  We present a comparative study of the accuracy and precision of correlation
function methods and full-field inference in cosmological data analysis. To do
so, we examine a Bayesian hierarchical model that predicts log-normal fields
and their two-point correlation function. Although a simplified analytic model,
the log-normal model produces fields that share many of the essential
characteristics of the present-day non-Gaussian cosmological density fields. We
use three different statistical techniques: (i) a standard likelihood-based
analysis of the two-point correlation function; (ii) a likelihood-free
(simulation-based) analysis of the two-point correlation function; (iii) a
field-level analysis, made possible by the more sophisticated data assimilation
technique. We find that (a) standard assumptions made to write down a
likelihood for correlation functions can cause significant biases, a problem
that is alleviated with simulation-based inference; and (b) analysing the
entire field offers considerable advantages over correlation functions, through
higher accuracy, higher precision, or both. The gains depend on the degree of
non-Gaussianity, but in all cases, including for weak non-Gaussianity, the
advantage of analysing the full field is substantial.
