Field-level vs summaries: convergence of information in non-Gaussian density fields

Beatriz Tucci; Fabian Schmidt; Ivana Nikolac

arxiv: 2606.18227 · v1 · pith:ZGDAN4ADnew · submitted 2026-06-16 · 🌌 astro-ph.CO

Field-level vs summaries: convergence of information in non-Gaussian density fields

Ivana Nikolac , Fabian Schmidt , Beatriz Tucci This is my paper

Pith reviewed 2026-06-26 23:15 UTC · model grok-4.3

classification 🌌 astro-ph.CO

keywords field-level inferencesummary statisticsnon-Gaussian fieldscosmological parameterspower spectrumbispectruminformation content

0 comments

The pith

Field-level inference captures more information than fixed summaries in non-Gaussian cosmological fields as nonlinearity increases.

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

The paper compares field-level inference to summary statistics in a controlled model of weakly non-Gaussian density fields. It finds that as the nonlinear coupling strength grows, uncertainties from power spectrum plus bispectrum and similar summaries increase faster than those from field-level methods, causing growing information loss. This loss is only partly recovered by adding composite operator correlators up to the six-point function, and the gap widens in lower-noise settings. A reader would care because cosmological surveys increasingly use field-level approaches to extract maximal information from data.

Core claim

In the Gaussian limit the power spectrum plus bispectrum, composite operator correlators and field-level inference yield equivalent constraints. As the nonlinear coupling λ increases, the summary-based uncertainties on the model parameters grow faster than the FLI ones, leading to an increasing information loss for a fixed set of summaries. This loss is largely but not completely recovered by adding OCs corresponding to up to the 6-point function, and becomes even more pronounced for lower-noise data.

What carries the argument

The minimal nonlinear forward model consisting of a linear density field with a single local quadratic coupling λ and Gaussian noise, used to compare MCMC-based field-level inference against simulation-based inference and Fisher forecasts on summaries.

If this is right

In the Gaussian limit (λ=0), P+B, OCs and FLI give equivalent constraints.
Summary uncertainties grow faster than FLI uncertainties with increasing λ.
Adding OCs up to the 6-point function recovers most but not all of the information.
The information loss is more pronounced for lower-noise data.

Where Pith is reading between the lines

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

Realistic cosmological analyses with low noise levels may require field-level methods to avoid significant information loss.
Future work could explore whether summaries beyond the 6-point function can close the gap with FLI in more complex models.
Similar comparisons could be performed in simulations with additional nonlinear effects like higher-order couplings.

Load-bearing premise

The forward model is a linear density field with a single local quadratic coupling λ and Gaussian noise.

What would settle it

Perform the MCMC field-level inference and the summary-based inference on simulated datasets with varying λ and noise levels, and check whether the ratio of summary to FLI uncertainties increases with λ as predicted.

read the original abstract

We elucidate the sources of information gain in weakly non-Gaussian cosmological fields at the field- vs. summary-statistic-level in a controlled setting. Specifically, we compare field-level inference (FLI) with the standard power spectrum plus bispectrum (P${+}$B), and a family of composite-operator correlators (OCs) built from auto- and cross-spectra of local powers of the galaxy density field. The forward model is a linear density field with a single local quadratic coupling $\lambda$ and Gaussian noise; this minimal nonlinear setup interpolates between a purely Gaussian dataset ($\lambda=0$) and a non-Gaussian one ($\lambda\sim 1$), while keeping the analytical structure tractable. FLI is performed by jointly sampling the initial conditions, bias and noise parameters via MCMC; the summary posteriors are obtained with simulation-based inference (SBI) as well as Fisher estimates. In the Gaussian limit, the P${+}$B, OCs and FLI yield equivalent constraints, in agreement with the perturbative expectation. As the nonlinear coupling $\lambda$ increases, the summary-based uncertainties on the model parameters grow faster than the FLI ones, leading to an increasing information loss for a fixed set of summaries. This loss is largely, but not completely, recovered by adding OCs corresponding to up to the 6-point function. The information loss over FLI becomes even more pronounced for lower-noise data, where summaries corresponding to up to the 6-point function still capture significantly less information than the field.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

In this toy model, field-level inference keeps tighter constraints than summaries as the quadratic coupling grows, with higher-order operators closing most but not all of the gap.

read the letter

The main thing to know is that this paper finds, inside a deliberately simple forward model, that field-level inference maintains better parameter constraints than summary statistics as the nonlinear coupling strength increases, with the gap growing at lower noise levels. Adding composite operators up to sixth order recovers most but not all of the lost information.

What works well is the controlled setup. They take a linear density field, add one local quadratic term with strength lambda, and Gaussian noise. This lets them go from Gaussian (lambda=0) to non-Gaussian (lambda around 1) while keeping everything solvable. They run joint MCMC on initial conditions, bias, and noise for the field-level case, and use simulation-based inference plus Fisher matrix for the power spectrum plus bispectrum and the operator correlators. The fact that all methods agree when lambda is zero is a solid sanity check. The observed trend—that summaries degrade faster—is a concrete result that can be checked by others.

The soft spot is the extreme simplicity of the model. With only one free parameter lambda and no other bias terms or complex nonlinear evolution, the quantitative information loss numbers are tied to this specific case. It is not obvious how much of the gap would persist in a more realistic simulation with multiple fields or different noise properties. The abstract does not give the actual numerical factors, so the practical size of the effect is hard to judge without the figures.

This kind of study is useful for anyone comparing analysis pipelines for future cosmological data sets where non-Gaussian signals will matter. It gives a clear example of the information trade-off. I would bring it to a reading group for discussion of the method comparison. It deserves peer review because the logic is internally consistent and the experiment is reproducible within the model.

Referee Report

0 major / 1 minor

Summary. The paper conducts a controlled comparison of field-level inference (FLI) via MCMC against summary-statistic approaches (P+B and composite operator correlators up to 6-point) using SBI and Fisher methods in a minimal forward model: linear density field with local quadratic coupling λ and Gaussian noise. It reports that constraints from summaries degrade faster than FLI as λ increases from the Gaussian limit, with only partial recovery from higher-order summaries, and the information loss is more severe at lower noise levels.

Significance. If the quantitative results hold, this work supplies a clean benchmark for information loss in fixed summaries versus FLI as nonlinearity grows, within an exactly solvable minimal model that isolates the effect without extraneous cosmological complications. The use of standard inference tools (MCMC, SBI, Fisher) on an explicitly defined forward model strengthens the internal consistency of the comparison.

minor comments (1)

[Abstract] Abstract: the description of the OCs as 'built from auto- and cross-spectra of local powers of the galaxy density field' would benefit from an explicit list or equation defining the operators (e.g., which powers and which cross terms) to allow immediate reproduction of the 6-point set.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the work, the clear summary of its scope, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper conducts a controlled numerical comparison of field-level MCMC inference against summary-statistic posteriors (via SBI and Fisher) inside an explicitly constructed minimal forward model (linear field + local quadratic coupling λ + Gaussian noise). All reported trends, including the Gaussian-limit equivalence and the differential information loss with rising λ, are direct outputs of applying standard inference algorithms to data generated from this model; no derivation step reduces a claimed prediction to a fitted input by construction, and no load-bearing self-citation or ansatz is invoked. The setup is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central comparison rests on the choice of a minimal forward model whose only free parameter is the quadratic coupling strength.

free parameters (1)

λ
Nonlinear coupling strength that is varied to control the degree of non-Gaussianity in the controlled setup.

axioms (1)

domain assumption Forward model consists of a linear density field plus single local quadratic coupling and Gaussian noise
Chosen to keep analytical structure tractable while allowing interpolation between Gaussian and non-Gaussian regimes.

pith-pipeline@v0.9.1-grok · 5806 in / 1227 out tokens · 32319 ms · 2026-06-26T23:15:06.581567+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 2 canonical work pages

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M.M. Ivanov, O.H.E. Philcox, T. Nishimichi, M. Simonović, M. Takada and M. Zaldarriaga, Precision analysis of the redshift-space galaxy bispectrum,PhRvD105(2022) 063512 [2110.10161]

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Ivanov, O.H.E

M.M. Ivanov, O.H.E. Philcox, G. Cabass, T. Nishimichi, M. Simonović and M. Zaldarriaga, Cosmology with the galaxy bispectrum multipoles: Optimal estimation and application to BOSS data,PhRvD107(2023) 083515 [2302.04414]

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Philcox and M.M

O.H.E. Philcox and M.M. Ivanov,BOSS DR12 full-shape cosmology:ΛCDM constraints from the large-scale galaxy power spectrum and bispectrum monopole,PhRvD105(2022) 043517 [2112.04515]

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T. Bakx, M.M. Ivanov, O.H.E. Philcox and Z. Vlah,One-Loop Galaxy Bispectrum: Consistent Theory, Efficient Analysis with COBRA, and Implications for Cosmological Parameters,arXiv e-prints(2025) arXiv:2507.22110 [2507.22110]

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A. Chudaykin, M.M. Ivanov and O.H.E. Philcox,Reanalyzing DESI DR1: 1.ΛCDM Constraints from the Power Spectrum and Bispectrum,arXiv e-prints(2025) arXiv:2507.13433 [2507.13433]

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N.-M. Nguyen, F. Schmidt, B. Tucci, M. Reinecke and A. Kostić,How much information can be extracted from galaxy clustering at the field level?,arXiv e-prints(2024) arXiv:2403.03220 [2403.03220]

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Pith/arXiv arXiv 2026
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M. Schmittfull, T. Baldauf and U. Seljak,Near optimal bispectrum estimators for large-scale structure,Physical Review D91(2015)

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Schmittfull and A

M. Schmittfull and A. Moradinezhad Dizgah,Galaxy skew-spectra in redshift-space,JCAP03 (2021) 020 [2010.14267]

arXiv 2021
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Dizgah, H

A.M. Dizgah, H. Lee, M. Schmittfull and C. Dvorkin,Capturing non-gaussianity of the large-scale structure with weighted skew-spectra,Journal of Cosmology and Astroparticle Physics2020(2020) 011-011

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Schmidt, F

F. Schmidt, F. Elsner, J. Jasche, N.M. Nguyen and G. Lavaux,A rigorous eft-based forward model for large-scale structure,Journal of Cosmology and Astroparticle Physics2019(2019) 042-042

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Rubira and F

H. Rubira and F. Schmidt,Non-Gaussian Galaxy Stochasticity and the Noise-Field Formulation,2511.05484

arXiv
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Cabass and F

G. Cabass and F. Schmidt,The eft likelihood for large-scale structure,Journal of Cosmology and Astroparticle Physics2020(2020) 042-042

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arXiv 2021
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Gao, A.M

Z. Gao, A.M. Dizgah and Z. Vlah,Parity in composite-field galaxy correlators, 2025

2025
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work page doi:10.1201/b10905 2011
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Tucci and F

B. Tucci and F. Schmidt,Eftoflss meets simulation-based inference:σ8 from biased tracers, Journal of Cosmology and Astroparticle Physics2024(2024) 063

2024
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Greenberg, M

D.S. Greenberg, M. Nonnenmacher and J.H. Macke,Automatic Posterior Transformation for Likelihood-Free Inference,arXiv e-prints(2019) arXiv:1905.07488 [1905.07488]

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A. Tejero-Cantero, J. Boelts, M. Deistler, J.-M. Lueckmann, C. Durkan, P. Gonçalves et al., sbi: A toolkit for simulation-based inference,The Journal of Open Source Software5(2020) 2505 [2007.09114]

arXiv 2020
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G. Papamakarios, T. Pavlakou and I. Murray,Masked Autoregressive Flow for Density Estimation,arXiv e-prints(2017) arXiv:1705.07057 [1705.07057]

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Andrews, J

A. Andrews, J. Jasche, G. Lavaux and F. Schmidt,Bayesian field-level inference of primordial non-Gaussianity using next-generation galaxy surveys,MNRAS520(2023) 5746 [2203.08838]

arXiv 2023
[34]

Sokal,Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms, NATO Sci

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A. Gelman and D.B. Rubin,Inference from Iterative Simulation Using Multiple Sequences, Statistical Science7(1992) 457 . – 33 –

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T. Akiba, S. Sano, T. Yanase, T. Ohta and M. Koyama,Optuna: A next-generation hyperparameter optimization framework,1907.10902

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Desjacques, D

V. Desjacques, D. Jeong and F. Schmidt,Large-scale galaxy bias,PhysRep733(2018) 1 [1611.09787]

Pith/arXiv arXiv 2018

[2] [2]

Desjacques, D

V. Desjacques, D. Jeong and F. Schmidt,The Galaxy Power Spectrum and Bispectrum in Redshift Space,JCAP12(2018) 035 [1806.04015]

Pith/arXiv arXiv 2018

[3] [3]

Matarrese, L

S. Matarrese, L. Verde and A.F. Heavens,Large-scale bias in the Universe: bispectrum method, MNRAS290(1997) 651 [astro-ph/9706059]

Pith/arXiv arXiv 1997

[4] [4]

Scoccimarro, S

R. Scoccimarro, S. Colombi, J.N. Fry, J.A. Frieman, E. Hivon and A. Melott,Nonlinear Evolution of the Bispectrum of Cosmological Perturbations,ApJ496(1998) 586 [astro-ph/9704075]

Pith/arXiv arXiv 1998

[5] [5]

Sefusatti, M

E. Sefusatti, M. Crocce, S. Pueblas and R. Scoccimarro,Cosmology and the bispectrum,PhRvD 74(2006) 023522 [astro-ph/0604505]

Pith/arXiv arXiv 2006

[6] [6]

Ivanov, O.H.E

M.M. Ivanov, O.H.E. Philcox, T. Nishimichi, M. Simonović, M. Takada and M. Zaldarriaga, Precision analysis of the redshift-space galaxy bispectrum,PhRvD105(2022) 063512 [2110.10161]

arXiv 2022

[7] [7]

Ivanov, O.H.E

M.M. Ivanov, O.H.E. Philcox, G. Cabass, T. Nishimichi, M. Simonović and M. Zaldarriaga, Cosmology with the galaxy bispectrum multipoles: Optimal estimation and application to BOSS data,PhRvD107(2023) 083515 [2302.04414]

arXiv 2023

[8] [8]

Philcox and M.M

O.H.E. Philcox and M.M. Ivanov,BOSS DR12 full-shape cosmology:ΛCDM constraints from the large-scale galaxy power spectrum and bispectrum monopole,PhRvD105(2022) 043517 [2112.04515]

arXiv 2022

[9] [9]

D’Amico, Y

G. D’Amico, Y. Donath, M. Lewandowski, L. Senatore and P. Zhang,The BOSS bispectrum analysis at one loop from the Effective Field Theory of Large-Scale Structure,arXiv e-prints (2022) arXiv:2206.08327 [2206.08327]

arXiv 2022

[10] [10]

Philcox, M.M

O.H.E. Philcox, M.M. Ivanov, G. Cabass, M. Simonović, M. Zaldarriaga and T. Nishimichi, Cosmology with the redshift-space galaxy bispectrum monopole at one-loop order,PhRvD106 (2022) 043530 [2206.02800]

arXiv 2022

[11] [11]

Bakx, M.M

T. Bakx, M.M. Ivanov, O.H.E. Philcox and Z. Vlah,One-Loop Galaxy Bispectrum: Consistent Theory, Efficient Analysis with COBRA, and Implications for Cosmological Parameters,arXiv e-prints(2025) arXiv:2507.22110 [2507.22110]

arXiv 2025

[12] [12]

Chudaykin, M.M

A. Chudaykin, M.M. Ivanov and O.H.E. Philcox,Reanalyzing DESI DR1: 1.ΛCDM Constraints from the Power Spectrum and Bispectrum,arXiv e-prints(2025) arXiv:2507.13433 [2507.13433]

arXiv 2025

[13] [13]

Nguyen, F

N.-M. Nguyen, F. Schmidt, B. Tucci, M. Reinecke and A. Kostić,How much information can be extracted from galaxy clustering at the field level?,arXiv e-prints(2024) arXiv:2403.03220 [2403.03220]

arXiv 2024

[14] [14]

Krause, Y

2pt Collaboration, E. Krause, Y. Kobayashi, A.N. Salcedo, M.M. Ivanov, T. Abel et al.,A parameter-masked mock data challenge for beyond-two-point galaxy clustering statistics, 2024. – 32 –

2024

[15] [15]

Akitsu, M

K. Akitsu, M. Simonović, S.-F. Chen, G. Cabass and M. Zaldarriaga,Cosmology inference with perturbative forward modeling at the field level: a comparison with joint power spectrum and bispectrum analyses,arXiv e-prints(2025) arXiv:2509.09673 [2509.09673]

arXiv 2025

[16] [16]

Cabass, M

G. Cabass, M. Simonović and M. Zaldarriaga,Cosmological information in perturbative forward modeling,2307.04706

arXiv

[17] [17]

Schmidt,On the connection between field-level inference andn-point correlation functions, 2504.15351

F. Schmidt,On the connection between field-level inference andn-point correlation functions, 2504.15351

arXiv

[18] [18]

Pietroni and F

M. Pietroni and F. Schmidt,On the Relation Between Field-Level Posteriors, Correlators, and their Likelihoods,arXiv e-prints(2026) arXiv:2604.25385 [2604.25385]

Pith/arXiv arXiv 2026

[19] [19]

Schmittfull, T

M. Schmittfull, T. Baldauf and U. Seljak,Near optimal bispectrum estimators for large-scale structure,Physical Review D91(2015)

2015

[20] [20]

Schmittfull and A

M. Schmittfull and A. Moradinezhad Dizgah,Galaxy skew-spectra in redshift-space,JCAP03 (2021) 020 [2010.14267]

arXiv 2021

[21] [21]

Dizgah, H

A.M. Dizgah, H. Lee, M. Schmittfull and C. Dvorkin,Capturing non-gaussianity of the large-scale structure with weighted skew-spectra,Journal of Cosmology and Astroparticle Physics2020(2020) 011-011

2020

[22] [22]

Schmidt, F

F. Schmidt, F. Elsner, J. Jasche, N.M. Nguyen and G. Lavaux,A rigorous eft-based forward model for large-scale structure,Journal of Cosmology and Astroparticle Physics2019(2019) 042-042

2019

[23] [23]

Rubira and F

H. Rubira and F. Schmidt,Non-Gaussian Galaxy Stochasticity and the Noise-Field Formulation,2511.05484

arXiv

[24] [24]

Cabass and F

G. Cabass and F. Schmidt,The eft likelihood for large-scale structure,Journal of Cosmology and Astroparticle Physics2020(2020) 042-042

2020

[25] [25]

Schmidt,Ann-th order Lagrangian Forward Model for Large-Scale Structure,JCAP04 (2021) 033 [2012.09837]

F. Schmidt,Ann-th order Lagrangian Forward Model for Large-Scale Structure,JCAP04 (2021) 033 [2012.09837]

arXiv 2021

[26] [26]

Gao, A.M

Z. Gao, A.M. Dizgah and Z. Vlah,Parity in composite-field galaxy correlators, 2025

2025

[27] [27]

Handbook of Markov Chain Monte Carlo , year = 2011, pages =

S. Brooks, A. Gelman, G. Jones and X.-L. Meng,Handbook of Markov Chain Monte Carlo, Chapman and Hall/CRC (May, 2011), 10.1201/b10905

work page doi:10.1201/b10905 2011

[28] [28]

Tucci and F

B. Tucci and F. Schmidt,Eftoflss meets simulation-based inference:σ8 from biased tracers, Journal of Cosmology and Astroparticle Physics2024(2024) 063

2024

[29] [29]

Greenberg, M

D.S. Greenberg, M. Nonnenmacher and J.H. Macke,Automatic Posterior Transformation for Likelihood-Free Inference,arXiv e-prints(2019) arXiv:1905.07488 [1905.07488]

Pith/arXiv arXiv 2019

[30] [30]

Tejero-Cantero, J

A. Tejero-Cantero, J. Boelts, M. Deistler, J.-M. Lueckmann, C. Durkan, P. Gonçalves et al., sbi: A toolkit for simulation-based inference,The Journal of Open Source Software5(2020) 2505 [2007.09114]

arXiv 2020

[31] [31]

Papamakarios, T

G. Papamakarios, T. Pavlakou and I. Murray,Masked Autoregressive Flow for Density Estimation,arXiv e-prints(2017) arXiv:1705.07057 [1705.07057]

Pith/arXiv arXiv 2017

[32] [32]

doi:10.1016/C2017-0-01943-2 , author =

S. Dodelson and F. Schmidt,Modern Cosmology, Academic Press (2020), 10.1016/C2017-0-01943-2

work page doi:10.1016/c2017-0-01943-2 2020

[33] [33]

Andrews, J

A. Andrews, J. Jasche, G. Lavaux and F. Schmidt,Bayesian field-level inference of primordial non-Gaussianity using next-generation galaxy surveys,MNRAS520(2023) 5746 [2203.08838]

arXiv 2023

[34] [34]

Sokal,Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms, NATO Sci

A. Sokal,Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms, NATO Sci. Ser. B361(1997) 131

1997

[35] [35]

Gelman and D.B

A. Gelman and D.B. Rubin,Inference from Iterative Simulation Using Multiple Sequences, Statistical Science7(1992) 457 . – 33 –

1992

[36] [36]

Akiba, S

T. Akiba, S. Sano, T. Yanase, T. Ohta and M. Koyama,Optuna: A next-generation hyperparameter optimization framework,1907.10902

Pith/arXiv arXiv 1907

[37] [37]

Durkan, A

C. Durkan, A. Bekasov, I. Murray and G. Papamakarios,Neural spline flows,1906.04032

arXiv 1906

[38] [38]

Kingma and J

D.P. Kingma and J. Ba,Adam: A Method for Stochastic Optimization,arXiv e-prints(2014) arXiv:1412.6980 [1412.6980]. – 34 –

Pith/arXiv arXiv 2014