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REVIEW 5 minor 109 references

An Introduction to Bayesian and Frequentist Simulation-Based Inference with Machine Learning

T0 review · 0 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This review argues that one machine-learning recipe—training neural surrogates on samples from the joint parameter-observation distribution—can serve Bayesian and frequentist inference and extends to Empirical Bayes and unfolding.

desk verdict A thorough, honest, and well-structured review of simulation-based inference that earns a serious referee, despite (or because of) a deliberately narrow definition of novelty. read the letter →

arxiv 2607.21702 v1 pith:6SAO4FHI submitted 2026-07-23 cs.LG astro-ph.COastro-ph.GAhep-exhep-phstat.ML

Maximilian Dax , Theo Heimel , Gilles Louppe This is my paper
classification cs.LGastro-ph.COastro-ph.GAhep-exhep-phstat.ML
keywords simulation-basedinferenceneuralposteriorestimationlikelihoodlikelihood-freeamortizedfrequentistcoverageunfoldingEmpiricalBayes
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 review sets out to establish that one machine-learning recipe—train a conditional density or classifier on samples drawn from the joint distribution of parameters and observations—can serve all major statistical goals that scientists encounter. If it is right, researchers who have a simulator but no tractable likelihood can obtain amortized Bayesian posteriors and frequentist confidence sets from the same trained network, and can also invert detector or forward effects for unfolding and empirical Bayes tasks. The paper argues that validation is available through calibration plots, classifier two-sample tests, importance-sampling efficiency, and posterior predictive checks, while making explicit that these checks are necessary but not sufficient. The practical payoff would be a shared workflow across astrophysics, particle physics, and other inverse problems: simulate, train once, then infer and validate cheaply.

What carries the argument

The central machinery is the joint distribution over parameters and observations, sampled by drawing parameters from a prior and running the simulator. From these pairs, neural posterior estimation minimizes the forward Kullback-Leibler divergence to fit a conditional density qθ(µ|x), while neural ratio estimation trains a classifier to separate joint samples from shuffled pairs, recovering the likelihood-to-evidence ratio r(x|µ) = p(x|µ)/p(x). The likelihood ratio between hypotheses follows by dividing these ratios, so the same object feeds Bayesian updates (multiply by the prior) and frequentist confidence intervals (via the asymptotic chi-squared behaviour of the log-ratio). For distribut

What would settle it

Take a simulator with a tractable normal likelihood and known posterior, train both a neural posterior estimator and a classifier-based ratio estimator on joint samples, and compare learned posteriors and likelihood-to-evidence ratios to the exact expressions on a grid of observations. A disagreement beyond Monte Carlo error while the paper's calibration and classifier two-sample diagnostics pass would show that the claimed validated, amortized inference does not hold for that case.

Watch

Extended reading notes

Core claim

The unifying claim is that Bayesian parameter estimation, frequentist parameter estimation, and distribution-level inference are the same inverse problem viewed through different statistical lenses. In each case the forward model is known only through a simulator that can sample x from p(x|µ), and pairs (µ, x) from the joint distribution are enough to train neural surrogates: a conditional density qθ(µ|x) approximating the posterior, or a classifier-based ratio approximating the likelihood-to-evidence ratio. The same learned ratio multiplies over independent observations to give frequentist confidence sets, and the same posterior machinery drives iterative distribution-level updates for unfo

Load-bearing premise

The claim rests on the premise that the simulator or likelihood correctly describes the real data-generating process; if it is misspecified, the trained surrogates can be biased and the validation tests in the paper may not detect it.

Editorial extensions

If this is right

  • A single trained network can report both a Bayesian posterior and frequentist confidence intervals, removing the need to maintain separate analysis pipelines for discovery and parameter estimation.
  • After initial training, inference on new observations is amortized: no new simulations are needed, enabling real-time or large-scale analyses.
  • Multiplication of learned likelihood ratios over independent observations extends frequentist inference to millions of events at low computational cost.
  • The same machinery solves unfolding, empirical Bayes, image deconvolution, and quantification learning, because they share the same distribution-inversion structure.
  • Validation diagnostics require only simulator samples, so they remain applicable even when no reference likelihood-based method is available.

Reading between the lines

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

  • The authors leave implicit that a single amortized network could be audited once on standardized stress tests and then treated as a reusable scientific instrument, shifting emphasis from per-analysis validation to community-wide benchmarks.
  • Because the likelihood-to-evidence ratio can also be read as a posterior-to-prior ratio, distribution-level methods could in principle be initialized from a learned classifier rather than an assumed prior, suggesting a route toward prior-independent unfolding beyond the paper's fixed-point argument.
  • A concrete extension not developed here: use the importance-sampling effective sample size not only as a diagnostic but as an online stopping rule for sequential inference, retraining only while efficiency remains low.
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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

0 major / 5 minor

Summary. This manuscript is an introductory review of machine-learning-based simulation-based inference (SBI), covering Bayesian and frequentist frameworks, neural posterior/likelihood estimation, ratio estimation, importance sampling, distribution-level inference (empirical Bayes/unfolding), data embedding and combination of observations, and validation diagnostics. The central claim is conditional: if a simulator or tractable likelihood is available, these methods can provide amortized Bayesian or frequentist inference, and the same machinery applies to distribution-level tasks. The paper does not claim robustness to simulator misspecification; it explicitly identifies this as an open challenge (Sec. 4.5, Sec. 5).

Significance. The paper's central claim is a conditional methodological-availability claim, not a claim of new theoretical results or universal validity of SBI. The derivations underpinning NPE (Eqs. 22--23), the likelihood-ratio trick (Eqs. 26--28), NPE-IS evidence estimation (Eqs. 32--34), and latent-likelihood ratio regression (Eqs. 36--38) are standard and are reproduced correctly. The paper is more careful than many introductions in separating necessary from sufficient validation (Sec. 4) and in flagging simulator misspecification (Sec. 4.5, Sec. 5) and the conditional nature of the methods. These strengths make it a useful community-review contribution even though it introduces no new algorithms or code.

minor comments (5)
  1. [Sec. 3.3.2, Eq. (41)] The score-matching objective is not self-contained as printed: the expectation is written over p(x,z|mu) while the score is evaluated at mu0 and the derivative is nabla_mu log r_theta(x|mu0,mu1). Based on Eq. (36), the expectation should presumably be over p(x,z|mu1) and the gradient taken with respect to the appropriate ratio argument, with t(x,z|mu1) rather than t(x,z|mu0). Please clarify, since this is an equation a reader may implement.
  2. [Sec. 2.2, Eqs. (7)-(10)] Please state explicitly that c=1,4,9 correspond to the chi-square quantiles for one degree of freedom and that Eq. (10) defines an asymptotic confidence set via Wilks' theorem. The surrounding text says 'in the limit of many independent observations,' but an explicit statement would prevent misapplication outside the asymptotic regime in a tutorial.
  3. [Sec. 3.3.1, after Eq. (34)] The formula sigma_log p(x) = sqrt((1-epsilon)/(N epsilon)) is quoted without derivation or the approximation underlying it. Since this is an introduction, add a brief caveat that this is an asymptotic/approximate expression for the log-evidence uncertainty, or point the reader to the derivation in Ref. [11].
  4. [Sec. 4.4.4] The statement that q_theta(mu|x)=p(mu) passes any joint calibration test is important but underexplained. One sentence explaining that the marginal of the joint samples is p(mu), so the prior-CDF transform is uniform, would make the limitation transparent.
  5. [Sec. 3.5.1] The sentence 'joint training is not possible in neural likelihood estimation' is too strong as written. Joint training is not possible with the standard log-likelihood loss without including the transformation Jacobian, but a conditional invertible embedding could in principle be trained jointly. Please qualify the statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the review's derivations are standard identities and its self-citations are not load-bearing.

full rationale

This paper is a review and tutorial, not an original derivation of new predictions from fitted inputs. Its central claim is conditional: given a simulator or tractable likelihood, ML-based SBI can provide Bayesian/frequentist parameter inference and can be extended to distribution-level tasks. The main technical derivations are self-contained mathematical identities: the NPE objective (Eqs. 22-23) follows from rewriting the marginalised forward KL divergence as an expectation over the joint distribution; the likelihood-ratio trick (Eqs. 26-28) is obtained by functional differentiation of the binary cross-entropy; importance-sampling corrections and evidence estimation (Eqs. 32-34) are standard Monte Carlo identities; the distribution-level EM fixed-point argument (Eqs. 43-46) is a direct algebraic check; and the known-latent-likelihood ratio identity (Eqs. 35-38) is an exact manipulation. None of these steps fits a parameter to one quantity and then predicts a closely related quantity. The paper makes no empirical prediction and performs no fitting, so there is no fitted-input-called-prediction pattern. Self-citations appear (e.g., Refs. 11, 49, 73, 74, 98, 99), but they are used as examples of existing implementations, applications, or published methods, not as the sole justification for the review's conceptual claims. The cited results are externally published and independently testable; the review does not invoke a self-authored uniqueness theorem or forbid alternatives through self-citation. The manuscript also explicitly acknowledges its main limitations, including simulator misspecification and the fact that validation diagnostics are necessary but not sufficient (Secs. 4.4.4 and 5). Thus no circular step can be identified by quoting the paper's own equations or derivation chain.

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

This review introduces no free parameters or invented entities. The axioms are the standard assumptions underlying all simulation-based inference methods described; they are explicitly or implicitly used throughout Secs. 2-3.

assumptions (6)
  • standard math Bayes' theorem defines the posterior as p(µ|x) ∝ p(µ) p(x|µ)
    Sec. 2.1, Eq. (3); fundamental to all Bayesian SBI methods.
  • domain assumption A forward model exists that either allows evaluation of p(x|µ) or sampling x ~ p(x|µ)
    Sec. 2, Introduction; the entire SBI paradigm assumes access to a simulator.
  • domain assumption Simulated samples are i.i.d. and representative of the true likelihood
    Sec. 3.5.2: combining observations via Eq. (8) relies on independence.
  • domain assumption The likelihood is sufficiently smooth in x for NPE to interpolate between training samples
    Sec. 3.1.2, last paragraph: 'This relies on sufficient smoothness of the likelihood in x.'
  • domain assumption A classifier trained with binary cross-entropy converges to the likelihood ratio given sufficient expressivity
    Sec. 3.2.1, Eq. (27)-(28); assumes universal approximation capacity.
  • domain assumption Normalizing flows with forward KLD optimization are expressive enough to represent the posterior
    Sec. 3.1; the authors acknowledge expressivity limits (Sec. 3.1, 'The expressivity of normalizing flows is ... limited').

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Cite this review

Pith. "Pith review of An Introduction to Bayesian and Frequentist Simulation-Based Inference with Machine Learning." pith.science (2026). https://pith.science/paper/6SAO4FHI

@misc{pith2026260721702,
  author       = {Pith},
  title        = {Pith review of: An Introduction to Bayesian and Frequentist Simulation-Based Inference with Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6SAO4FHI}},
  note         = {Machine review of arXiv:2607.21702}
}
read the original abstract

Simulation-based inference (SBI) with machine learning is an increasingly important tool for solving inverse problems in science and engineering, including parameter inference and the inversion of detector effects. We provide an overview of the Bayesian and frequentist statistical frameworks, describe how machine-learning-based SBI methods, such as neural posterior estimation and neural likelihood estimation, can be used for parameter estimation within these frameworks, and show that the same methods can also be applied to Empirical Bayes or unfolding tasks. We also discuss how to validate inference results and the limitations of SBI with machine learning.

Figures

Figures reproduced from arXiv: 2607.21702 by the authors.

Figure 1
Figure 1. The likelihood p(x|µ) (orange) and prior p(µ) (blue) together define the joint distribution p(µ, x). Bayesian inference aims to sample the posterior µ ∼ p(µ|x0 ) (green; horizontal slice through joint distribution) or evaluate the evi￾dence p(x0 ) (red; horizontal projection of joint distribution) for an observation x0 . 2 Statistical framework In this section, we first outline parameter inference in the Bayesian (S… view at source ↗
Figure 2
Figure 2. The left panel shows an example likelihood [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The left panel shows an example likelihood conditional on parameters of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: A Gaussian fitted to a bimodal distribution (gray) covers both modes with [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Gaussian mixture model qθ fitted to a bimodal distribution p via the for￾ward KLD, using batch size 1. In each iteration n, a gradient step is taken to de￾crease the loss L = −log qθ (µn ), increasing the density around the training sample µn ∼ p(µ). The learning rate …
Figure 6
Figure 6. Figure 6: Illustration of samples used for the Monte Carlo estimation of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: An NPE model qθ (µ|x) trained with all samples from the joint distribu￾tion (gray, N = 2000) outperforms an unconditional model qϵ (µ) trained with only the subset of samples accepted in an ϵ-neighborhood of the observation of interest (green, N = 30). The latter corre…
Figure 8
Figure 8. Figure 8: Larger learning rates typically lead to faster convergence in NPE training. [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Left panel: illustration of likelihood ratio trick, where a classifier [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Left: importance sampling reweights samples from the NPE proposal [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Illustration of the iterative procedure to find the distribution [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Effects of biases and under/overdispersion of a learned qθ (µ) (blue) compared to the target distribution p(µ) (gray) on the validation plots described in Secs. 4.3 and 4.4. We display the importance sampling efficiency as a function of the sample size N (column 2), t…
Figure 13
Figure 13. Figure 13: PP plots for an underconfident estimate qθ of a Gaussian target p. The PP test has intrinsic statistical uncertainty inherited from sampling µ ∼ p(µ) (10 independent tests shown in coloured lines), which decreases with increasing number of samples n. The gray bands in…

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Works this paper leans on

109 extracted references · 2 canonical work pages

  1. [99]

    2012 , publisher=

    Density ratio estimation in machine learning , author=. 2012 , publisher=

    2012

  2. [6]

    Statistica sinica , pages=

    Posterior predictive assessment of model fitness via realized discrepancies , author=. Statistica sinica , pages=. 1996 , publisher=

    1996

  3. [97]

    Bayesian Analysis , volume=

    Likelihood-free inference by ratio estimation , author=. Bayesian Analysis , volume=. 2022 , publisher=

    2022

  4. [1]

    2018 , publisher=

    Handbook of approximate Bayesian computation , author=. 2018 , publisher=

    2018

  5. [2]

    Genetics , volume=

    Approximate Bayesian computation in population genetics , author=. Genetics , volume=. 2002 , publisher=

    2002

  6. [3]

    , author=

    Population growth of human Y chromosomes: a study of Y chromosome microsatellites. , author=. Molecular biology and evolution , volume=. 1999 , publisher=

    1999

  7. [4]

    TBA , year=

    Unknown Unknowns in Machine Learning for Physics , author=. TBA , year=

  8. [5]

    arXiv preprint arXiv:2011.01808 , year=

    Bayesian workflow , author=. arXiv preprint arXiv:2011.01808 , year=

    arXiv 2011

Show all 109 references
  1. [7]

    and Gair, Jonathan and Macke, Jakob H

    Dax, Maximilian and Green, Stephen R. and Gair, Jonathan and Macke, Jakob H. and Buonanno, Alessandra and Sch. Real-Time Gravitational Wave Science with Neural Posterior Estimation. Phys. Rev. Lett. 2021. doi:10.1103/PhysRevLett.127.241103. arXiv:2106.12594

    2021 arXiv

  2. [8]

    Evidence for eccentricity in the population of binary black holes observed by LIGO-Virgo-KAGRA

    Gupte, Nihar and others. Evidence for eccentricity in the population of binary black holes observed by LIGO-Virgo-KAGRA. Phys. Rev. D. 2025. doi:10.1103/vpyp-nvfp. arXiv:2404.14286

    2025 arXiv

  3. [9]

    arXiv preprint arXiv:2508.12939 , year=

    Simulation-based inference: A practical guide , author=. arXiv preprint arXiv:2508.12939 , year=

    arXiv

  4. [10]

    Cranmer, Kyle and Lo, Jason , title =

  5. [11]

    Mishra-Sharma, Siddharth , title =

  6. [12]

    Simulation-based inference methods for particle physics

    Brehmer, Johann and Cranmer, Kyle. Simulation-based inference methods for particle physics. 2020. arXiv:2010.06439

    2020 arXiv

  7. [13]

    Machine Learning Techniques for Astrophysics and Cosmology: Simulation-Based Inference

    Thiele, Leander. Machine Learning Techniques for Astrophysics and Cosmology: Simulation-Based Inference. 2026. arXiv:2605.10719

    2026 arXiv

  8. [14]

    James Bradbury and Roy Frostig and Peter Hawkins and Matthew James Johnson and Yash Katariya and Chris Leary and Dougal Maclaurin and George Necula and Adam Paszke and Jake Vander

  9. [15]

    Advances in neural information processing systems , volume=

    Pytorch: An imperative style, high-performance deep learning library , author=. Advances in neural information processing systems , volume=

  10. [16]

    Radev, Stefan T and Schmitt, Marvin and Schumacher, Lukas and Elsemüller, Lasse and Pratz, Valentin and Schälte, Yannik and Köthe, Ullrich and Bürkner, Paul-Christian , journal =

  11. [17]

    swyft: Truncated Marginal Neural Ratio Estimation in Python

    Miller, Benjamin Kurt and Cole, Alex and Weniger, Christoph and Nattino, Francesco and Ku, Ou and Grootes, Meiert W. swyft: Truncated Marginal Neural Ratio Estimation in Python. J. Open Source Softw. 2022. doi:10.21105/joss.04205

    2022 doi

  12. [18]

    doi:10.5281/zenodo.8405782 , license =

    Rozet, François and Delaunoy, Arnaud and Miller, Benjamin and others , year =. doi:10.5281/zenodo.8405782 , license =

    doi

  13. [19]

    Jan Boelts and Michael Deistler and Manuel Gloeckler and Álvaro Tejero-Cantero and Jan-Matthis Lueckmann and Guy Moss and Peter Steinbach and Thomas Moreau and Fabio Muratore and Julia Linhart and Conor Durkan and Julius Vetter and Benjamin Kurt Miller and Maternus Herold and ...

    2025

  14. [20]

    arXiv preprint arXiv:2405.08719 , year=

    Addressing misspecification in simulation-based inference through data-driven calibration , author=. arXiv preprint arXiv:2405.08719 , year=

    arXiv

  15. [21]

    arXiv preprint arXiv:2209.01845 , year=

    Investigating the impact of model misspecification in neural simulation-based inference , author=. arXiv preprint arXiv:2209.01845 , year=

    arXiv

  16. [22]

    International conference on machine learning , pages=

    On the difficulty of training recurrent neural networks , author=. International conference on machine learning , pages=. 2013 , organization=

    2013

  17. [23]

    International Conference on Learning Representations , year=

    Revisiting Classifier Two-Sample Tests , author=. International Conference on Learning Representations , year=

  18. [24]

    2004 , institution=

    On multivariate goodness-of-fit and two-sample testing , author=. 2004 , institution=

    2004

  19. [25]

    A crisis in simulation-based inference? Beware, your posterior approximations can be unfaithful , author=

  20. [26]

    International conference on artificial intelligence and statistics , pages=

    Benchmarking simulation-based inference , author=. International conference on artificial intelligence and statistics , pages=. 2021 , organization=

    2021

  21. [27]

    Test , volume=

    Assessing probabilistic forecasts of multivariate quantities, with an application to ensemble predictions of surface winds , author=. Test , volume=. 2008 , publisher=

    2008

  22. [28]

    The annals of mathematical statistics , volume=

    Remarks on a multivariate transformation , author=. The annals of mathematical statistics , volume=. 1952 , publisher=

    1952

  23. [29]

    Statistics and Computing , volume=

    Graphical test for discrete uniformity and its applications in goodness-of-fit evaluation and multiple sample comparison , author=. Statistics and Computing , volume=. 2022 , publisher=

    2022

  24. [30]

    arXiv preprint arXiv:1804.06788 , year=

    Validating Bayesian inference algorithms with simulation-based calibration , author=. arXiv preprint arXiv:1804.06788 , year=

    arXiv

  25. [31]

    Science , volume=

    Swope Supernova Survey 2017a (SSS17a), the optical counterpart to a gravitational wave source , author=. Science , volume=. 2017 , publisher=

    2017

  26. [32]

    Owen , year = 2013, title =

    Art B. Owen , year = 2013, title =

    2013

  27. [33]

    Truncated proposals for scalable and hassle-free simulation-based inference , url =

    Deistler, Michael and Goncalves, Pedro J and Macke, Jakob H , booktitle =. Truncated proposals for scalable and hassle-free simulation-based inference , url =

  28. [34]

    The Journal of Machine Learning Research , volume=

    Normalizing flows for probabilistic modeling and inference , author=. The Journal of Machine Learning Research , volume=. 2021 , publisher=

    2021

  29. [35]

    International Conference on Machine Learning , pages=

    Variational Inference with Normalizing Flows , author=. International Conference on Machine Learning , pages=. 2015 , eprint=

    2015

  30. [36]

    IEEE transactions on neural networks and learning systems , volume=

    BayesFlow: Learning complex stochastic models with invertible neural networks , author=. IEEE transactions on neural networks and learning systems , volume=. 2020 , publisher=

    2020

  31. [37]

    NeurIPS 2023 , eprint=

    Flow Matching for Scalable Simulation-Based Inference , author=. NeurIPS 2023 , eprint=

    2023

  32. [38]

    arXiv preprint arXiv:2312.05440 , year=

    Consistency Models for Scalable and Fast Simulation-Based Inference , author=. arXiv preprint arXiv:2312.05440 , year=

    arXiv

  33. [39]

    International Conference on Learning Representations , year=

    Poornima Ramesh and Jan-Matthis Lueckmann and Jan Boelts and. International Conference on Learning Representations , year=

  34. [40]

    Proceedings of the 40th International Conference on Machine Learning , journal =

    Compositional Score Modeling for Simulation-Based Inference , author =. Proceedings of the 40th International Conference on Machine Learning , journal =. 2023 , editor =

    2023

  35. [41]

    Proceedings of the 41st International Conference on Machine Learning , journal =

    Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models , author =. Proceedings of the 41st International Conference on Machine Learning , journal =. 2024 , editor =

    2024

  36. [42]

    The Eleventh International Conference on Learning Representations , journal=

    Flow Matching for Generative Modeling , author=. The Eleventh International Conference on Learning Representations , journal=. 2023 , url=

    2023

  37. [43]

    International Conference on Machine Learning , pages=

    Consistency Models , author=. International Conference on Machine Learning , pages=. 2023 , organization=

    2023

  38. [44]

    Advances in neural information processing systems , volume=

    Generative adversarial nets , author=. Advances in neural information processing systems , volume=

  39. [45]

    Advances in neural information processing systems , volume=

    Denoising diffusion probabilistic models , author=. Advances in neural information processing systems , volume=

  40. [46]

    Advances in neural information processing systems , volume=

    Generative modeling by estimating gradients of the data distribution , author=. Advances in neural information processing systems , volume=

  41. [47]

    International conference on machine learning , pages=

    Deep unsupervised learning using nonequilibrium thermodynamics , author=. International conference on machine learning , pages=. 2015 , organization=

    2015

  42. [48]

    The Eleventh International Conference on Learning Representations , year=

    Building Normalizing Flows with Stochastic Interpolants , author=. The Eleventh International Conference on Learning Representations , year=

  43. [49]

    The Eleventh International Conference on Learning Representations , year=

    Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow , author=. The Eleventh International Conference on Learning Representations , year=

  44. [50]

    The 22nd International Conference on Artificial Intelligence and Statistics , pages=

    Sequential neural likelihood: Fast likelihood-free inference with autoregressive flows , author=. The 22nd International Conference on Artificial Intelligence and Statistics , pages=. 2019 , organization=

    2019

  45. [51]

    Symposium on Advances in Approximate Bayesian Inference , pages=

    Likelihood-free inference with emulator networks , author=. Symposium on Advances in Approximate Bayesian Inference , pages=. 2019 , organization=

    2019

  46. [52]

    Statistical inference for noisy nonlinear ecological dynamic systems , volume =

    Wood, Simon , year =. Statistical inference for noisy nonlinear ecological dynamic systems , volume =. Nature , doi =

  47. [53]

    2018 , eprint=

    Approximating the Likelihood in Approximate Bayesian Computation , author=. 2018 , eprint=

    2018

  48. [54]

    International Conference on Machine Learning , pages=

    Automatic posterior transformation for likelihood-free inference , author=. International Conference on Machine Learning , pages=. 2019 , organization=

    2019

  49. [55]

    Proceedings of the 31st International Conference on Neural Information Processing Systems , pages=

    Flexible statistical inference for mechanistic models of neural dynamics , author=. Proceedings of the 31st International Conference on Neural Information Processing Systems , pages=

  50. [56]

    Fast -free Inference of Simulation Models with

    George Papamakarios and Iain Murray , year=2016, eprint=. Fast -free Inference of Simulation Models with

    2016

  51. [57]

    The Fourteenth International Conference on Learning Representations , year=

    Robust Amortized Bayesian Inference with Self-Consistency Losses on Unlabeled Data , author=. The Fourteenth International Conference on Learning Representations , year=

  52. [58]

    International Conference on Machine Learning , pages=

    Leveraging Self-Consistency for Data-Efficient Amortized Bayesian Inference , author=. International Conference on Machine Learning , pages=. 2024 , organization=

    2024

  53. [59]

    and Gair, Jonathan and Gupte, Nihar and P

    Dax, Maximilian and Green, Stephen R. and Gair, Jonathan and Gupte, Nihar and P. Real-time inference for binary neutron star mergers using machine learning. Nature. 2025. doi:10.1038/s41586-025-08593-z. arXiv:2407.09602

    2025 arXiv

  54. [60]

    International conference on machine learning , pages=

    Stochastic backpropagation and approximate inference in deep generative models , author=. International conference on machine learning , pages=. 2014 , organization=

    2014

  55. [61]

    arXiv preprint arXiv:1312.6114 , year=

    Auto-encoding variational bayes , author=. arXiv preprint arXiv:1312.6114 , year=

    arXiv

  56. [62]

    and Gair, Jonathan and P

    Dax, Maximilian and Green, Stephen R. and Gair, Jonathan and P. Neural Importance Sampling for Rapid and Reliable Gravitational-Wave Inference. Phys. Rev. Lett. 2023. doi:10.1103/PhysRevLett.130.171403. arXiv:2210.05686

    2023 arXiv

  57. [63]

    Bayesian Analysis , number =

    John Skilling , title =. Bayesian Analysis , number =. 2006 , doi =

    2006

  58. [64]

    The journal of chemical physics , volume=

    Equation of state calculations by fast computing machines , author=. The journal of chemical physics , volume=. 1953 , publisher=

    1953

  59. [65]

    Journal of the optical society of America , volume=

    Bayesian-based iterative method of image restoration , author=. Journal of the optical society of America , volume=. 1972 , publisher=

    1972

  60. [66]

    Astronomical Journal, Vol

    An iterative technique for the rectification of observed distributions , author=. Astronomical Journal, Vol. 79, p. 745 (1974) , volume=

    1974

  61. [67]

    Hastings, W. K. , title = ". Biometrika , volume =. 1970 , month =

    1970

  62. [68]

    The frontier of simulation-based inference

    Cranmer, Kyle and Brehmer, Johann and Louppe, Gilles. The frontier of simulation-based inference. Proc. Nat. Acad. Sci. 2020. doi:10.1073/pnas.1912789117. arXiv:1911.01429

    2020 arXiv

  63. [69]

    , title =

    D'Agostini, G. , title =. Nucl. Instrum. Meth. , volume =. doi:10.1016/0168-9002(95)00274-X , reportnumber =

    doi

  64. [70]

    Advances in neural information processing systems , volume=

    Deep sets , author=. Advances in neural information processing systems , volume=

  65. [71]

    Dagm german conference on pattern recognition , pages=

    Detecting model misspecification in amortized Bayesian inference with neural networks , author=. Dagm german conference on pattern recognition , pages=. 2023 , organization=

    2023

  66. [72]

    International Conference on Learning Representations , year=

    An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale , author=. International Conference on Learning Representations , year=

  67. [73]

    and Metodiev, Eric M

    Andreassen, Anders and Komiske, Patrick T. and Metodiev, Eric M. and Nachman, Benjamin and Thaler, Jesse. OmniFold: A Method to Simultaneously Unfold All Observables. Phys. Rev. Lett. 2020. doi:10.1103/PhysRevLett.124.182001. arXiv:1911.09107

    2020 arXiv

  68. [74]

    Statistical issues in unfolding methods for high energy physics , author=

  69. [75]

    Journal of the American statistical Association , volume=

    A statistical model for positron emission tomography , author=. Journal of the American statistical Association , volume=. 1985 , publisher=

    1985

  70. [76]

    The Annals of statistics , pages=

    On the convergence properties of the EM algorithm , author=. The Annals of statistics , pages=. 1983 , publisher=

    1983

  71. [77]

    Journal of the royal statistical society: series B (methodological) , volume=

    Maximum likelihood from incomplete data via the EM algorithm , author=. Journal of the royal statistical society: series B (methodological) , volume=. 1977 , publisher=

    1977

  72. [78]

    Invertible Networks or Partons to Detector and Back Again

    Bellagente, Marco and Butter, Anja and Kasieczka, Gregor and Plehn, Tilman and Rousselot, Armand and Winterhalder, Ramon and Ardizzone, Lynton and K. Invertible Networks or Partons to Detector and Back Again. SciPost Phys. 2020. doi:10.21468/SciPostPhys.9.5.074. arXiv:2006.06685

    2020 arXiv

  73. [79]

    An unfolding method based on conditional invertible neural networks (cINN) using iterative training

    Backes, Mathias and Butter, Anja and Dunford, Monica and Malaescu, Bogdan. An unfolding method based on conditional invertible neural networks (cINN) using iterative training. SciPost Phys. Core. 2024. doi:10.21468/scipostphyscore.7.1.007. arXiv:2212.08674

    2024 arXiv

  74. [80]

    International Conference on Artificial Intelligence and Statistics , pages=

    Neural empirical Bayes: Source distribution estimation and its applications to simulation-based inference , author=. International Conference on Artificial Intelligence and Statistics , pages=. 2021 , organization=

    2021

  75. [81]

    Simulation-Prior Independent Neural Unfolding Procedure

    Butter, Anja and Heimel, Theo and Huetsch, Nathan and Kagan, Michael and Plehn, Tilman. Simulation-Prior Independent Neural Unfolding Procedure. 2025. arXiv:2507.15084

    2025 arXiv

  76. [82]

    Mining gold from implicit models to improve likelihood-free inference

    Brehmer, Johann and Louppe, Gilles and Pavez, Juan and Cranmer, Kyle. Mining gold from implicit models to improve likelihood-free inference. Proc. Nat. Acad. Sci. 2020. doi:10.1073/pnas.1915980117. arXiv:1805.12244

    2020 arXiv

  77. [83]

    A Guide to Constraining Effective Field Theories with Machine Learning

    Brehmer, Johann and Cranmer, Kyle and Louppe, Gilles and Pavez, Juan. A Guide to Constraining Effective Field Theories with Machine Learning. Phys. Rev. D. 2018. doi:10.1103/PhysRevD.98.052004. arXiv:1805.00020

    2018 arXiv

  78. [84]

    , author=

    Survey sampling. , author=. 1965 , publisher=

    1965

  79. [85]

    Journal of Official Statistics , volume=

    Weighting for unequal Pi , author=. Journal of Official Statistics , volume=. 1992 , publisher=

    1992

  80. [86]

    The landscape of unfolding with machine learning

    Huetsch, Nathan and others. The landscape of unfolding with machine learning. SciPost Phys. 2025. doi:10.21468/SciPostPhys.18.2.070. arXiv:2404.18807

    2025 arXiv

  81. [87]

    Approximating Likelihood Ratios with Calibrated Discriminative Classifiers

    Cranmer, Kyle and Pavez, Juan and Louppe, Gilles. Approximating Likelihood Ratios with Calibrated Discriminative Classifiers. 2015. arXiv:1506.02169

    2015 arXiv

  82. [88]

    Information , volume=

    Recurrent neural networks: A comprehensive review of architectures, variants, and applications , author=. Information , volume=. 2024 , publisher=

    2024

  83. [89]

    Advances in neural information processing systems , volume=

    Attention is all you need , author=. Advances in neural information processing systems , volume=

  84. [90]

    Electronic Journal of Statistics , volume=

    Likelihood-free frequentist inference: Bridging classical statistics and machine learning for reliable simulator-based inference , author=. Electronic Journal of Statistics , volume=. 2024 , publisher=

    2024

  85. [91]

    AI open , volume=

    Graph neural networks: A review of methods and applications , author=. AI open , volume=. 2020 , publisher=

    2020

  86. [92]

    Philosophical Transactions of the Royal Society of London

    Outline of a theory of statistical estimation based on the classical theory of probability , author=. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences , volume=. 1937 , publisher=

    1937

  87. [93]

    An implementation of neural simulation-based inference for parameter estimation in ATLAS

    Aad, Georges and others. An implementation of neural simulation-based inference for parameter estimation in ATLAS. Rept. Prog. Phys. 2025. doi:10.1088/1361-6633/add370. arXiv:2412.01600

    2025 arXiv

  88. [94]

    International conference on machine learning , pages=

    On contrastive learning for likelihood-free inference , author=. International conference on machine learning , pages=. 2020 , organization=

    2020

  89. [95]

    International conference on machine learning , pages=

    Likelihood-free MCMC with amortized approximate ratio estimators , author=. International conference on machine learning , pages=. 2020 , organization=

    2020

  90. [96]

    Artificial intelligence and statistics , pages=

    High-dimensional density ratio estimation with extensions to approximate likelihood computation , author=. Artificial intelligence and statistics , pages=. 2014 , organization=

    2014

  91. [98]

    Advances in Neural Information Processing Systems , volume=

    Contrastive neural ratio estimation , author=. Advances in Neural Information Processing Systems , volume=

  92. [100]

    2009 , publisher=

    The elements of statistical learning: data mining, inference, and prediction , author=. 2009 , publisher=

    2009

  93. [101]

    Learning likelihood ratios with neural network classifiers

    Rizvi, Shahzar and Pettee, Mariel and Nachman, Benjamin. Learning likelihood ratios with neural network classifiers. JHEP. 2024. doi:10.1007/JHEP02(2024)136. arXiv:2305.10500

    2024 arXiv

  94. [102]

    arXiv preprint arXiv:1605.08803 , year=

    Density estimation using Real NVP , author=. arXiv preprint arXiv:1605.08803 , year=

    arXiv

  95. [103]

    Advances in neural information processing systems , volume=

    Masked autoregressive flow for density estimation , author=. Advances in neural information processing systems , volume=

  96. [104]

    Advances in neural information processing systems , volume=

    Neural ordinary differential equations , author=. Advances in neural information processing systems , volume=

  97. [105]

    ACM Computing Surveys (CSUR) , volume=

    A review on quantification learning , author=. ACM Computing Surveys (CSUR) , volume=. 2017 , publisher=

    2017

  98. [106]

    Advances in Neural Information Processing Systems , volume=

    L-c2st: Local diagnostics for posterior approximations in simulation-based inference , author=. Advances in Neural Information Processing Systems , volume=

  99. [107]

    How to understand limitations of generative networks

    Das, Ranit and Favaro, Luigi and Heimel, Theo and Krause, Claudius and Plehn, Tilman and Shih, David. How to understand limitations of generative networks. SciPost Phys. 2024. doi:10.21468/SciPostPhys.16.1.031. arXiv:2305.16774

    2024 arXiv

  100. [108]

    International Conference on Machine Learning , pages=

    Sampling-based accuracy testing of posterior estimators for general inference , author=. International Conference on Machine Learning , pages=. 2023 , organization=

    2023

  101. [109]

    arXiv preprint arXiv:2402.04355 , year=

    Pqmass: Probabilistic assessment of the quality of generative models using probability mass estimation , author=. arXiv preprint arXiv:2402.04355 , year=

    arXiv

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.