The role of the likelihood for elastic scattering uncertainty quantification

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.