  When fitting cosmological models to data, a Bayesian framework is commonly
used, requiring assumptions on the form of the likelihood and model prior. In
light of current tensions between different data, it is interesting to
investigate the robustness of cosmological measurements to statistical
assumptions about the likelihood distribution from which the data was drawn. We
consider the impact of changes to the likelihood caused by uncertainties due to
the finite number of mock catalogs used to estimate the covariance matrix,
leading to the replacement of the standard Gaussian likelihood with a
multivariate $t$-distribution. These changes to the likelihood have a
negligible impact on recent cosmic microwave background (CMB) lensing and
baryon acoustic oscillation (BAO) measurements. We then extend our analysis to
perform a sensitivity test on the Gaussian likelihoods typically adopted,
considering how increasing the size of the tails of the likelihood (again using
a $t$-distribution) affects cosmological inferences. For an open $\Lambda$CDM
model constrained by BAO alone, we find that increasing the weight in the tails
shifts and broadens the resulting posterior on the parameters, with a
$\sim$0.2-0.4$\sigma$ effect on $\Omega_\Lambda$ and $\Omega_{\rm k}$. In
contrast, the CMB temperature and polarization constraints in $\Lambda$CDM
showed less than 0.03$\sigma$ changes in the parameters, except for $\{\tau$,
ln($10^{10}A_{\rm s})$, $\sigma_8$, $S_8$, $\sigma_8\Omega_{\rm m}^{0.25}$,
$z_{\rm re}$, $10^9A_{\rm s}e^{-2\tau}\}$ which shifted by around
0.1-0.2$\sigma$. If we use solely $\ell < 30$ data, the amplitude $A_{\rm s}
e^{-2\tau}$ varies in the posterior mean by 0.7$\sigma$ and the error bars
increase by 6%. We conclude, at least for current-generation CMB and BAO
measurements, that uncertainties in the shape and tails of the likelihood do
not contribute to current tensions.
