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

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.