. This structure keeps the population density analytically tractable for the selection-function integral while letting the data determine each marginal and correlation. Around it sits a normalizing-flow variational guide that approximates the hyperparameter posterior by maximizing an evidence lower bound; the selection integral is evaluated by importance sampling over injected signals for ground-based data and by a differe

Load-bearing premise

For the real-event GWTC-3 analysis, the load-bearing premise is that the mass-ratio distribution rises toward equal masses: the inference applies a penalty that suppresses negative slopes in p(q), so the reported preference for q close to 1 is partly imposed rather than purely learned.

What would settle it

Rerun the GWTC-3 inference with the monotonicity regularization removed: if the inferred p(q) flattens or falls toward q = 1, the reported feature is regularization-driven. Alternatively, simulate a catalog with a known non-monotonic mass-ratio distribution and check whether the penalty biases the recovery; a bias would violate the framework's model-agnostic claim in that regime.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Future space-based missions can extract intrinsic three-dimensional population distributions and absolute merger rates from tens of detected events, even when the underlying astrophysical model is unknown.
  • Large upcoming ground-based catalogs can be re-analyzed quickly as new detections arrive, since the trained network generates posterior samples without restarting a sampling chain.
  • Data-driven reconstructions can expose features such as mass-gap edges, formation-channel bimodality, or redshift turnover that rigid parametric fits would miss.
  • The same framework transfers across detector types, so joint analyses of space-based and ground-based observations are feasible without redesigning the inference machinery.
  • Selection-effect corrections based on official injection sets make the method applicable to real pipeline outputs, not just idealized simulations.

Where Pith is reading between the lines

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

  • A direct test suggests itself: rerun the GWTC-3 analysis with the monotonicity penalty on p(q) removed; if the rise toward equal masses survives, it is a data-driven feature, and if not, it is prior-driven.
  • Because the copula separates marginals from correlations, the framework is a natural platform for measuring mass-spin or mass-ratio–effective-spin correlations in the same catalog, a step the paper mentions for future work.
  • The amortized setup could be repurposed as an online population monitor: update the inferred population continuously as events stream in, which would be useful for electromagnetic follow-up of unusual mergers.
  • The same mixture-copula variational recipe applies to any sparse astronomical catalog with a known selection function, such as supernova surveys or X-ray transient samples, beyond gravitational waves.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a deep-generative, model-agnostic framework for gravitational-wave population inference. A flexible mixture/copula population model (the Correlated Compound Mixture Density Network) is coupled to a normalizing-flow variational guide, which approximates the posterior over population hyperparameters; a two-stage procedure then infers the merger rate. The method is first tested on a toy density-estimation problem and on simulated LISA catalogs of SMBH mergers, and is then applied to the GWTC-3 stellar-mass BBH catalog using injection-based selection corrections. The central claim is that the framework accurately recovers three-dimensional intrinsic populations and absolute rates while correcting selection effects and measurement uncertainties.

Significance. If correct, the framework would provide a scalable, flexible alternative to parametric hierarchical Bayesian inference and would be useful for both current LVK analyses and future LISA-era population studies. The paper has several genuine strengths: the architecture separates an interpretable physics-facing mixture/copula density from a neural variational guide; the toy-model density test and the comparison with a correctly specified and a misspecified MCMC model are informative; and the GWTC-3 analysis uses public injection data. However, the central selection-effect calculation contains a load-bearing error, and the validation evidence is weaker than claimed. The real-data results, especially the mass-ratio distribution, are partly driven by an explicitly imposed prior and are then reported as recovered features. These issues undercut the main claims as currently presented.

major comments (3)
  1. [§2.A, Eq. (3); Algorithm 1 line 6] The per-event term in Eq. (3) and in Algorithm 1 line 6 incorrectly includes Pdet(θ) in the numerator. In standard GW population inference, after conditioning on detection (or equivalently in the marked Poisson process where detection is determined by the data), the per-event likelihood is ∫ p(D_i|θ) ppop(θ|Λ) dθ, and the selection function enters only through the detection efficiency α(Λ) in the denominator. Rewriting with PE posterior samples, the term is average_i [p(θ_i|D_i)/π(θ_i)] ppop(θ_i|Λ), with no extra Pdet factor. The factor Pdet in Eq. (3) and Algorithm 1 line 6 double-counts selection and will bias the inferred population toward low-Pdet regions. This is not a minor typo: Algorithm 1 explicitly multiplies the posterior-sample weight by Pdet, and Eq. (10) in §4.B inconsistently defines the weight as ppop/π without Pdet. Because the paper's central claim is correct selection-
  2. [§3.C, Table I and Fig. 4] The LISA validation is based on one realization per catalog size and reports only failure-to-reject p-values from KS and energy-distance tests. With N_obs as small as 17, the tests have very limited power, so p>0.05 provides weak evidence that the learned distribution matches the ground truth. Moreover, the simulated detected events have very high SNR (and hence Pdet≈1), so the LISA setup does not stress the selection-correction mechanism; it is therefore unlikely to expose the Eq. (3) error. The Fisher-Gaussian posterior approximation and the SNR surrogate are also used without validation of their impact on the population-level result. The claim that the method 'accurately recovers complex three-dimensional distributions' is stronger than the evidence from a single realization and low-power tests.
  3. [§4.B and §4.C] The monotonicity regularization on p(q) is described as a physics-informed prior motivated by comparable-mass formation, but the same section then reports the resulting rise toward q=1 as a 'robust feature, reinforced by our physics-informed monotonicity regularization' (Sec. 4.C). This is circular for the q claim: the posterior predictive p(q) is guaranteed to favor nondecreasing shapes by construction, so it cannot be presented as independent data-driven evidence of a preference for equal-mass binaries. The paper should clearly separate the prior assumption from the inference result and quantify how strongly the regularization, rather than the 69 GWTC-3 events, determines the q posterior. This concern is specific to the q result and does not by itself invalidate the framework, but it weakens the GWTC-3 validation claim.
minor comments (4)
  1. [Eq. (1) vs Eq. (3)] Eq. (1) writes the per-event term with p(D_i|θ), while Eq. (3) and the surrounding text use p(θ|D_i). The posterior p(θ|D_i) is not the likelihood; the paper must consistently write the likelihood or explicitly include the PE prior division. This is related to the major comment above but should be fixed even after the Pdet issue is resolved.
  2. [Eq. (4)] The Gamma posterior for Ryr is stated to follow from an 'uninformative prior', but the Gamma(N_obs+1, T_obs α) form corresponds to a specific prior (e.g., a Gamma(1,0) prior); please state the prior explicitly.
  3. [§2.B-§2.D] The selection of the number of mixture components K and the regularization strength is described qualitatively ('balancing model expressiveness with computational efficiency'). Since these choices affect the flexibility and the q prior, a concrete criterion or sensitivity check would strengthen the presentation.
  4. [Figures 8-9] The captions of Figures 8 and 9 appear to contain raw Unicode escape sequences (e.g., '/uni000...') in the typeset version; please correct the PDF/renderer issue.

Circularity Check

2 steps flagged

Partial circularity: the GWTC-3 mass-ratio rise is imposed by a monotonicity regularizer and then reported as a robust recovered feature; the Fig. 5 consistency 'prediction' of Nobs is tied by construction to the input count. The core LISA simulation validation is independent.

specific steps
  1. self definitional [Sec. IV B (inference methodology) and Sec. IV C (inferred population distributions, Panel c)]
    "motivated by the astrophysical expectation that binary black holes preferentially form with comparable masses [75], we apply a monotonicity regularization term that penalizes negative slopes in the marginal distribution of the mass ratio p(q). ... the distribution displays a monotonic rise towards q = 1 ... This strong preference for symmetric binaries is a robust feature, reinforced by our physics-informed monotonicity regularization."

    The monotonic rise toward q=1 is inserted into the objective function as a penalty on negative slopes, so presenting it as an independently recovered inference is circular: the input constraint already guarantees the qualitative feature. The paper's own wording ('reinforced by our physics-informed monotonicity regularization') concedes that the reported feature is not purely data-driven. This does not invalidate the LISA simulated-validation or the general architecture, but it makes the GWTC-3 mass-ratio claim partially constructed rather than inferred.

  2. fitted input called prediction [Sec. III C, Fig. 5 (Test 2), following Eq. (4)]
    "Second, we perform an observable consistency check by predicting the expected number of observed events, Nobs, to ensure the model is consistent with the actual data fed into it. ... Test 2 ... The model precisely predicts the number of events that should be observed given the inferred population, matching the actual catalog size. This consistency check is an indicator that the selection effects are being properly modeled and corrected for."

    The rate posterior in Eq. (4), p(R|Nobs,Λ) ∝ e^{-R T α}(R T α)^{Nobs}, is conditioned on the observed count Nobs. Drawing the expected observed count from this posterior therefore reproduces the input Nobs by construction, regardless of whether α or the selection model is accurate. Calling this match a validation of the selection correction treats the input count as if it were a free prediction.

full rationale

The framework's central LISA demonstration is not circular: it is validated against independently generated mock catalogs with known ground-truth populations, so the claimed recovery of three-dimensional shapes and rates is an empirical test rather than a consequence of the model definition. The core hierarchical Poisson structure is also standard in the field. However, two sub-claims reduce to inputs by construction. First, the GWTC-3 mass-ratio rise toward q=1 is explicitly produced by a monotonicity regularizer and then reported as a 'robust feature,' which is a self-definitional inference for that specific feature. Second, the Fig. 5 'prediction' of the observed event count is conditioned on the observed count through Eq. (4), so matching it is statistically forced and cannot independently validate selection-effect modeling. The skeptic's additional concern about Pdet appearing in the numerator of Eq. (3) is a likelihood-correctness issue rather than a circularity; it is load-bearing for the claimed selection correction but is not a case of a prediction equaling its input by construction. No self-citation chain or imported uniqueness theorem is load-bearing. Overall, the central method retains independent content, but the two reduced claims justify a moderate partial-circularity score.

Axiom & Free-Parameter Ledger

4 free parameters · 7 axioms · 0 invented entities

No new physical entities are postulated; the network components are computational constructs. The main load-bearing inputs are the hand-chosen mixture complexity, the imposed monotonicity prior, and the assumptions about Fisher-approximated posteriors and injection-based selection estimates.

free parameters (4)
  • Number of mixture components K = 2 (LISA), 4 (GWTC-3), 5 (MCMC comparison), 6 (toy validation)
    K is chosen by hand per task to balance expressiveness and overfitting; no data-driven model selection is performed, and K constrains the flexibility of the 'model-agnostic' density.
  • Monotonicity regularization strength = not stated
    The penalty on negative slopes of p(q) is introduced in Sec. IV B but its magnitude is never given; it directly shapes the reported rise toward q=1.
  • Variational guide architecture = 8 affine coupling layers, 128 hidden units, ReLU, Adam lr=1e-3, 1000 epochs
    Network capacity and training schedule are chosen ad hoc; they affect the fidelity of the posterior approximation and hence the inferred population.
  • Effective sample size threshold = 4 × Nobs
    Threshold for penalizing low-Neff selection integrals is chosen from empirical tests; it influences which regions of hyperparameter space are allowed.
axioms (7)
  • standard math Inhomogeneous Poisson process likelihood for detections (Eq. 1)
    Hierarchical Bayesian framework for GW catalogs, standard in the field.
  • standard math Sklar's theorem and Gaussian copula representation (Eq. 6)
    Justifies factorizing the joint density into marginals and a copula.
  • standard math Variational ELBO with a normalizing-flow guide approximates the true posterior
    Assumes the amortized variational posterior q_phi(Λ) is flexible enough and that optimization converges; not proven for this problem.
  • domain assumption LISA event posteriors are multivariate Gaussians from the Fisher Information Matrix
    Used to generate mock PE samples (Sec. III B); known to break down at low SNR, and the simulated catalog includes SNR values from tens to tens of thousands.
  • domain assumption The differentiable SNR surrogate MLP faithfully reproduces the true SNR and selection function
    Accelerates selection integration (Sec. III B) but any surrogate error propagates into α(Λ) and the rate.
  • ad hoc to paper p(q) is nondecreasing for stellar-mass BBHs
    Introduced as a 'physics-informed' regularization in Sec. IV B; not derived from data, and it manufactures the reported rise to q=1.
  • domain assumption Official LIGO-Virgo-KAGRA O1+O2+O3 injection file is an unbiased estimate of the selection function
    Selection correction (Eq. 9) assumes the injection sampling distribution and recovery are correctly characterized.

pith-pipeline@v1.3.0-alltime-deepseek · 20166 in / 20792 out tokens · 210145 ms · 2026-08-03T06:25:34.856869+00:00 · methodology

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read the original abstract

Inferring the intrinsic population of compact binary mergers is complicated by detector selection biases and measurement uncertainties. Traditional parametric methods are limited by the need to presuppose functional forms, introducing model-dependent biases. To overcome these limitations, we introduce an inference framework powered by deep generative modeling. We develop a flexible, data-driven population model using a Correlated Compound-Mixture Density Network. This architecture integrates mixture models to handle multimodality, Gaussian copulas for parameter dependencies, and a library of flexible marginal distributions. The network is trained to approximate the posterior distribution of the population's hyperparameters using amortized variational inference with Normalizing Flows on catalogs of gravitational-wave events. We demonstrate the framework's capabilities in two distinct regimes. First, using simulated catalogs of supermassive black hole binary mergers for the Laser Interferometer Space Antenna (LISA), we show that the method accurately recovers complex three-dimensional distributions and absolute merger rates from sparse datasets, effectively correcting for selection effects and measurement uncertainties. Second, we validate the framework on real observational data from the LIGO-Virgo-KAGRA GWTC-3 catalog, successfully inferring the population of stellar-mass binary black holes using an injection-based selection effect correction. Our results confirm that the method is robust, scalable, and applicable across different detector sensitivities and source populations.

Figures

Figures reproduced from arXiv: 2602.00195 by Cheng cheng, Lang Cui, Yi-kun Li, Yun Fang.

Figure 1
Figure 1. Figure 1: Technical Architecture of the Flow-Guided Correlated Compound Mixture Density Network. The network [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Validation of the density estimation network [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Projected distributions of the intrinsic ( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Scalability validation across LISA catalog sizes ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Validation of event count inference. Each panel shows posteriors for intrinsic count [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Computational cost comparison. Left: Neural [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Overview of the 69 binary black hole events from GWTC-3 used in this analysis. The panels show the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Two-dimensional joint posterior predictive distributions for the LIGO BBH population inferred from [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Reconstructed marginal distributions for LIGO BBH population. [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

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Reference graph

Works this paper leans on

88 extracted references · 10 linked inside Pith

  1. [1]

    The Galaxy Stellar Mass Function (GSMF) uses a dou- ble Schechter function with redshift-dependent parame- ters interpolated from Table IV (based on COSMOS2020 [54])

    Astrophysical Model Parameters We assume a ΛCDM cosmology with H0 = 67.4 km s −1Mpc−1, Ωm = 0 .315, and ΩΛ = 0 .685 [41]. The Galaxy Stellar Mass Function (GSMF) uses a dou- ble Schechter function with redshift-dependent parame- ters interpolated from Table IV (based on COSMOS2020 [54]). Table IV: Discrete data points for the redshift evolution of the GSM...

  2. [2]

    α2 0.35 10.93 -3.22 -1.45 -2.97 -0.66 0.65 10.98 -3.32 -1.44 -3.04 -0.79 0.95 11.01 -3.15 -1.35 -3.11 -0.72 1.30 11.01 -3.16 -1.34 -3.49 -0.56 1.75 10.88 -3.64 -1.52 -3.17 -0.55 2.25 10.80 -3.59 -1.46 -3.54 0.04 2.75 11.00 -3.66 -1.46 -9.00 0.00 3.25 10.85 -3.70 -1.46 -9.00 0.00 4.00 10.46 -3.70 -1.46 -9.00 0.00 5.00 10.33 -3.89 -1.46 -9.00 0.00 6.00 10.1...

  3. [3]

    A. Abac, I. Abouelfettouh, F. Acernese, K. Ackley, C. Adamcewicz, S. Adhicary, D. Adhikari, N. Adhikari, R. Adhikari, V. Adkins, et al. , Gwtc-4.0: Population properties of merging compact binaries, arXiv preprint arXiv:2508.18083 (2025)

    Pith/arXiv arXiv 2025

  4. [4]

    Observational Model Parameters We use the LISA noise PSD from [ 67] with arm length L = 2 .5 × 109 m, high-frequency OMS noise (1.5 × 10−11 m)2Hz−1, and acceleration noise (3 × 10−15 ms−2)2Hz−1

  5. [5]

    B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016)

    2016

  6. [6]

    Abbott, T

    R. Abbott, T. Abbott, F. Acernese, K. Ackley, C. Adams, N. Adhikari, R. Adhikari, V. Adya, C. Affeldt, D. Agar- wal, et al. , Population of merging compact binaries in- ferred using gravitational waves through gwtc-3, Physical Review X 13, 011048 (2023)

    2023

  7. [7]

    Amaro-Seoane et al

    P. Amaro-Seoane et al. , Laser interferometer space an- tenna (2017), arXiv:1702.00786 [astro-ph.IM]

    Pith/arXiv arXiv 2017

  8. [8]

    Zevin, S

    M. Zevin, S. S. Bavera, C. P. Berry, V. Kalogera, T. Fra- gos, P. Marchant, C. L. Rodriguez, F. Antonini, D. E. Holz, and C. Pankow, One channel to rule them all? con- straining the origins of binary black holes using multiple formation pathways, The Astrophysical Journal 910, 152 (2021)

    2021

  9. [9]

    Stevenson, C

    S. Stevenson, C. P. Berry, and I. Mandel, Hierarchical analysis of gravitational-wave measurements of binary black hole spin–orbit misalignments, Monthly Notices of the Royal Astronomical Society 471, 2801 (2017)

    2017

  10. [10]

    Mandel and F

    I. Mandel and F. S. Broekgaarden, Rates of compact object coalescences, Living Reviews in Relativity 25, 10.1007/s41114-021-00034-3 (2022)

    doi:10.1007/s41114-021-00034-3 2022

  11. [11]

    Evans, R

    M. Evans, R. X. Adhikari, C. Afle, S. W. Ballmer, S. Bis- coveanu, S. Borhanian, D. A. Brown, Y. Chen, R. Eisen- stein, A. Gruson, et al. , A horizon study for cosmic explorer: science, observatories, and community, arXiv preprint arXiv:2109.09882 (2021). 15

    Pith/arXiv arXiv 2021

  12. [12]

    Hu and Y.-L

    W.-R. Hu and Y.-L. Wu, The taiji program in space for gravitational wave physics and the nature of gravity, Na- tional Science Review 4, 685 (2017)

    2017

  13. [13]

    Luo, L.-S

    J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, et al. , Tianqin: a space-borne gravitational wave detector, Classical and Quantum Gravity 33, 035010 (2016)

    2016

  14. [14]

    Punturo, M

    M. Punturo, M. Abernathy, F. Acernese, B. Allen, N. Andersson, K. Arun, F. Barone, B. Barr, M. Bar- suglia, M. Beker, et al. , The einstein telescope: A third- generation gravitational wave observatory, Classical and Quantum Gravity 27, 194002 (2010)

    2010

  15. [15]

    T. J. Loredo, Promise of bayesian inference for astro- physics, in Statistical Challenges in Modern Astronomy , edited by E. D. Feigelson and G. J. Babu (Springer New York, New York, NY, 1992) pp. 275–297

    1992

  16. [16]

    D. W. Hogg, J. Bovy, and D. Lang, Data analysis recipes: Fitting a model to data, arXiv preprint arXiv:1008.4686 (2010)

    Pith/arXiv arXiv 2010

  17. [17]

    Mandel, W

    I. Mandel, W. M. Farr, and J. R. Gair, Extracting distri- bution parameters from multiple uncertain observations with selection biases, Monthly Notices of the Royal As- tronomical Society 486, 1086 (2019)

    2019

  18. [18]

    Thrane and C

    E. Thrane and C. Talbot, An introduction to bayesian in- ference in gravitational-wave astronomy: parameter esti- mation, model selection, and hierarchical models, Publi- cations of the Astronomical Society of Australia 36, e010 (2019)

    2019

  19. [19]

    Guttman, E

    N. Guttman, E. Payne, P. D. Lasky, and E. Thrane, Trends in the population of binary black holes following the fourth gravitational-wave transient catalog: a data- driven analysis, arXiv preprint arXiv:2509.09876 (2025)

    Pith/arXiv arXiv 2025

  20. [20]

    Talbot and J

    C. Talbot and J. Golomb, Growing pains: understand- ing the impact of likelihood uncertainty on hierarchi- cal bayesian inference for gravitational-wave astronomy, Monthly Notices of the Royal Astronomical Society 526, 3495 (2023)

    2023

  21. [21]

    Mancarella and D

    M. Mancarella and D. Gerosa, Sampling the full hierar- chical population posterior distribution in gravitational- wave astronomy, Physical Review D 111, 103012 (2025)

    2025

  22. [22]

    K. W. Wong, G. Contardo, and S. Ho, Gravitational- wave population inference with deep flow-based genera- tive network, Physical Review D 101, 123005 (2020)

    2020

  23. [23]

    A. M. Farah, B. Edelman, M. Zevin, M. Fishbach, J. M. Ezquiaga, B. Farr, and D. E. Holz, Things that might go bump in the night: Assessing structure in the binary black hole mass spectrum, The Astrophysical Journal 955, 107 (2023)

    2023

  24. [24]

    Tiwari, Exploring features in the binary black hole population, The Astrophysical Journal 928, 155 (2022)

    V. Tiwari, Exploring features in the binary black hole population, The Astrophysical Journal 928, 155 (2022)

    2022

  25. [25]

    curse of dimensionality,

    face the “curse of dimensionality,” making inference computationally prohibitive [ 13, 26]. To overcome these limitations, the field is increas- ingly exploring semi-parametric and non-parametric ap- proaches. Methods using cubic splines [ 19, 22], binned Gaussian processes [ 27], or Dirichlet process mixture models [ 28] allow the data to dictate the com...

  26. [26]

    C. M. Fabbri, D. Gerosa, A. Santini, M. Mould, A. Toubiana, and J. Gair, Reconstructing parametric gravitational-wave population fits from nonparametric results without refitting the data, Physical Review D 111, 104053 (2025)

    2025

  27. [27]

    Edelman, B

    B. Edelman, B. Farr, and Z. Doctor, Cover your ba- sis: Comprehensive data-driven characterization of the binary black hole population, The Astrophysical Journal 946, 16 (2023)

    2023

  28. [28]

    Gennari, S

    V. Gennari, S. Mastrogiovanni, N. Tamanini, S. Marsat, and G. Pierra, Searching for additional structure and red- shift evolution in the observed binary black hole popula- tion with a parametric time-dependent mass distribution, Physical Review D 111, 123046 (2025)

    2025

  29. [29]

    Foreman-Mackey, D

    D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Good- man, emcee: the mcmc hammer, Publications of the As- tronomical Society of the Pacific 125, 306 (2013)

    2013

  30. [30]

    Talbot and E

    C. Talbot and E. Thrane, Measuring the binary black hole mass spectrum with an astrophysically motivated parameterization, The Astrophysical Journal 856, 173 (2018)

    2018

  31. [31]

    T. A. Callister and W. M. Farr, Parameter-free tour of the binary black hole population, Physical Review X 14, 021005 (2024)

    2024

  32. [32]

    Rinaldi and W

    S. Rinaldi and W. Del Pozzo, Figaro: hierarchical non- parametric inference for population studies, Journal of Open Source Software 9, 6589 (2024)

    2024

  33. [33]

    Golomb and C

    J. Golomb and C. Talbot, Searching for structure in the binary black hole spin distribution, Physical Review D 108, 103009 (2023)

    2023

  34. [34]

    Rezende and S

    D. Rezende and S. Mohamed, Variational inference with normalizing flows, in International conference on ma- chine learning (PMLR, 2015) pp. 1530–1538

    2015

  35. [35]

    Papamakarios, E

    G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mo- hamed, and B. Lakshminarayanan, Normalizing flows for probabilistic modeling and inference, Journal of Machine Learning Research 22, 1 (2021)

    2021

  36. [36]

    Lanchares, O

    D. Lanchares, O. G. Freitas, J. González-Nuevo, and J. A. Font, Assessment of normalizing flows for pa- rameter estimation on time-frequency representations of gravitational-wave data, arXiv preprint arXiv:2505.08089 (2025)

    Pith/arXiv arXiv 2025

  37. [37]

    Colloms, C

    S. Colloms, C. P. Berry, J. Veitch, and M. Zevin, Exploring the evolution of gravitational-wave emitters with efficient emulation: Constraining the origins of bi- nary black holes using normalising flows, arXiv preprint arXiv:2503.03819 (2025)

    Pith/arXiv arXiv 2025

  38. [38]

    D. Shih, M. Freytsis, S. R. Taylor, J. A. Dror, and N. Smyth, Fast parameter inference on pulsar timing ar- rays with normalizing flows, Physical review letters 133, 011402 (2024)

    2024

  39. [39]

    Leyde, S

    K. Leyde, S. R. Green, A. Toubiana, and J. Gair, Grav- itational wave populations and cosmology with neural posterior estimation, Physical Review D 109, 064056 (2024)

    2024

  40. [40]

    Jiang, H.-L

    J.-Q. Jiang, H.-L. Huang, J. He, Y.-T. Wang, and Y.-S. Piao, A fast deep-learning approach to probing primor- dial black hole populations in gravitational wave events, arXiv preprint arXiv:2505.15530 (2025)

    Pith/arXiv arXiv 2025

  41. [41]

    Mould, N

    M. Mould, N. E. Wolfe, and S. Vitale, Rapid inference and comparison of gravitational-wave population mod- els with neural variational posteriors, Physical Review D 111, 123049 (2025)

    2025

  42. [42]

    D. Ruhe, K. Wong, M. Cranmer, and P. Forré, Nor- malizing flows for hierarchical bayesian analysis: A gravitational wave population study, arXiv preprint arXiv:2211.09008 (2022)

    Pith/arXiv arXiv 2022

  43. [43]

    Adamcewicz and E

    C. Adamcewicz and E. Thrane, Do unequal-mass binary black hole systems have larger χeff? probing correlations with copulas in gravitational-wave astronomy, Monthly Notices of the Royal Astronomical Society 517, 3928 (2022)

    2022

  44. [44]

    Adamcewicz, P

    C. Adamcewicz, P. D. Lasky, and E. Thrane, Evidence for a correlation between binary black hole mass ratio and black hole spins, The Astrophysical Journal 958, 13 (2023)

    2023

  45. [45]

    Aghanim, Y

    N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. Barreiro, N. Bartolo, S. Basak, et al. , Planck 2018 results-vi. cos- mological parameters, Astronomy & Astrophysics 641, A6 (2020)

    2018

  46. [46]

    Gregory, Bayesian Logical Data Analysis for the Phys- ical Sciences: A Comparative Approach with Mathemat- ica® Support (Cambridge University Press, 2005)

    P. Gregory, Bayesian Logical Data Analysis for the Phys- ical Sciences: A Comparative Approach with Mathemat- ica® Support (Cambridge University Press, 2005)

    2005

  47. [47]

    C. M. Bishop and N. M. Nasrabadi, Pattern recognition and machine learning , Vol. 4 (Springer, 2006)

    2006

  48. [48]

    R. B. Nelsen, An introduction to copulas (Springer, 2006)

    2006

  49. [49]

    Chen and M

    Y. Chen and M. U. Gutmann, Adaptive gaussian copula abc, in The 22nd International Conference on Artificial Intelligence and Statistics (PMLR, 2019) pp. 1584–1592. 16

    2019

  50. [50]

    Sklar, Fonctions de répartition à n dimensions et leurs marges, in Annales de l’ISUP , Vol

    M. Sklar, Fonctions de répartition à n dimensions et leurs marges, in Annales de l’ISUP , Vol. 8 (1959) pp. 229–231

    1959

  51. [51]

    Kumaraswamy, A generalized probability density function for double-bounded random processes, Journal of hydrology 46, 79 (1980)

    P. Kumaraswamy, A generalized probability density function for double-bounded random processes, Journal of hydrology 46, 79 (1980)

    1980

  52. [52]

    M. D. Hoffman, D. M. Blei, C. Wang, and J. Paisley, Stochastic variational inference, the Journal of machine Learning research 14, 1303 (2013)

    2013

  53. [53]

    D. M. Blei, A. Kucukelbir, and J. D. McAuliffe, Varia- tional inference: A review for statisticians, Journal of the American statistical Association 112, 859 (2017)

    2017

  54. [54]

    L. Dinh, J. Sohl-Dickstein, and S. Bengio, Density esti- mation using real nvp, arXiv preprint arXiv:1605.08803 (2016)

    Pith/arXiv arXiv 2016

  55. [55]

    Paszke, S

    A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al. , Pytorch: An imperative style, high-performance deep learning library, Advances in neural information processing systems 32 (2019)

    2019

  56. [56]

    Bingham, J

    E. Bingham, J. P. Chen, M. Jankowiak, F. Obermeyer, N. Pradhan, T. Karaletsos, R. Singh, P. Szerlip, P. Hors- fall, and N. D. Goodman, Pyro: Deep universal prob- abilistic programming, Journal of machine learning re- search 20, 1 (2019)

    2019

  57. [57]

    G. J. Székely and M. L. Rizzo, Energy statistics: A class of statistics based on distances, Journal of statis- tical planning and inference 143, 1249 (2013)

    2013

  58. [58]

    Weaver, I

    J. Weaver, I. Davidzon, S. Toft, O. Ilbert, H. Mc- Cracken, K. Gould, C. Jespersen, C. Steinhardt, C. La- gos, P. Capak, et al. , Cosmos2020: The galaxy stellar mass function-the assembly and star formation cessation of galaxies at 0.2< z ￿ 7.5, Astronomy & Astrophysics 677, A184 (2023)

    2023

  59. [59]

    Davidzon, O

    I. Davidzon, O. Ilbert, C. Laigle, J. Coupon, H. Mc- Cracken, I. Delvecchio, D. Masters, P. Capak, B. Hsieh, O. Le Fèvre, et al. , The cosmos2015 galaxy stellar mass function-thirteen billion years of stellar mass assembly in ten snapshots, Astronomy & astrophysics 605, A70 (2017)

    2017

  60. [60]

    Rodriguez-Gomez, S

    V. Rodriguez-Gomez, S. Genel, M. Vogelsberger, D. Si- jacki, A. Pillepich, L. V. Sales, P. Torrey, G. Snyder, D. Nelson, V. Springel, et al. , The merger rate of galax- ies in the illustris simulation: a comparison with obser- vations and semi-empirical models, Monthly Notices of the Royal Astronomical Society 449, 49 (2015)

    2015

  61. [61]

    Fakhouri, C.-P

    O. Fakhouri, C.-P. Ma, and M. Boylan-Kolchin, The merger rates and mass assembly histories of dark matter haloes in the two millennium simulations, Monthly No- tices of the Royal Astronomical Society 406, 2267 (2010)

    2010

  62. [62]

    Lacey and S

    C. Lacey and S. Cole, Merger rates in hierarchical mod- els of galaxy formation, Monthly Notices of the Royal Astronomical Society 262, 627 (1993)

    1993

  63. [63]

    Springel et al., Simulations of the formation, evolution and clustering of galaxies and quasars, Nature 435, 629 (2005)

    V. Springel et al., Simulations of the formation, evolution and clustering of galaxies and quasars, Nature 435, 629 (2005)

    2005

  64. [64]

    Kormendy and L

    J. Kormendy and L. C. Ho, Coevolution (or not) of su- permassive black holes and host galaxies, Annual Review of Astronomy and Astrophysics 51, 511–653 (2013)

    2013

  65. [65]

    A. E. Reines and M. Volonteri, Relations between central black hole mass and total galaxy stellar mass in the local universe, The Astrophysical Journal 813, 82 (2015)

    2015

  66. [66]

    Pacucci and A

    F. Pacucci and A. Loeb, The redshift evolution of the m•–m￿ relation for jwst’s supermassive black holes at z> 4, The Astrophysical Journal 964, 154 (2024)

    2024

  67. [67]

    Volonteri, The formation and evolution of massive black holes, Science 337, 544–547 (2012)

    M. Volonteri, The formation and evolution of massive black holes, Science 337, 544–547 (2012)

    2012

  68. [68]

    Langen, N

    V. Langen, N. Tamanini, S. Marsat, and E. Bortolas, Hi- erarchical bayesian inference on an analytical toy model of the lisa mbhb population (2025)

    2025

  69. [69]

    Ajith, S

    P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krish- nan, J. Whelan, B. Bruegmann, P. Diener, J. Gonzalez, M. Hannam, et al. , A phenomenological template fam- ily for black-hole coalescence waveforms, Classical and Quantum Gravity 24, S689 (2007)

    2007

  70. [70]

    Ajith, M

    P. Ajith, M. Hannam, S. Husa, Y. Chen, B. Brüg- mann, N. Dorband, D. Müller, F. Ohme, D. Pollney, C. Reisswig, et al. , Inspiral-merger-ringdown waveforms for black-hole binaries with nonprecessing spins, Physical Review Letters 106, 241101 (2011)

    2011

  71. [71]

    Robson, N

    T. Robson, N. J. Cornish, and C. Liu, The construction and use of lisa sensitivity curves, Classical and Quantum Gravity 36, 105011 (2019)

    2019

  72. [72]

    Cornish and T

    N. Cornish and T. Robson, Galactic binary science with the new lisa design, in Journal of Physics: Conference Series, Vol. 840 (IOP Publishing, 2017) p. 012024

    2017

  73. [73]

    Klein, E

    A. Klein, E. Barausse, A. Sesana, A. Petiteau, E. Berti, S. Babak, J. Gair, S. Aoudia, I. Hinder, F. Ohme, et al. , Science with the space-based interferometer elisa: Su- permassive black hole binaries, Physical Review D 93, 024003 (2016)

    2016

  74. [74]

    Khan and R

    S. Khan and R. Green, Gravitational-wave surrogate models powered by artificial neural networks, Physical Review D 103, 064015 (2021)

    2021

  75. [75]

    Cutler and E

    C. Cutler and E. E. Flanagan, Gravitational waves from merging compact binaries: How accurately can one ex- tract the binary’s parameters from the inspiral wave- form?, Physical Review D 49, 2658 (1994)

    1994

  76. [76]

    M. Vallisneri, Use and abuse of the fisher informa- tion matrix in the assessment of gravitational-wave parameter-estimation prospects, Physical Review D— Particles, Fields, Gravitation, and Cosmology 77, 042001 (2008)

    2008

  77. [77]

    T. B. Littenberg and N. J. Cornish, Prototype global analysis of lisa data with multiple source types, Physical Review D 107, 063004 (2023)

    2023

  78. [78]

    Srinivasan, E

    R. Srinivasan, E. Barausse, N. Korsakova, and R. Trotta, Simulation-based population inference of lisa’s galactic binaries: Bypassing the global fit (2025)

    2025

  79. [79]

    Fishbach, D

    M. Fishbach, D. E. Holz, and B. Farr, Are ligo’s black holes made from smaller black holes?, The Astrophysical Journal Letters 840, L24 (2017)

    2017

  80. [80]

    Gabbard, M

    H. Gabbard, M. Williams, F. Hayes, and C. Messen- ger, Matching matched filtering with deep networks for gravitational-wave astronomy, Physical review letters 120, 141103 (2018)

    2018

Showing first 80 references.