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

This paper establishes a unified, non-perturbative framework for cosmological inference by deriving a field-level posterior that explicitly connects the full information content of large-scale structure data to its decomposition into connected correlators, thereby quantifying the information loss inherent in compressing data into summary statistics like the power and bispectrum while enabling optimal initial condition reconstruction.

Original authors: Massimo Pietroni, Fabian Schmidt

Published 2026-04-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Massimo Pietroni, Fabian Schmidt

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, complex tapestry. Cosmologists want to understand how this tapestry was woven, but they can only see the final pattern (the galaxies and matter we observe today), not the original threads (the initial conditions).

This paper introduces a new, powerful way to study that tapestry. Instead of squinting at the whole thing or just counting a few specific patterns, the authors propose a method to analyze the entire field of data at once. Here is a simple breakdown of their ideas using everyday analogies:

1. The Problem: Looking at the Whole vs. Looking at Parts

Traditionally, scientists study the universe by compressing the massive amount of data into a few "summary statistics," like the Power Spectrum (which tells you how much "clumping" exists at different sizes) or the Bispectrum (which tells you about the shapes of those clumps).

The Analogy: Imagine trying to understand a symphony. The traditional method is to write down just the volume of the music (Power Spectrum) or the rhythm (Bispectrum). You get a good idea of the song, but you lose the specific melody, the harmony, and the unique texture of the instruments. You are throwing away information.
The Question: How much information are we losing by only looking at these summaries?

2. The Solution: The "Field-Level Posterior" (FLP)

The authors developed a mathematical framework called the Field-Level Posterior (FLP). Instead of compressing the data first, this method looks at the raw data field directly, accounting for how the universe evolved from its beginning to now.

The Analogy: Instead of just listening to the volume of the symphony, the FLP is like having a recording of the entire orchestra playing, including every instrument, every note, and every silence. It tries to reconstruct the original sheet music (the initial conditions) by listening to the full performance, even if there is static (noise) in the recording.

3. Breaking Down the Complexity: The "Gaussian" Base

The universe isn't perfectly simple; it's messy and non-linear. To handle this, the authors break their complex equation into two parts:

The Gaussian Part: This is the "smooth" baseline, like the steady hum of a fan. In their math, this part already contains all the standard "clumping" information (the Power Spectrum) but treated in a fully nonlinear way.
The Non-Gaussian Part: This is the "messy" part, the extra wrinkles and twists. They organize this mess by looking at how different points in the universe are connected (correlators).

The Analogy: Think of a smooth cake (Gaussian) with sprinkles on top (Non-Gaussian). The authors say, "Let's first understand the cake perfectly, and then we'll add the sprinkles layer by layer." This allows them to see exactly which "sprinkles" (higher-order correlations) contain new information that the "cake" (Power Spectrum) missed.

4. Connecting the Dots: The Fisher Matrix

The authors use a tool called the Fisher Matrix to measure how much information is available. They show that the information in their "Full Orchestra" (Field-Level) approach can be rebuilt by combining the information from the "Volume" (Power Spectrum), the "Rhythm" (Bispectrum), and the "Harmony" (Trispectrum), provided you also know how these elements influence each other (cross-covariance).

The Analogy: If you only listen to the drums (Power Spectrum), you miss the melody. If you only listen to the melody (Bispectrum), you miss the rhythm. But if you listen to the drums and melody together, and you understand how the drummer changes the melody, you can reconstruct the whole song. The paper proves that you need to know how these summaries "talk" to each other to get the full picture.

5. Real-World Application: The BAO Scale

The paper tests this method on the Baryon Acoustic Oscillation (BAO) scale. This is a specific "ruler" imprinted in the universe from the Big Bang, used to measure cosmic distances.

The Finding: They show that the full field contains all the necessary information to measure this ruler perfectly, even with noise. They identify exactly which parts of their complex equation are needed to get this perfect measurement.
The Analogy: Imagine trying to measure the length of a room with a blurry tape measure. The traditional method might give you a rough estimate. The authors' method shows you how to use the entire room's layout (walls, furniture, shadows) to calculate the length with perfect precision, identifying exactly which clues you need to ignore the blur.

6. Reconstructing the Past

Finally, the paper shows that their method naturally leads to a way to "rewind" the universe. By analyzing the current field, they can mathematically derive the optimal estimate of what the universe looked like at the very beginning.

The Analogy: If you see a puddle on the sidewalk, you can guess where the rain fell. The authors' method is like a super-advanced detective that looks at the puddle, the wind, and the temperature to perfectly reconstruct exactly where every single raindrop fell, even if the wind was blowing hard.

Summary

In short, this paper provides a unified language to compare two ways of studying the universe:

The Compressed Way: Looking at summaries (Power Spectrum, etc.).
The Full Way: Looking at the entire data field.

They prove that the "Full Way" is the gold standard. They also show exactly how much information is lost when you choose the "Compressed Way" and demonstrate that to get close to the full truth, you must combine multiple summaries and understand how they are linked. This helps scientists know when a simple summary is enough and when they need the full, complex picture to get the right answer.

1. Problem Statement

The statistical analysis of Large Scale Structure (LSS) in cosmology has traditionally relied on summary statistics, primarily the power spectrum ( $P$ ) and the bispectrum ( $B$ ). While computationally efficient, this approach involves compressing the full data field into a finite set of observables, raising a critical question: What information is lost during this compression?

Conversely, field-level inference (analyzing the full density field $\delta_d$ directly) promises access to the complete information content. However, field-level methods are often viewed as tied to specific perturbative expansions, reconstruction algorithms, or high-dimensional numerical implementations, making it difficult to compare them systematically with standard correlator-based analyses. There is a lack of a unified framework to:

Quantify the information loss induced by compressing the field into summary statistics.
Understand how cross-covariances between different statistics (e.g., $P$ and $B$ ) reconstruct the full field information.
Clarify the relationship between field-level inference and the reconstruction of initial conditions.

2. Methodology

The authors develop a Field-Level Posterior (FLP) for cosmological data that is non-perturbative with respect to the forward model. The core methodology involves:

General Forward Model: The observed field $\delta_d$ is modeled as a function of initial conditions ( $\delta_{in}$ ) and a noise realization ( $\epsilon$ ). The FLP is derived by marginalizing over these latent variables.
Expansion in Connected Correlators: Instead of expanding in perturbation theory (SPT), the authors expand the FLP around its Gaussian limit.
- The "Gaussian" part ( $L_g$ ) is defined using the fully nonlinear power spectrum ( $W_2$ ) of the forward model, not the linear one. This absorbs the full two-point information into the Gaussian baseline.
- Non-Gaussianities are organized as an expansion in connected higher-order correlators ( $W_3, W_4, \dots$ ) using Hermite functionals ( $H_j$ ).
Fisher Matrix Decomposition: The authors derive the Fisher matrix for the FLP and reorganize it into a series of contributions labeled by the connected correlators ( $W_j$ ). This allows for a term-by-term comparison between the full field information and the information contained in specific summary statistics.
Comparison with Compressed Likelihoods: They construct likelihoods for summary statistics (Power Spectrum, Bispectrum, and joint $P+B$ ) by integrating the FLP over the data field constrained by the estimators. They then compare the resulting Fisher matrices to the decomposed field-level Fisher matrix.

3. Key Contributions

A. Unified Framework for Information Quantification

The paper establishes that the information in the FLP can be reorganized as the information contained in an infinite hierarchy of correlators, provided their full covariance matrix (including all cross-covariances) is retained.

Power Spectrum ( $P$ ): The standard Gaussian likelihood for $P$ recovers only the Gaussian part of the field-level Fisher matrix. It misses terms arising from the cross-correlation between $P$ and higher-order statistics (like the bispectrum).
Bispectrum ( $B$ ): A Gaussian likelihood for $B$ recovers the Gaussian contribution to the bispectrum block but fails to capture the cross-covariance terms between $P$ and $B$ .
Cross-Covariance is Crucial: The authors demonstrate that including the cross-covariance between the power spectrum and bispectrum is necessary to recover the terms in the field-level Fisher matrix that are missing from single-statistic analyses. The full field information is recovered only when the hierarchy is truncated at a certain order and all cross-covariances up to that order are included.

B. Non-Perturbative Construction

Unlike previous field-level approaches (e.g., inverse perturbation theory) that rely on specific grid resolutions or noiseless limits, this formulation:

Is valid for any weakly non-Gaussian observable.
Applies equally to perturbative calculations, simulation-based predictions, and effective field theories (EFTofLSS).
Treats noise at the correlator level rather than requiring explicit noise realizations, allowing for a direct connection to EFTofLSS stochastic terms.

C. Optimal Reconstruction of Initial Conditions

The formalism naturally yields the optimal estimator for the initial conditions ( $\delta_{in}$ ) given the observed data and noise model.

The reconstructed field emerges as a byproduct of the Legendre transform of the generating functional.
At lowest order (Gaussian approximation), this recovers the standard Wiener filter.
At higher orders, it includes non-linear corrections (involving $W_3, W_4$ , etc.), providing a systematic way to improve reconstruction beyond linear theory.

4. Key Results

Fisher Matrix Decomposition: The field-level Fisher matrix element $F_{ab}$ $F_{ab}$ is decomposed into contributions $F^{(i,j)}_{ab}$ $F_{ab}^{(i, j)}$ involving derivatives of correlators $W_i$ $W_{i}$ and $W_j$ $W_{j}$ .
- The "2-2" block (Power Spectrum) contains a Gaussian term, a term present in the $P$ -only Fisher matrix (involving $W_4$ ), and a "genuine" field-level term involving higher-order connected correlators ( $W_j$ for $j \ge 5$ ) that are lost if only $P$ is used.
- These "lost" terms are recovered in the $P+B$ analysis only when the cross-covariance between $P$ and $B$ is included.
BAO Scale Recovery: The authors apply the framework to the Baryon Acoustic Oscillation (BAO) scale. They show that the damping of the BAO signal due to long-wavelength modes (which usually requires reconstruction techniques) is naturally captured in the FLP.
- The information loss in the power spectrum due to mode coupling is exactly compensated by the information gained from the bispectrum (specifically the squeezed limit configurations).
- This proves that the full field contains all information required for optimal BAO reconstruction in the presence of noise, and identifies the specific correlator terms needed to saturate this limit.
Validity Regime: The expansion is controlled by the magnitude of higher-order correlators relative to the power spectrum. The paper provides criteria for when Gaussian approximations for summary statistics are sufficient and when non-Gaussian covariance corrections (or full field-level analysis) are required.

5. Significance

Theoretical Unification: The paper bridges the gap between "field-level" and "correlator-based" cosmology. It shows they are not competing approaches but rather different truncations of the same underlying information hierarchy.
Quantifying Information Loss: It provides a rigorous, quantitative method to determine exactly how much information is discarded when moving from the full field to summary statistics like $P$ or $P+B$ .
Practical Guidance: It clarifies that simply adding the bispectrum to the power spectrum is insufficient; one must include the cross-covariance to avoid biased parameter errors.
Reconstruction Insights: It offers a new perspective on initial condition reconstruction, showing it as a natural consequence of field-level inference rather than a separate algorithmic step.
Future Applications: The framework is designed to be compatible with simulation-based inference (emulators, ML surrogates) and EFTofLSS, making it a versatile tool for next-generation galaxy surveys (e.g., Euclid, DESI, LSST) pushing into the mildly non-linear regime.

In summary, Pietroni and Schmidt provide a rigorous mathematical foundation that unifies field-level and correlator-based inference, demonstrating that the "missing" information in compressed analyses is precisely the cross-correlated information between different orders of the correlation hierarchy.