In the present study, we have pointed out the necessity of estimating errors along with estimating values of parameters that define nuclear mass models. Such errors not only allow for quantifying quality of models in terms of confidence intervals instead of fit residuals, but also for putting theoretical error bars on mass predictions.
A crucial element in the error analysis is the fact that the nuclear mass models belong to the class of inaccurate models, which describe data with accuracy that is much lower than that of the data themselves. For such models, standard least-square methods to estimate errors and values of parameters are not based on statistical assumptions, but rather pertain to analyzing the sensitivity of the model parameters to the data. Consequently, results may, and do depend on weights that are used when defining the rms deviations between the model results and the data.
The discussion of error analysis was illustrated by using a simple mass model that includes a global liquid-drop part and a locally fluctuating shell-effect part, with a number of model parameters. A set of metadata masses was generated by fitting the most complex variant of the model with the fourth-order shell-effect polynomials to experimental nuclear binding energies. The metadata were then used as an "experimental" input for performing fits that used less sophisticated second- and third-order polynomials. In this way, we had at our disposal the exact model of the metadata and two inaccurate models that mimicked realistic mass fits.
Within such a scheme, we were able to illustrate many properties of nuclear mass fits. In particular, we showed explicitly the relationship between the statistical noise in the metadata and error estimates. We also presented methods to differentiate between important and unimportant model parameters, which are based on the singular value decomposition of the regression matrix. By performing mass fits with mass-number dependent weights, we showed that values of the model parameters may involve much larger uncertainties than those given by standard error estimates. Finally, we have shown the role of confidence intervals and fit residuals in evaluating the quality of exact and inaccurate models.
Fruitful discussions with Andrzej Majhofer are gratefully acknowledged. This work was supported in part by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme and by the Polish Ministry of Science under Contract No. N N202 328234.