Estimating Marginal Likelihoods in Likelihood-Free Inference via Neural Density Estimation

Arnaud Estoup; Jean-Michel Marin; Julien Stoehr; Paul Bastide

REVIEW 2 major objections 5 minor 25 references

Estimating Marginal Likelihoods in Likelihood-Free Inference via Neural Density Estimation

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper shows that SNLE's own outputs—successive neural likelihood surrogates and posterior samples—are enough to estimate the marginal likelihood, via a product of per-round ratios or an importance-sampling flow with dilated tails.

desk verdict The IS estimator is a working tool, but the new SIS estimator draws its Monte Carlo samples from the wrong distribution, so the paper's main novelty needs a repair before it can support the evidence-from-SNLE-outputs claim. read the letter →

arxiv 2507.08734 v1 pith:WG2D72WO submitted 2025-07-11 stat.CO

Paul Bastide , Arnaud Estoup , Jean-Michel Marin , Julien Stoehr This is my paper

classification stat.CO MSC 62F1562-0865C05

keywords marginallikelihoodBayesianevidencemodelselectionsimulation-basedinferencesequentialneuralestimationimportancesamplingnormalizingflowslikelihood-free

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

The paper proposes a way to estimate the marginal likelihood, or Bayesian evidence, using only the outputs that Sequential Neural Likelihood Estimation (SNLE) already produces: a sequence of neural surrogates $q^{(\ell)}(x^*|\theta)$ and posterior samples. The first estimator writes the evidence as a product of per-round ratios, each computable by averaging the ratio of consecutive surrogate likelihoods over samples from the previous round. The second trains a normalizing flow on the final posterior sample and uses it as an importance-sampling proposal with deliberately dilated tails. In the paper's Gaussian experiments, the importance-sampling estimator stays close to the true evidence up to dimension 50 and is less sensitive to tuning than the retargeted harmonic-mean estimator. If the claim holds, Bayesian model selection becomes available in likelihood-free settings without any additional simulator calls.

What carries the argument

The load-bearing object is the ratio identity $$ R_\ell = \frac{C_\ell}{C_{\ell-1}} = \int \frac{$q^{{(\ell)}}$(x^*|\$\theta$)}{$q^{{(\ell-1)}}$(x^*|\$\theta$)}\,\hat{\pi}^{(\ell-1)}(\$\theta$|x^*)\,d\$\theta$, $$ obtained by the change of measure from the prior to the previous round's approximate posterior. Because Algorithm 1 has already sampled $\theta_i^{(\ell-1)}$ from $\hat{\pi}^{(\ell-1)}$, the ratio is estimated by a plain Monte Carlo average, and the evidence estimate is the product of these averages across rounds. The companion machinery is a normalizing flow—a tractable neural density that can be sampled from and evaluated pointwise—fitted to the final posterior sample and used as an importance proposal; its base-variance temperature $T \ge 1$ dilates the proposal so the IS weights have heavier-tailed support, mirroring the concentrated flow used by the harmonic-mean approach with $T \le 1$.

What would settle it

Take a generative model with closed-form evidence and a prior much wider than the posterior; run SNLE, then recompute the product estimator after replacing $q^{(\ell)}(x^*|\theta)$ with its true value outside the region the rounds actually visited. If the estimate moves by more than Monte Carlo error, the method is inheriting its accuracy from the surrogate's tail behavior rather than from the training rounds; a second check is to compare IS-SNLE against a gold-standard nested-sampling or brute-force estimate on a heavy-tailed likelihood.

Watch

Extended reading notes

Core claim

The central claim is that the marginal likelihood $C = \int f(x^*|\theta)\pi(\theta)\,d\theta$ can be recovered from SNLE's own training record. Denoting by $q^{(\ell)}$ the neural likelihood surrogate after round $\ell$ and by $(\theta_i^{(\ell-1)})$ the samples drawn from the previous approximate posterior, the paper defines $$ \widehat{C}_L = \prod_{\ell=1}^{L} \frac{1}{N}\sum_{i=1}^{N} \frac{$q^{{(\ell)}}$(x^*|\$theta_i^{{(\ell-1)}}$)}{$q^{{(\ell-1)}}$(x^*|\$theta_i^{{(\ell-1)}}$)}, $$ with $q^{(0)}\equiv 1$ and $C_0 = 1$. This product telescopes the normalizing constants of successive approximate posteriors, so it can be evaluated offline from the artifacts of Algorithm 1. The paper also introduces an importance-sampling estimator that fits a normalizing flow $h(\theta)$ to the final MCMC posterior sample, dilates its base variance by a temperature $T\ge 1$ to obtain heavier tails, and estimates $$ \widehat{C}_L = \frac{1}{N_{\mathrm{IS}}}\sum_{j=1}^{N_{\mathrm{IS}}} \frac{$q^{{(L)}}$(x^*|\$theta_j^{{\mathrm{IS}}$})\pi(\$theta_j^{{\mathrm{IS}}$})}{h(\$theta_j^{{\mathrm{IS}}$})}. $$ On the Gaussian toy model with dimensions 3, 10, 20, and 50, the IS-SNLE estimates stay close to the closed-form evidence and outperform the sequential product in higher dimensions, while the retargeted harmonic-mean estimator degrades. The paper positions these estimators as a general bridge from SNLE posterior inference to model comparison.

Load-bearing premise

The estimate is only as good as the surrogate likelihood $q^{(L)}(x^*|\theta)$ over the whole prior support, but SNLE spends its simulations where the posterior is high, so errors in the tails of the prior can bias the evidence without any diagnostic catching them.

Editorial extensions

If this is right

Bayesian model comparison in simulator-based models can be done offline after an SNLE run: no new simulations, no ABC rejection step, and no separate evidence pass.
The IS-SNLE estimator applies beyond likelihood-free inference: any posterior sample plus a pointwise-evaluable surrogate likelihood yields an evidence estimate.
On the paper's benchmarks, IS-SNLE with a dilated-flow proposal is the more accurate variant in higher dimensions, while the sequential product remains a zero-extra-cost alternative that degrades with dimension.
Because the estimator uses only the final surrogate $q^{(L)}$, the prior, and the flow $h$, practitioners can pair it with any MCMC or sampling scheme that produces posterior draws from SNLE.
The same machinery points toward extending evidence estimation to SNPE and SNRE, the two other neural SBI families, as the paper identifies as future work.

Reading between the lines

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

The product formulation is structurally a stepping-stone/SMC estimator of a ratio of normalizing constants; if that analogy holds, its variance should be governed by how much consecutive surrogate posteriors overlap, so an overlap diagnostic computed from existing samples would predict when SIS-SNLE will fail.
An inexpensive robustness check the paper does not run is to compare $q^{(L)}(x^*|\theta)$ evaluated at prior draws against its values at posterior draws; a large gap would flag tail error before paying for a gold-standard evidence estimate.
In misspecified models the neural surrogate may be overconfident, and the evidence estimate could then favor the model whose surrogate happens to be sharpest; this bias would be visible in calibration tests across replicated simulated datasets.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things you should know about this one. The IS estimator in Eq. (3) is clean and useful: after SNLE, fit a flow to the posterior MCMC sample, dilute it a bit, and importance-sample the surrogate evidence. That is textbook importance sampling, but the packaging around SNLE outputs fills a real practical gap, and the toy results support it. The more novel SIS estimator in Eq. (2), however, is invalid as written.

The problem is indexing. Algorithm 1 draws theta_i^(ell) from pi-hat^(ell-1), so the samples it labels theta_i^(ell-1) come from pi-hat^(ell-2), not pi-hat^(ell-1). The ratio C_ell/C_{ell-1} is an expectation under pi-hat^(ell-1) of q^(ell)/q^(ell-1), so Eq. (2) uses draws from the wrong distribution and the product does not telescope. The first factor references theta_i^(0), which Algorithm 1 never generates. Simply replacing ell-1 with ell matches the distribution but makes q^(ell) train on the same samples used for the MC average, so in-sample bias appears; the honest fix, fresh MCMC draws from each pi-hat^(ell-1), costs simulations. The claim that the estimator works 'directly from the sequence of density approximations and posterior samples produced by Algorithm 1' is therefore not true as stated.

Credit where it's due: the paper knows its other main weakness. Section 7 explicitly flags support mismatch and 'propagation of density estimation errors,' which matters because q^(L) is trained in high-posterior regions while the evidence integral needs the prior's tails. No diagnostic or bound is provided, and the abstract's promise of 'the marginal likelihood' reads as an overclaim, but the limitation is not hidden. The HM estimator is clearly credited to McEwen et al. and Spurio Mancini et al., and the experiments are honest about the fact that SIS underperforms IS once the dimension grows. Also, no code ships, which limits reproducibility for a methods note like this.

Who gets value: practitioners who already run SNLE and want a cheap evidence estimate have a working tool in IS-SNLE. The SIS idea is worth fixing rather than discarding, but it is currently the weakest part of the paper.

Recommendation: send to peer review, with the expectation of major revision. The indexing error is specific and repairable, the IS contribution is sound, and the topic matters to the SBI community. A referee can do something with this.

Referee Report

2 major / 5 minor

Summary. The manuscript proposes three estimators of the marginal likelihood from the outputs of Sequential Neural Likelihood Estimation (SNLE): a sequential importance sampling estimator (SIS-SNLE, Eq. (2)) that multiplies ratios of consecutive surrogate-evidence constants; an importance sampling estimator (IS-SNLE, Eq. (3)) using a normalizing-flow proposal fitted to the final posterior MCMC sample; and a retargeted harmonic mean estimator (HM-SNLE, Eq. (4)) following Spurio Mancini et al. (2023). The methods are evaluated on a Gaussian toy model in dimensions 3, 10, 20 and 50, with 25 replications and a temperature parameter for flow tail adjustment. The central claim is that the evidence can be computed from SNLE outputs alone, without additional simulator calls.

Significance. The IS estimator is a genuinely useful idea: it turns the SNLE surrogate likelihood into a tractable importance-sampling target and requires no extra simulations, and the toy experiments show it to be more accurate and less temperature-sensitive than the harmonic mean variant. The HM retargeting is also competently adapted from prior work. However, the SIS estimator as written relies on an incorrect change-of-measure identity (see Major Comment 1), and all estimators share a gap between the surrogate evidence and the true evidence that is acknowledged but not quantified. With the SIS issue repaired and the surrogate-evidence gap addressed, the IS and HM parts could form a solid contribution to SBI model comparison.

major comments (2)

[§2, Eq. (2) and Algorithm 1] The change-of-measure identity leading to R_ell = E_{hat-pi^(ell-1)}[q^(ell)/q^(ell-1)] is correct, but the Monte Carlo estimator uses samples theta_i^(ell-1) that Algorithm 1 draws from hat-pi^(ell-2), not from hat-pi^(ell-1); for ell=1, no samples theta_i^(0) exist. Consequently, the average in Eq. (2) is not an unbiased estimator of R_ell, and the product does not telescope to C_L. The assertion that Eq. (2) can be computed directly from the sequence of density approximations and posterior samples produced by Algorithm 1 is therefore unsubstantiated. The missing importance weight would involve hat-pi^(ell-1)/hat-pi^(ell-2), which is proportional to q^(ell-1) C_{ell-2} / (q^(ell-2) C_{ell-1}) and is not available in the simple average. The SIS claim is repairable only by drawing fresh MCMC samples from each hat-pi^(ell-1) (independent of the training of q^(ell)) or by explicitly accounting for this weight; as written, the central SIS result is invalid.
[§3, Eq. (3) and §7] The IS estimator is unbiased for the surrogate evidence C_L = int q^(L)(x*|theta) pi(theta) dtheta, not for the true evidence C = int f(x*|theta) pi(theta) dtheta. The paper claims that C_L closely approximates C, but SNLE's sequential training concentrates simulations in high-posterior regions, and there is no guarantee that q^(L) is accurate over the entire prior support where the evidence integral has mass. Section 7 acknowledges 'propagation of density estimation errors' but provides no diagnostic or error bound. Since the stated purpose is model selection, where even small log-evidence errors can change conclusions, the manuscript should either provide a diagnostic (e.g., comparing q^(L) against held-out simulations in the tails, or a sensitivity analysis over the prior) or explicitly restrict the claim to the surrogate evidence.

minor comments (5)

[§4] There are typographical errors: 'The learning set is used to to fit a NF' and 'the evaluating set to fitted to a subset of the posterior samples' should be corrected.
[Figures 1 and 2] The captions contain 'he dashed line' where 'the dashed line' is intended; please correct.
[Notation] The notation is inconsistent: Algorithm 1 and Section 2 use hat-pi^(ell), while Section 4 uses pi^(L); please unify the notation throughout.
[Algorithm 1] Algorithm 1 does not define or store q^(0) or the initial prior samples theta_i^(0), even though Eq. (2) refers to them; the algorithm should be extended accordingly if the SIS estimator is to be used.
[§6] For reproducibility, the manuscript should report MCMC burn-in, thinning, random-seed handling, and the precise flow training details; a code/data availability statement would also be helpful.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction of the evidence estimate; the SIS estimator in Eq. (2) contains an indexing flaw that is a correctness issue, not a self-referential fit.

full rationale

The paper's estimators are not circular. The SIS and IS procedures are plug-in Monte Carlo estimators of the normalizing constant of the SNLE surrogate likelihood q^(L) pi; q^(L) is trained on simulator output, never on the evidence value itself, and the closed-form Gaussian benchmark provides independent external content. The author self-citations (Grelaud et al. 2009; Pudlo et al. 2016; Marin et al. 2018) are contextual and are not load-bearing for the proposed estimator, so they do not raise the circularity score. The one serious derivation defect is in Section 2, Eq. (2): the factor for R_ell is written with theta_i^(ell-1) drawn from bpi^(ell-1), but Algorithm 1 generates theta_i^(ell-1) from bpi^(ell-2); consequently the product does not telescope to C_L as written and the central SIS claim is currently unsubstantiated. This is an indexing/correctness flaw, not a reduction of a prediction to its inputs, so it is not a circularity. The replacement of the true evidence by C_L = integral q^(L) pi also rests on the surrogate-error assumption acknowledged in Section 7; that is an approximation risk, not a self-definitional step.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard Monte Carlo theory plus the strong domain assumption that the SNLE surrogate likelihood is accurate over the entire prior support. The temperature and network architectures are hand-chosen free parameters. No new physical or model entities are introduced.

free parameters (4)

IS temperature T = 1.25 (default), tested 1.0 and 2.0
Variance dilation of the normalizing flow base distribution to give heavier tails; chosen by hand and not derived from first principles.
HM temperature T = 0.8 (default), tested 0.5 and 1.0
Variance contraction for the learned harmonic mean proposal; chosen by hand.
Flow and SNLE architectures = MAF with 5 transforms and 2 hidden layers of 64 for SNLE; MAF with 3 transforms and 2 hidden layers of 32 for IS…
Hand-selected network sizes; no sensitivity analysis for the architecture choices.
Number of rounds and simulations = L=5 rounds, N=1,000 per round; N_IS=1,000; HM N'=2,000
Experimental setup settings, not fitted to data, but they directly affect estimator accuracy and computational cost.

assumptions (4)

standard math Importance sampling is unbiased when the proposal support contains the target support.
Used in justifying the IS estimator (Eq 3) and the SIS ratio estimator (Eq 2).
standard math The telescoping product of normalizing constants is valid when the intermediate distributions are known up to a normalizing constant.
Underpins the SIS estimator C_L = product of C_ell/C_{ell-1} in Section 2.
domain assumption The SNLE surrogate likelihood q^(ell)(x*|theta) converges to the true likelihood f(x*|theta) on the support of the prior as ell and N grow.
Implicit throughout; the paper equates C_L with the true evidence without quantifying surrogate error.
domain assumption The normalizing flow h_T(theta) has sufficiently heavy tails so that q^(L)(x*|theta) pi(theta) / h_T(theta) has finite variance.
Required for the IS estimator to be stable; enforced heuristically by temperature T but not proven.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Estimating Marginal Likelihoods in Likelihood-Free Inference via Neural Density Estimation." pith.science (2026). https://pith.science/paper/WG2D72WO

@misc{pith2026250708734,
  author       = {Pith},
  title        = {Pith review of: Estimating Marginal Likelihoods in Likelihood-Free Inference via Neural Density Estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WG2D72WO}},
  note         = {Machine review of arXiv:2507.08734}
}

read the original abstract

The marginal likelihood, or evidence, plays a central role in Bayesian model selection, yet remains notoriously challenging to compute in likelihood-free settings. While Simulation-Based Inference (SBI) techniques such as Sequential Neural Likelihood Estimation (SNLE) offer powerful tools to approximate posteriors using neural density estimators, they typically do not provide estimates of the evidence. In this technical report presented at BayesComp 2025, we present a simple and general methodology to estimate the marginal likelihood using the output of SNLE.

Figures

Figures reproduced from arXiv: 2507.08734 by the authors.

**Figure 2.** Estimated log-marginal likelihoods with HM and varying temperature. Each panel corresponds [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Estimated log-marginal likelihoods with IS and varying temperature. Each panel corresponds to [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

25 extracted references · 18 canonical work pages

[1]

Boelts, M

J. Boelts, M. Deistler, M. Glöckler, Á. Tejero-Cantero, J.-M. Lueckmann, G. Moss, P. Steinbach, T. Moreau, F. Muratore, J. Linhart, C. Durkan, J. Vetter, B. K. Miller, M. Herold, A. Ziaeemehr, M. Pals, T. Gruner, S. Bischoff, N. Krouglova, R. Gao, J. K. Lappalainen, B. Mucsányi, F. Pei, A. Schulz, Z. Stefanidi, P. Rodrigues, C. Schröder, F. Abu Zaid, J. B...

2025
[2]

The frontier of simulation-based inference

Kyle Cranmer, Johann Brehmer, and Gilles Louppe. The frontier of simulation-based inference. Proceedings of the National Academy of Sciences, 117 0 (48): 0 30055--30062, 2020. doi:10.1073/pnas.1912789117

doi:10.1073/pnas.1912789117 2020
[3]

Density estimation using Real NVP

Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using Real NVP . February 2017. URL https://openreview.net/forum?id=HkpbnH9lx

work page 2017
[4]

David Greenberg, Marcel Nonnenmacher, and Jakob H. Macke. Automatic posterior transformation for likelihood-free inference. In ICML, 2019

work page 2019
[5]

Grelaud, J.-M

A. Grelaud, J.-M. Marin, C. P. Robert, F. Rodolphe, and F. Taly. Likelihood-free methods for model choice in gibbs random fields. Bayesian Analysis, 3 0 (2): 0 427--442, 2009. doi:10.1214/09-BA412

doi:10.1214/09-ba412 2009
[6]

Likelihood-free mcmc with amortized approximate likelihood ratios

Joeri Hermans, Vincent Begy, and Gilles Louppe. Likelihood-free mcmc with amortized approximate likelihood ratios. In ICML, 2020

work page 2020
[7]

Jan-Matthis Lueckmann, Jan Boelts, David Greenberg, Pedro Gonçalves, and Jakob H. Macke. Benchmarking simulation-based inference. In AISTATS, 2021

work page 2021
[8]

Marin, P

J.-M. Marin, P. Pudlo, A. Estoup, and C.P. Robert. Likelihood-free model choice. In S. A. Sisson, Y. Fan, and M. Beaumont, editors, Handbook of Approximate Bayesian Computation, chapter 6. Chapman & Hall/CRC, 2018

work page 2018

Show all 25 references

[9]

McEwen, Christopher G

Jason D. McEwen, Christopher G. R. Wallis, Matthew A. Price, and Alessio Spurio Mancini. Machine learning assisted bayesian model comparison: learnt harmonic mean estimator. November 2021. doi:10.48550/ARXIV.2111.12720. URL http://arxiv.org/abs/2111.12720. arXiv:2111.12720 [stat]

2021 doi arXiv
[10]

Sequential monte carlo samplers

Pierre Del Moral, Arnaud Doucet, and Ajay Jasra. Sequential monte carlo samplers. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68 0 (3): 0 411--436, 2006. doi:10.1111/j.1467-9868.2006.00553.x

2006
[11]

Newton and Adrian E

Michael A. Newton and Adrian E. Raftery. Approximate bayesian inference with the weighted likelihood bootstrap. Journal of the Royal Statistical Society: Series B (Methodological), 56 0 (1): 0 3--26, 1994

1994
[12]

Art B. Owen. Monte carlo theory, methods and examples. 2013. Book manuscript, available at https://statweb.stanford.edu/ owen/mc/

2013
[13]

Fast -free inference of simulation models with bayesian conditional density estimation

George Papamakarios and Iain Murray. Fast -free inference of simulation models with bayesian conditional density estimation. NeurIPS, 2016

2016
[14]

Masked Autoregressive Flow for Density Estimation

George Papamakarios, Theo Pavlakou, and Iain Murray. Masked Autoregressive Flow for Density Estimation . In Advances in Neural Information Processing Systems , volume 30. Curran Associates, Inc., 2017

2017
[15]

Sequential neural likelihood: Fast likelihood-free inference with autoregressive flows

George Papamakarios, David Sterratt, and Iain Murray. Sequential neural likelihood: Fast likelihood-free inference with autoregressive flows. In AISTATS, 2019

2019
[16]

Normalizing flows for probabilistic modeling and inference

George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshminarayanan. Normalizing flows for probabilistic modeling and inference. Journal of Machine Learning Research, 22 0 (57): 0 1--64, 2021. URL http://jmlr.org/papers/v22/19-1028.html

2021
[17]

Price, Davide Piras, Alessio Spurio Mancini, and Jason D

Alicja Polanska, Matthew A. Price, Davide Piras, Alessio Spurio Mancini, and Jason D. McEwen. Learned harmonic mean estimation of the bayesian evidence with normalizing flows. 2024. doi:10.48550/ARXIV.2405.05969. URL https://arxiv.org/pdf/2405.05969

2024 doi
[18]

Pritchard, M.T

Jonathan K. Pritchard, M.T. Seielstad, A. Perez-Lezaun, and M.W. Feldman. Population growth of human y chromosomes: a study of y chromosome microsatellites. Molecular biology and evolution, 16 0 (12): 0 1791--1798, 1999

1999
[19]

Pierre Pudlo, Jean-Michel Marin, Arnaud Estoup, Jean-Marie Cornuet, Mathieu Gautier, and Christian P. Robert. Reliable abc model choice via random forests. Bioinformatics, 32 0 (6): 0 859--866, 2016

2016
[20]

Robert and George Casella

Christian P. Robert and George Casella. Monte Carlo Statistical Methods. Springer, 2nd edition, 2004. ISBN 9780387212395

2004
[21]

Zuko : Normalizing flows in pytorch, 2022

François Rozet et al. Zuko : Normalizing flows in pytorch, 2022. URL https://pypi.org/project/zuko

2022
[22]

Sara Sheehan and Yun S. Song. Deep learning for population genetic inference. PLOS Computational Biology, 12 0 (3): 0 e1004845, 2016

2016
[23]

Spurio Mancini, M.M

A. Spurio Mancini, M.M. Docherty, M.A. Price, and J.D. McEwen. Bayesian model comparison for simulation-based inference. RAS Techniques and Instruments, 2 0 (1): 0 710--722, 2023. doi:10.1093/rasti/rzad051

2023 doi
[24]

Balding, Robert C

Simon Tavare, David J. Balding, Robert C. Griffiths, and Peter Donnelly. Inferring coalescence times from dna sequence data. Genetics, 145 0 (2): 0 505--518, 1997

1997
[25]

Lewis, Yu Fan, Lynn Kuo, and Ming-Hui Chen

Wangang Xie, Paul O. Lewis, Yu Fan, Lynn Kuo, and Ming-Hui Chen. Improving marginal likelihood estimation for bayesian phylogenetic model selection. Systematic Biology, 60 0 (2): 0 150--160, 2011. doi:10.1093/sysbio/syq085

2011 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Boelts, M

J. Boelts, M. Deistler, M. Glöckler, Á. Tejero-Cantero, J.-M. Lueckmann, G. Moss, P. Steinbach, T. Moreau, F. Muratore, J. Linhart, C. Durkan, J. Vetter, B. K. Miller, M. Herold, A. Ziaeemehr, M. Pals, T. Gruner, S. Bischoff, N. Krouglova, R. Gao, J. K. Lappalainen, B. Mucsányi, F. Pei, A. Schulz, Z. Stefanidi, P. Rodrigues, C. Schröder, F. Abu Zaid, J. B...

2025

[2] [2]

The frontier of simulation-based inference

Kyle Cranmer, Johann Brehmer, and Gilles Louppe. The frontier of simulation-based inference. Proceedings of the National Academy of Sciences, 117 0 (48): 0 30055--30062, 2020. doi:10.1073/pnas.1912789117

doi:10.1073/pnas.1912789117 2020

[3] [3]

Density estimation using Real NVP

Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using Real NVP . February 2017. URL https://openreview.net/forum?id=HkpbnH9lx

work page 2017

[4] [4]

David Greenberg, Marcel Nonnenmacher, and Jakob H. Macke. Automatic posterior transformation for likelihood-free inference. In ICML, 2019

work page 2019

[5] [5]

Grelaud, J.-M

A. Grelaud, J.-M. Marin, C. P. Robert, F. Rodolphe, and F. Taly. Likelihood-free methods for model choice in gibbs random fields. Bayesian Analysis, 3 0 (2): 0 427--442, 2009. doi:10.1214/09-BA412

doi:10.1214/09-ba412 2009

[6] [6]

Likelihood-free mcmc with amortized approximate likelihood ratios

Joeri Hermans, Vincent Begy, and Gilles Louppe. Likelihood-free mcmc with amortized approximate likelihood ratios. In ICML, 2020

work page 2020

[7] [7]

Jan-Matthis Lueckmann, Jan Boelts, David Greenberg, Pedro Gonçalves, and Jakob H. Macke. Benchmarking simulation-based inference. In AISTATS, 2021

work page 2021

[8] [8]

Marin, P

J.-M. Marin, P. Pudlo, A. Estoup, and C.P. Robert. Likelihood-free model choice. In S. A. Sisson, Y. Fan, and M. Beaumont, editors, Handbook of Approximate Bayesian Computation, chapter 6. Chapman & Hall/CRC, 2018

work page 2018