The role of the likelihood for elastic scattering uncertainty quantification
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| Format: | Preprint |
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2024
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| _version_ | 1866909125384863744 |
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| author | Pruitt, C. D. Lovell, A. E. Hebborn, C. Nunes, F. M. |
| author_facet | Pruitt, C. D. Lovell, A. E. Hebborn, C. Nunes, F. M. |
| contents | 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 $χ^{2}$ minimization with Markov Chain Monte Carlo sampling to obtain parameter posteriors. Following previous work, we consider how results are affected when $χ^{2}$/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 $χ^{2}$/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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_00753 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The role of the likelihood for elastic scattering uncertainty quantification Pruitt, C. D. Lovell, A. E. Hebborn, C. Nunes, F. M. Nuclear Theory 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 $χ^{2}$ minimization with Markov Chain Monte Carlo sampling to obtain parameter posteriors. Following previous work, we consider how results are affected when $χ^{2}$/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 $χ^{2}$/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. |
| title | The role of the likelihood for elastic scattering uncertainty quantification |
| topic | Nuclear Theory |
| url | https://arxiv.org/abs/2403.00753 |