  Background: Analyses of elastic scattering with the optical model (OMP) are
widely used in nuclear reactions.
  Purpose: Previous work compared a traditional frequentist approach and a
Bayesian approach to quantify uncertainties in the OMP. In this study, we
revisit this comparison and consider the role of the likelihood used in the
analysis.
  Method: We compare the Levenberg-Marquardt algorithm for $\chi^{2}$
minimization with Markov Chain Monte Carlo sampling to obtain parameter
posteriors. Following previous work, we consider how results are affected when
$\chi^{2}$/N is used for the likelihood function, N being the number of data
points, to account for possible correlations in the model and underestimation
of the error in the data.
  Results: We analyze a simple linear model and then move to OMP analysis of
elastic angular distributions using a) a 5-parameter model and b) a 6-parameter
model. In the linear model, the frequentist and Bayesian approaches yield
consistent optima and uncertainty estimates. The same is qualitatively true for
the 5-parameter OMP analysis. For the 6-parameter OMP analysis, the parameter
posterior is no longer well-approximated by a Gaussian and a covariance-based
frequentist prediction becomes unreliable. In all cases, when the Bayesian
approach uses $\chi^{2}$/N in the likelihood, uncertainties increase by
$\sqrt{N}$.
  Conclusions: When the parameter posterior is near-Gaussian and the same
likelihood is used, the frequentist and Bayesian approaches recover consistent
parameter uncertainty estimates. If the parameter posterior has significant
higher moments, the covariance-only frequentist approach becomes unreliable and
the Bayesian approach should be used. Empirical coverage can serve as an
important internal check for uncertainty estimation, providing red flags for
uncertainty analyses.
