  Field-level inference is emerging as a promising technique for optimally
extracting information from cosmological datasets. Indeed, previous analyses
have shown field-based inference produces tighter parameter constraints than
power spectrum analyses. However, estimates of the detailed quantitative gain
in constraining power differ. Here, we demonstrate the gain in constraining
power depends on the parameter space being constrained. As a specific example,
we find that field-based analysis of an LSST Y1-like mock data set only
marginally improves constraints relative to a 2-point function analysis in
$\Lambda$CDM, yet it more than doubles the constraining power of the data in
the context of $w$CDM models. This effect reconciles some, but not all, of the
discrepant results found in the literature. Our results demonstrate the
importance of using a full systematics model when quantifying the information
gain for realistic field-level analyses of future data sets.
