Field-Level Inference in Cosmology

• Field-level inference is a method that uses complete cosmic field data without compressing information, enabling unbiased and accurate cosmological parameter estimation.
• It employs a Bayesian hierarchical framework combined with advanced MCMC techniques to efficiently explore high-dimensional posterior distributions.
• This approach outperforms summary statistic methods by capturing non-Gaussian features and subtle one-point statistics crucial for modern cosmological analyses.

Field-level inference in cosmology refers to the direct statistical analysis of cosmic fields—such as the evolved matter density or related tracer fields—without data compression into summary statistics like the two-point correlation function or power spectrum. This approach, typically embedded in a hierarchical Bayesian framework and combined with advanced data assimilation techniques, is designed to optimally extract cosmological information by leveraging every data element (e.g., pixel or voxel) and their spatial correlations. The following sections provide a rigorous account of the principles, methodology, advantages, and consequences of field-level inference as established in (Leclercq et al., 2021).

1. Comparative Methodologies in Cosmic Field Inference

The paper contrasts three principal statistical approaches within a Bayesian hierarchical modeling (BHM) context:

• Likelihood-Based Analysis (LBA) of 2PCF: The observed field is compressed to its two-point correlation function (2PCF), assuming the summary vector is Gaussian-distributed (or modeled as a multivariate t-distribution when the covariance is simulation-estimated). The likelihood for parameters $(\alpha, \beta)$ is

$$P(\xi_\text{obs} \mid \alpha, \beta) \propto \left[1 + \frac{(\xi_\text{obs} - \xi_\text{LN})^\top C^{-1} (\xi_\text{obs} - \xi_\text{LN})}{N_\text{sims} - 1}\right]^{-N_\text{sims} / 2}$$

where $\xi_\text{LN}$ denotes the analytic expectation and $C$ the estimated covariance.

• Simulation-Based Inference (SBI) of 2PCF: This “likelihood-free” approach utilizes forward simulations and data compression via optimal score functions (e.g., MOPED-type techniques), yielding summary statistics $\tilde{\alpha}$ and $\tilde{\beta}$ and employing rejection sampling in the compressed parameter space. SBI avoids explicit likelihood assumptions but still operates on reduced data representation.
• Field-Level Inference (FLI): This assimilates the full field data (no compression) within a joint posterior,

$$P(\alpha, \beta, g \mid f_\text{obs}) \propto P(f_\text{obs} \mid g, \alpha) P(g \mid \beta) P(\alpha) P(\beta)$$

and marginalizes the latent field $g$ to determine the posterior of $(\alpha, \beta)$. Markov Chain Monte Carlo (including Hamiltonian Monte Carlo and NUTS) are used for efficient exploration of the high-dimensional posterior.

The key technical distinction is the absence of any compression in FLI, directly connecting the cosmological parameters and the latent field to the observed data.

2. Bayesian Hierarchical Log-Normal Model

A single-parameter log-normal field model is adopted to encapsulate essential non-Gaussian features found in the contemporary cosmic density field. The generative structure is:

• Latent Gaussian Field: Zero-mean, covariance given by a parameterized 2PCF,

$$P(g \mid \beta) = \mathcal{G}[0, C(\beta)] \qquad \xi_G(r) = \exp\left(-\frac{1}{4}\frac{r^2}{\beta^2}\right)$$

• Log-Normal Transformation: The field $g$ is mapped to the log-normal field $f$ as

$$f = \frac{1}{\alpha}\left[\exp(\alpha g - \frac{1}{2}\alpha^2) - 1\right]$$

with $-\frac{1}{2}\alpha^2$ ensuring $\langle f \rangle = 0$, and the prefactor enforcing constant SNR in the $\alpha \to 0$ limit.

• Noisy Observation: The final observed field is modeled as a noisy version of $f$ with additive Gaussian noise.

The analytic two-point function of the log-normal field is

$$\xi_\text{LN}(r) = \frac{1}{\alpha^2} \big\{ \exp\left[\alpha^2 \exp\left(-\frac{r^2}{4\beta^2}\right)\right] - 1 \big\}$$

encoding non-Gaussianity in its higher-order parameter derivatives.

3. Evaluation and Performance Comparison

Table 1. Summary of Method Properties and Performance

Method	Compression Level	Bias/Accuracy in (α, β)	Sensitivity to Non-Gaussianity
LBA (2PCF)	Strong	Substantial bias; imprecise/confident inference	α poorly constrained especially at small α; 2PCF insensitive
SBI (2PCF)	Moderate	Unbiased (α, β); low precision for α	Improvements over LBA, but α precision remains limited
Field-Level	None	Unbiased, highest precision in (α, β)	Robust even for weak non-Gaussianity; fully leverages one-point statistics

LBA relies on potentially invalid Gaussianity assumptions; substantial bias (several sigma discrepancy) can be introduced. SBI improves accuracy via forward simulation and optimal compression but cannot recover information lost in the 2PCF itself—the summary remains insufficient, especially for one-point statistics and higher-order moments. Field-level inference achieves both unbiasedness and superior precision, even for α values close to the Gaussian regime.

4. Role and Impact of Non-Gaussianity

The severity of information loss with summary statistics becomes most pronounced when the underlying field is non-Gaussian. In the log-normal model,

• The parameter $\alpha$ modulates non-Gaussianity; as $\alpha\to 0$, the field becomes Gaussian.
• The analytic derivative of $\xi_\text{LN}$ with respect to $\alpha$ decays rapidly with separation $r$, so for small $\alpha$ the 2PCF barely constrains this parameter.
• FLI recovers α precisely, even with weak non-Gaussianity (α = 0.2), while LBA and SBI essentially return prior-level uncertainties in α but can still constrain β.

The conclusion is that any cosmological parameter whose effect is predominantly local or encoded in higher moments (such as non-Gaussianity) cannot be robustly constrained by statistics like the 2PCF. Only pixel-level, field-based approaches can guarantee unbiased and precise extraction under non-Gaussian deviations.

5. Technical Aspects of Field-Level Inference

• Data Assimilation via MCMC: High-dimensional sampling is made tractable by advanced algorithms (HMC, NUTS), exploiting analytic gradients when available.
• Joint Posterior Structure:

$$P_\text{DA}(\alpha,\beta | f_\text{obs}) = \int P(\alpha, \beta, g|f_\text{obs})\, dg$$

• Marginalization Over Latent Fields: FLI integrates over the full, possibly non-Gaussian field ensemble, not just summary statistics.
• Robustness to Likelihood Mis-specification: By avoiding the derivation of a reduced-data likelihood and working with the generative model directly, FLI is less susceptible to specification errors, provided the generative process itself is adequate.

FLI requires full knowledge of the data-generating process and careful modeling of observational noise and transformations. Computational cost is higher due to the nontrivial dimensionality, but algorithmic advances make this tractable for realistic cosmological datasets.

6. Implications for Future Cosmological Analyses

Upcoming surveys (Euclid, LSST, DESI) deliver high-quality, high-resolution datasets where information loss from compression is no longer negligible relative to statistical errors. In these regimes:

• FLI guarantees the extraction of all available cosmological information, subsuming both spatial (correlation) and one-point (local) structure.
• For non-Gaussian fields or parameter inference where sensitivity arises from pixel-level features, FLI outperforms any summary statistic method, independent of compression strategy.
• The technique is particularly important for accurate determination of parameters related to non-Gaussianity, stochasticity, or nonlocal bias.

However, the accuracy and unbiasedness of field-level inference depend on the fidelity of the generative model (e.g., forward model for structure formation and observation). If the forward model is incomplete or mis-specified, biases in parameter inference can still arise, underscoring the need for continued development of physically accurate and computationally efficient forward modeling in cosmology.

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In summary, field-level inference overcomes the accuracy and precision limitations of summary statistics-based cosmological analyses by leveraging all information encoded in the observed field. The superiority of this approach is most apparent in the presence of non-Gaussian features or when constraining parameters primarily imprinted in one-point or higher-than-second-order statistics. The transition to FLI is well-motivated for ongoing and future deep-field cosmological surveys (Leclercq et al., 2021).