Role of the likelihood for elastic scattering uncertainty quantification

Background: In the last decade, uncertainty quantification (UQ) for optical model potentials (OMPs) has become a focal point for nuclear reaction theory, and several competing approaches for OMP UQ have recently been developed.Purpose: We clarify recent efforts to compare frequentist and Bayesian approaches in the context of OMP UQ [G. B. King et al., Phys. Rev. Lett. 122, 232502 (2019)].Method: We replicate a portion of that OMP UQ study but use independent statistical tools. Specifically, we compare two methods for OMP parameter inference from elastic scattering data: the Levenberg-Marquardt algorithm for ${\ensuremath{\chi}}^{2}$ minimization on one hand and Markov chain Monte Carlo (MCMC) sampling on the other. Separately, we assess the common practice of using a renormalized likelihood (${\ensuremath{\chi}}^{2}/N$), $N$ being the number of data points, instead of the canonical weighted-least-squares likelihood (${\ensuremath{\chi}}^{2}$), as a way of accounting for unknown data correlations.Results: We show that for a generic linear model and for a five-parameter OMP analysis, frequentist and uniform-prior Bayesian approaches recover the same optimum and uncertainty estimates---not systematically larger uncertainties for the Bayesian approach, as was concluded in G. B. King et al., Phys. Rev. Lett. 122, 232502 (2019). Further, we show that if an additional, near-degenerate parameter is introduced into the same OMP analysis such that the parameter posterior becomes non-Gaussian, then covariance-based estimates of uncertainty become unreliable. Finally, we show that regardless of optimization approach, if ${\ensuremath{\chi}}^{2}/N$ is used for the likelihood, the resulting parametric uncertainties increase by $\sqrt{N}$, and that this is responsible for the conclusions drawn in the revisited study.Conclusions: Based on our replication results, we find that a fortuitous cancellation of unreported errors and the renormalization factor can lead to improvement in empirical coverages, as was the case in the original comparative study. We emphasize that developing and applying a realistic likelihood function is an essential task in a UQ analysis, and that several recent UQ studies that employed a renormalized likelihood (i.e., including a $1/N$ factor) may have yielded unrealistically large uncertainties for elastic-scattering observables. If the parameter posterior deviates from multivariate-normal, a sampling-based approach like MCMC has a clear advantage over methods that assume the Laplace approximation holds. We note that empirical coverage can serve as an important internal check for the analyst whose model or data may have additional, unaccounted-for uncertainties.