  Standard present day large-scale structure (LSS) analyses make a major
assumption in their Bayesian parameter inference --- that the likelihood has a
Gaussian form. For summary statistics currently used in LSS, this assumption,
even if the underlying density field is Gaussian, cannot be correct in detail.
We investigate the impact of this assumption on two recent LSS analyses: the
Beutler et al. (2017) power spectrum multipole ($P_\ell$) analysis and the
Sinha et al. (2017) group multiplicity function ($\zeta$) analysis. Using
non-parametric divergence estimators on mock catalogs originally constructed
for covariance matrix estimation, we identify significant non-Gaussianity in
both the $P_\ell$ and $\zeta$ likelihoods. We then use Gaussian mixture density
estimation and Independent Component Analysis on the same mocks to construct
likelihood estimates that approximate the true likelihood better than the
Gaussian $pseudo$-likelihood. Using these likelihood estimates, we accurately
estimate the true posterior probability distribution of the Beutler et al.
(2017) and Sinha et al. (2017) parameters. Likelihood non-Gaussianity shifts
the $f\sigma_8$ constraint by $-0.44\sigma$, but otherwise, does not
significantly impact the overall parameter constraints of Beutler et al.
(2017). For the $\zeta$ analysis, using the pseudo-likelihood significantly
underestimates the uncertainties and biases the constraints of Sinha et al.
(2017) halo occupation parameters. For $\log M_1$ and $\alpha$, the posteriors
are shifted by $+0.43\sigma$ and $-0.51\sigma$ and broadened by $42\%$ and
$66\%$, respectively. The divergence and likelihood estimation methods we
present provide a straightforward framework for quantifying the impact of
likelihood non-Gaussianity and deriving more accurate parameter constraints.
