  Observations of the cosmic 21-cm power spectrum (PS) are starting to enable
precision Bayesian inference of galaxy properties and physical cosmology,
during the first billion years of our Universe. Here we investigate the impact
of common approximations about the likelihood used in such inferences,
including: (i) assuming a Gaussian functional form; (ii) estimating the mean
from a single realization; and (iii) estimating the (co)variance at a single
point in parameter space. We compare "classical" inference that uses an
explicit likelihood with simulation based inference (SBI) that estimates the
likelihood from a training set. Our forward-models include: (i) realizations of
the cosmic 21-cm signal computed with 21cmFAST by varying UV and X-ray galaxy
parameters together with the initial conditions; (ii) realizations of the
telescope noise corresponding to a 1000 h integration with SKA1-Low; (iii) the
excision of Fourier modes corresponding to a foreground-dominated, horizon
"wedge". We find that the 1D PS likelihood is well described by a Gaussian
accounting for covariances between wavemodes and redshift bins (higher order
correlations are small). However, common approaches of estimating the
forward-modeled mean and (co)variance from a random realization or at a single
point in parameter space result in biased and over-constrained posteriors. Our
best results come from using SBI to fit a non-Gaussian likelihood with a
Gaussian mixture neural density estimator. Such SBI can be performed with up to
an order of magnitude fewer simulations than classical, explicit likelihood
inference. Thus SBI provides accurate posteriors at a comparably low
computational cost.
