On the Connection between Field-Level Inference and $n$-point Correlation Functions

TL;DR

This paper establishes a precise link between field-level inference (FLI) for galaxy clustering and conventional $n$-point statistics within an EFT forward model. It shows that an $m$-th order forward model captures information from all $n$-point functions with $n\leqslant m+1$, and this holds in both tree-level and loop regimes, provided the forward model is perturbative and uses appropriate cutoffs. The authors derive the marginalized posterior in the zero-noise limit, develop an iterative inverse-solution, and obtain MAP relations for bias and cosmological parameters at multiple orders, revealing how higher-order forward-model terms recruit information from higher-point functions such as the bispectrum and trispectrum. They also discuss the impact of incomplete forward models, estimate systematic shifts in inferred parameters, and compare the information content with explicit $n$-point likelihoods, offering guidance for robust FLI analyses and future extensions to include noise and more complete loop effects.

Abstract

Bayesian field-level inference of galaxy clustering guarantees optimal extraction of all cosmological information, provided that the data are correctly described by the forward model employed. The latter is unfortunately never strictly the case. A key question for field-level inference approaches then is where the cosmological information is coming from, and how to ensure that it is robust. In the context of perturbative approaches such as effective field theory, some progress on this question can be made analytically. We derive the parameter posterior given the data for the field-level likelihood given in the effective field theory, marginalized over initial conditions in the zero-noise limit. Particular attention is paid to cutoffs in the theory, the generalization to higher orders, and the error made by an incomplete forward model at a given order. The main finding is that, broadly speaking, an $m$-th order forward model captures the information in $n$-point correlation functions with $n \leqslant m+1$. Thus, by adding more terms to the forward model, field-level inference is made to automatically incorporate higher-order $n$-point functions. Also shown is how the effect of an incomplete forward model (at a given order) on the parameter inference can be estimated.