arxiv: 2604.25385 · v1 · submitted 2026-04-28 · 🌌 astro-ph.CO

Recognition: unknown

On the Relation Between Field-Level Posteriors, Correlators, and their Likelihoods

Massimo Pietroni , Fabian Schmidt

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:58 UTC · model grok-4.3

classification 🌌 astro-ph.CO

keywords cosmologylarge-scale structurefield-level inferenceposteriorscorrelatorslikelihoodsinformation compressionBAO reconstruction

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The pith

A field-level posterior for cosmological data, when expanded around its Gaussian limit, reorganizes all information into contributions from the connected correlators of the evolved field.

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

The paper constructs a field-level posterior by marginalizing over initial conditions and noise in a general forward model for large-scale structure observations. Expanding this posterior around the Gaussian limit produces a Fisher matrix whose terms are grouped by the connected correlators of the final field. The resulting expressions recover the familiar Gaussian-covariance formulas for power-spectrum and bispectrum likelihoods while making explicit which additional pieces of information are lost when data are compressed into finite sets of summary statistics. The same construction shows how cross-covariances between summaries can successively recover more of the full field-level content and naturally yields the optimal reconstruction of the initial conditions.

Core claim

We develop a field-level posterior for cosmological data by marginalizing over initial conditions and noise in a general forward model. Expanding the FLP around its Gaussian limit, we derive a general expression for the Fisher matrix and reorganize the field-level information into contributions associated with the connected correlators of the evolved field. This makes explicit which terms are captured by likelihood analyses based on the power spectrum, the bispectrum, or finite sets of summary statistics, and which are lost under compression.

What carries the argument

The field-level posterior (FLP) obtained by marginalizing initial conditions and noise, then expanded around its Gaussian limit to express Fisher information as sums over connected correlators of the evolved field.

Load-bearing premise

The marginalization over initial conditions and noise in a general forward model is feasible and the expansion around the Gaussian limit remains valid for the weakly non-Gaussian observables of interest.

What would settle it

Compute the full field-level Fisher matrix on a set of simulated density fields and compare it with the Fisher matrix obtained from a power-spectrum-only likelihood on the identical fields; any systematic difference should match the higher-order connected-correlator terms predicted by the expansion.

read the original abstract

We develop a field-level posterior for cosmological data by marginalizing over initial conditions and noise in a general forward model. While our focus is on large-scale structure data, the results generalize to any weakly non-Gaussian observable. Moreover, the construction is non-perturbative with respect to the forward model and applies equally well to perturbative calculations, simulation-based predictions, and more general effective descriptions. Expanding the FLP around its Gaussian limit, we derive a general expression for the Fisher matrix and reorganize the field-level information into contributions associated with the connected correlators of the evolved field. This makes explicit which terms are captured by likelihood analyses based on the power spectrum, the bispectrum, or finite sets of summary statistics, and which are lost under compression. We recover the standard Gaussian-covariance result for the power spectrum, show that the Gaussian bispectrum likelihood reproduces the corresponding field-level contribution, and show how cross-covariances among summaries progressively reconstruct more of the full field-level information. As an application to the BAO scale, we show how the field contains all the information required for its optimal reconstruction in the presence of noise, and identify the contributions in the FLP needed to attain this limit. We also show that the reconstruction of the initial field arises naturally as a byproduct of our approach, yielding the optimal estimate of the initial conditions given the data and the noise. Our results provide a unified framework to compare field-level and correlator-based inference, to quantify the information loss induced by compression, and to explore the role of stochasticity.

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

This paper gives a formal reorganization of field-level posterior information into connected-correlator pieces, recovering the usual Gaussian power-spectrum and bispectrum results without obvious internal gaps.

read the letter

The main thing to know is that this work expands the field-level posterior around its Gaussian limit and shows how the Fisher information splits into contributions from the connected correlators of the evolved field. That split makes explicit what a power-spectrum analysis captures, what the bispectrum adds, and what gets lost when you compress to finite summaries. It also recovers the standard Gaussian-covariance Fisher matrix for the power spectrum and shows that the Gaussian bispectrum likelihood exactly matches the corresponding field-level term. The BAO reconstruction example then illustrates that the full field contains all the information needed for optimal reconstruction in noise, with the initial conditions emerging as a byproduct of the same marginalization step. The construction stays non-perturbative in the forward model, which is a plus for people who want to use simulations or effective descriptions rather than pure perturbation theory. The marginalization over initial conditions and noise is treated as a general formal step whose practical feasibility is assumed in the weakly non-Gaussian regime. That assumption is reasonable for the paper's scope but will still need concrete approximations or sampling methods before anyone can run it on real data. No hidden contradictions appear in the reorganization itself, and the recovery of known limits provides a useful check. This is for cosmologists already working on field-level versus summary-statistic comparisons in large-scale structure. A reader comfortable with Fisher matrices and likelihoods for power spectra and bispectra will follow the steps and see the information-budget accounting clearly. I would send it to peer review. The formal mapping is new enough and the checks against standard results are solid enough that referees can usefully test the details and the BAO application.

Referee Report

2 major / 2 minor

Summary. The paper develops a field-level posterior (FLP) for cosmological data by marginalizing over initial conditions and noise in a general forward model. Expanding the FLP around its Gaussian limit yields a general Fisher matrix expression that reorganizes field-level information into contributions from the connected correlators of the evolved field. The framework recovers the standard Gaussian-covariance result for the power spectrum and the Gaussian bispectrum likelihood, identifies information recovered or lost under compression to finite summary statistics, and is applied to BAO-scale reconstruction to show that the field contains all information needed for optimal reconstruction in the presence of noise. It also yields the optimal estimate of the initial conditions as a byproduct.

Significance. If the central reorganization holds, the work supplies a formal bridge between field-level inference and correlator-based analyses in large-scale structure cosmology. It makes explicit which information is captured by power-spectrum or bispectrum likelihoods versus lost under compression, and demonstrates recovery of known Gaussian limits plus an application to optimal BAO reconstruction. The non-perturbative stance with respect to the forward model and the explicit recovery of standard results are clear strengths that could help quantify information loss in summary-statistic pipelines.

major comments (2)

[§3] The marginalization over initial conditions and noise is stated to be general and feasible for the weakly non-Gaussian regime, yet the derivation of the Fisher matrix and the subsequent reorganization into connected-correlator contributions (abstract and §3) provides no explicit intermediate equations or steps. Without these, it is impossible to verify that no additional assumptions are introduced during the expansion around the Gaussian limit or that the reorganization is free of post-hoc choices.
[BAO application section] The BAO application claims that the field contains all information required for optimal reconstruction and identifies the specific FLP contributions needed to attain this limit. This central claim is load-bearing for the practical utility of the framework, but the manuscript does not supply the concrete mapping or explicit expression showing how the correlator terms achieve the optimal limit.

minor comments (2)

[Abstract] The abstract is information-dense; splitting the description of the reorganization and the BAO application into separate sentences would improve readability.
[§2] Notation for the forward model, noise term, and connected correlators should be introduced with explicit definitions in the main text before the expansion is performed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help strengthen the clarity of our presentation. We address each major comment below and have revised the manuscript to incorporate additional explicit steps and mappings.

read point-by-point responses

Referee: [§3] The marginalization over initial conditions and noise is stated to be general and feasible for the weakly non-Gaussian regime, yet the derivation of the Fisher matrix and the subsequent reorganization into connected-correlator contributions (abstract and §3) provides no explicit intermediate equations or steps. Without these, it is impossible to verify that no additional assumptions are introduced during the expansion around the Gaussian limit or that the reorganization is free of post-hoc choices.

Authors: We agree that the intermediate steps in deriving the Fisher matrix from the field-level posterior and reorganizing it into connected-correlator contributions were not presented with sufficient explicitness. In the revised manuscript we expand §3 to include the full sequence of intermediate equations: beginning from the general definition of the FLP after marginalization over initial conditions and noise, proceeding through the Gaussian-limit expansion of the log-posterior, and arriving at the final expression in terms of connected correlators of the evolved field. These added steps make transparent that the reorganization follows directly from the cumulant expansion without additional assumptions or post-hoc choices beyond the stated weakly non-Gaussian regime. revision: yes
Referee: [BAO application section] The BAO application claims that the field contains all information required for optimal reconstruction and identifies the specific FLP contributions needed to attain this limit. This central claim is load-bearing for the practical utility of the framework, but the manuscript does not supply the concrete mapping or explicit expression showing how the correlator terms achieve the optimal limit.

Authors: We acknowledge that while the BAO section identifies the relevant FLP contributions, an explicit mapping from the correlator terms to the optimal reconstruction limit was not supplied. In the revised manuscript we add a concrete mapping and explicit expressions in the BAO application section. These show how the connected-correlator contributions combine to recover the full field-level Fisher information, thereby attaining the optimal reconstruction limit in the presence of noise and confirming that the field contains all required information. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from posterior definition and expansion

full rationale

The paper starts from the explicit definition of the field-level posterior obtained by marginalizing over initial conditions and noise in a general forward model. It then expands this posterior around its Gaussian limit to obtain a Fisher matrix expression and reorganize field-level information into connected-correlator contributions. The derivation recovers the standard Gaussian power-spectrum Fisher matrix and the Gaussian bispectrum likelihood exactly as consistency checks, without any reduction of the central claims to fitted parameters, self-citations, or ansatzes. No load-bearing step equates a derived quantity to its own input by construction; the reorganization follows directly from the formal marginalization and perturbative expansion around the Gaussian case.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard cosmological assumptions about forward models and weak non-Gaussianity; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The forward model allows marginalization over initial conditions and noise
Central to constructing the field-level posterior as stated in the abstract.
domain assumption The observable is weakly non-Gaussian
Required for the generalization and for the validity of the Gaussian-limit expansion.

pith-pipeline@v0.9.0 · 5579 in / 1302 out tokens · 43649 ms · 2026-05-07T14:58:52.980218+00:00 · methodology

discussion (0)

Reference graph

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